Structural System Reliability Analysis of Jack-up Platforms under Extreme Environmental Conditions

Platforms under Extreme Environmental Conditions

Thesis

In fulfillment of the requirements for the degree of Doctor in technical sciences, at Deift University of Technology, under the authority of the Rector Magnificus, Prof. dr. ir. J. Blaauwendraad, and to be defended in

public for a con-iniittee appointed by the board of Deans on Tuesday 28 January 1997 at 10.30 o'clock

by

Mohammad DAGHIGH

Master of Science, Tabriz University Born at Tabriz, Iran

This thesis is approved by the dissertation directors: Prof. ir. S. Hengst

Prof. ir. A.C.W.M. Vrouwenvelder

Composition of promotion committiee:

Rector Magnificus

Prof. ir. S. Hengst,

Prof. ir. A.C.W.M.Vrouwenvelder, Ir. H. Boonstra,

Prof. M.J. Baker, Prof. ir. B. Boon,

Prof. dr. ir. J.M.G. Kerstens, Prof. dr. ir. J.H. Vugts, Prof. dr. ir. J.A. Pinkster,

The chairman

Technical University of Deift Technical University of Deift Technical University of Deift University of Aberdeen, Schotland Technical University of Delft Technical University of Eindhoven Technical University of Deift Technical University of Deift

Prof. ir. S. Hengst

Prof. ir. A.C.W.M. Vrouwenvelder

Samenstelling promotiecommissie: Rector Magnificus,

Prof. ir. S. Hengst,

Prof. ir. A.C.W.M.Vrouwenvelder, Ir. H. Boonstra,

Prof. M.J. Baker, Prof. ir. B. Boon,

Prof. dr. ir. J.M.G. Kerstens, Prof. dr. ir. J.H. Vugts, Prof. dr. ir. JA. Pinkster,

CIP-GEGEVENS KONINKLIJKE BIBLIOTHEEK, DEN HAAG

Daghigh, Mohammad

Structural System Reliability Analysis of Jack-up Platforms under Extreme Environmental Conditions / Mohammad Daghigh. Thesis Delft.- With ref.- With summary in Dutch.

ISBN 90-370-0155-6

Subject headings: extreme loads / system reliability / nonlinear dynamics Printed by: Universiteitsdrukkerij T.U. Delft

Copyright © 1997 by Mohammad Daghigh All rights reserved.

No part of the material protected by this. copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrival system, without permission from the publisher. PRINTED IN THE NETHERLANDS

De voorzitter

Technische Universiteit Delft, promotor Technische Universiteit Delft, promotor Technische Universiteit Delft

University of Aberdeen, Schotland Technische Universiteit Deift Technische Universiteit Eindhoven Technische Universiteit Deift Technische Universiteit Delft

Structural System Reliability Analysis of Jack-up

Platforms under Extreme Environmental Conditions

Proefschrift

ter verkrijging van de graad van doctor aan de Technische Universiteit Delft, op gezag van de Rector Magnificus, Prof. dr. ir. J. Blaauwendraad,

in het openbaar te verdedigen ten overstaan van een commissie, door bet College van Dekanen aangewezen,

op dinsdag 28 januari 1997 te 10.30 uur

door

Mohammad DAGHIGH

Master of Science, Tabriz University geboren te Tabriz, Iran

my Parents, my Educators,

(I)

ACKNOWLEDGEMENTS

in the name of God, the most compassionate and the most merciful.

All my thanks to God for keeping me healthy and making me able to successfully finish this work.

I wish to place on record my deepest gratitude to Deift University of Technology for availing me the opportunity to research this subject towards my Ph.D. degree.

My deepest gratitude must be expressed to my promoters Prof.ir. S. Hengst and Prof.ir.

A.C.W.M. Vrouwenvelder for their excellent supervision of this study. Their valuable

comments together with their hospitality facilitated the realization objectives in a pleasurable setting. I also express my sincere thanks to my mentor Ir. H. Boonstra for his guidance and friendly relation towards me during this study.

I am respectfully grateful to Ir. M.J. Koster for his help, guidance and fruitful discussions during the first year of my research. I enjoyed very much all those daily

greetings, talks, discussions during working days which made a pleasant environment for my study.

I am sincerely grateful to Ing. R.L.C. Wouters, Ir. J.H. Vink and Ir. N. Wisse for their help, interest and continuous support in the course of my work. I would like to thank all related Colleagues of the Laboratory for Ship Construction in the Faculty of Mechanical Engineering and Marine Engineering for their helpful cooperations throughout the present

study.

I would like to thank Ir. C.J. Mommaas and Ir. G.J. GÑndlehner from MSC office for their valuable comments during the first period of this research and for their continuous interest on my work.

I express my best appreciations to the Islamic Republic of Iran for providing such an

excellent opportunity to participate this subject as my Ph.D. research. I am also indebted to

the embassy of the islamic republic of Iran in the Netherlands.

Finally recognition is expressed for the admirable patience of my wife Mrs. F. Zarrinkar

on my full schedule during the whole time of the research. I am much obliged to my son Saleh, with whom I should have spent more time together in the evenings and weekends.

1. Introduction 1.1

1.1 Foreword 1.1

1.2 Objectives and plan of work 1.3

1.3 Organization of work 1.4

1.4 Jack-ups risk versus fixed platforms 1.5

2. Finite element analysis of structural dynamics in jack-ups 2.1

2.1 Introduction 2.1

2.2 Developments in structural analysis of jack-up platforms 2.2

2.3 DampiÏg models for jäck-up structures 2.3

2.4 Overall structural modeling for a three leg platform 2.5

2.5 Hull Module modeling 2.7

2.6 The three-dimensional beam-column models for a leg module 2.10

2.7 The selection of leg-hull connection model 2.11

2.8 Numerical tests with proposed jacking model 2.13

3. Strength analysis of jack-up platforms 3.1

3.1 Introduction 3.1

3.2 Nonlinear structural analysis of offshore platforms 3.1

3.3 Elastic stability - buckling analysis 3.3

3.4 Plastic stability 3.4

3.5 Limit state design for tubular elements 3.6

3.6 Non-tubular members 3.7

3.7 Typical elements for jacking system 3.8

3.8 Force-deflection model of jacking pinions 3.8

3.9 Shock-pads and/or fixation system 3.10

3.10 Horizontal guides 3.10

4. Structural reliability assessment of jack-up platforms 4.1

4.1 Introduction 4.1

4.2 Background and problem statement for reliability analysis 4.1

4.4 Solution techniques for system reliability ' '4.6

4.5-Reliability analysis procedirefor a 'single element

'"

4.8

4.5.1 Short term reliábility 4.8

4.4.2 Long term reliability

''

4.9

4.6 System reliability analysis 4.10

4.7 Reliability analysis for a single element úsing the detailed model 4.15 4.7.1 Short term reliability of the detailed model 4.15 4.7.2 Long term reliability of the detailed model 4.16 4.8 System reliability analysis of the detailed model ' ' ' 4.17

4.9 Examples of system reliability analysis 4.19

Example 1- System reliability analysis for one group components with equal 4.19

limit state functiOns of single events

Example 2- System reliability for a number of groups components with 4.20

equal limit state functions of single events

Example 3- System reliability of two member parallel system with ductile 4.21

behaviour

Example 4- System reliability of three member parallel system with ductile 4.24

behaviour

5. Stochastic dynamic analysis of jack-ups with elastic behaviour using 5.1

stick models

5. 1 Introduction 5.1

5.2 Stick model of the three leg Neka platform' 5.1

5.3 General strategy for short term response 5.4

5.4 Probabilistic modeling of short term responss, application of three- 5.5 parameter Weibull model to response peaks

5.5 Autospectral densities of short term responses 5.6

5.6 Effect of different hydrodynamic damping models on the response spectra' s 5.8

5.7 Estimation of Dynamic Amplification Factors (DAF' s) and spatial variation 5.10

of DAF's in jack-up legs

5.8 Calibration Fàctors (CF) for single design wave analysis 5.13

5.9 Barrier crossing and extreme value statistics 5.15

5.10 Extreme value statistics by time variant formulation (extreme response 5.16 distribution in certain period of time)

5.14 Probabilistic formulation of dynamic amplification factor 5.24

5.15 Conventional long term reliability analysis with the short term analysis 5.26 using absolute velocity formulation in Morison equation

5.16 Nested technique in long term reliability analysis with the short term 5.28 analysis using absolute velocity formulation in Morison equation

5.17 Full long term reliability _5.30

5.17.1 Results of full long term reliability analysis with the short term 5.31 analysis using absolute velocity formulation in Morison equation

5.17.2 Results of full long term reliability analysis with the short term 5.32 analysis using relative velocity formulation for high sea-states (!i> 10 m).

5.18 Effect of time invariant random variables on the results of long term 5.33 reliability analysis with the short term analysis using absolute velocity

formulation in Morison equation

5.19 Summary and concluding remarks 5.35

6. Structural system reliability analysis of jack-up platforms with detailed 6.1

models

6.1 IntroductiOn 6.1

Part a) Component reliability 6.1

6.2 Description of detailed structural models 6.1

6.3 Environmental models; conventional design wave distribution versus nested 6.4

technique

6.4 Reliability analysis of chords and bracings of jack-up unit equipped with 6.7 fixed type jacking system and fixation system

6.5 Reliability analysis of chords and bracing jackup unit with floating 6.9 jacking system and without fixation system

6.6 Failure probability analysis of jacking system 6.11

6.7 Reliability analysis of hull structure 6.14

Part b) System reliability 6.16

6.8 Methods of the System reliability analysis 6.16

6.9 Equivalent total dampiñg ratio, effect of water depth and high sea-state 6.18

6.10 Structural analysis of jack-up systems including post-failure material 6.19

behaviour

6.11 Estimation of dynamic effects and calibration factors for first failure 6.21 elements in elasto-plastic models

(V) failure states

6.13 Nonlinear dynamics - Equivalent Duffing system; an approximate strategy 6.24 6.14 Nonlinear dynamics - Direct numerical integration 6.27 6.15 Classification of random variables for system reliability analysis 6.30 6.16 Sequences of multiple failures - reliability in higher levels 6.32 6.17 System reliability versus component reliability 6.36

6.18 Concluding remarks 6.38

6.19 Recoimnendations for calibration of safety levels 6.39

7. Principal results and concluding remarks 7.1

7.1 General 7.1

7.2 Short term extreme responses and reliability analysis for extreme responses 7.2 7.3 Methods of reliability analysis using design wave, quasi-static and

dynamic simulations

7.3

7.4 Long term structural response and reliability analysis 7.4 7.5 Structural modelling and nonlinear material behaviour 7.5

7.6 System reliability procedure for large systems 7.6 7.7 Future developments in jack-up design (structural analysis, reliability

analysis

7.8

Nomenclature Nom.!

References Ref.I

Appendix A. Description of example jack-up structure A.1

A. 1 Structural description of Neka jack-up platform A. 1

A.2 Environmental condition A.2

A.3 Natural periods and the effects of boundary conditions A.3

Appendix B. Sea scatter diagram and random variables B.1

B. 1 Sea scatter diagram B. i

B .2 Time variant and time-invariant random variables B. 1

Appendix C. Idealization of lattice legs C.!

C.1 Equivalent hydrodynamic leg model C.1

C. 2 Equivalent structural leg model C.3

Appendix E. A functional model for DAF and CF E.1 The polynomial model

E.2 Linear dynamic analysis

E.3 Elasto-plastic dynamic analysis for firt failure elements Appendix F. Formulation of the finite element method

I Finite element formulation of stutural dynamics

F.2 Formulation of damping models in dynamic simulation of jac-up structures Appendix G. Computer programs for structural and reliability analysis

i Hydrodynamic load calculation

G.2 Structural analysis programs G.3 Random data analysis programs

G.3.1 Available random data analysis programs; RANDA, ORIGIN G.3.2. Simple programs; MOMENT, FILTER and WEIBULL G.4 Reliability analysis by FORM routine

G. 5 Reliability analysis by Importance Sampling technique

G.6 Combined FORM and numerical integration technique Summary

Summary in Dutch (Samenvatting) Summary in Persian E.1 E..1 E. i E.3 F. i F.1 F.3 G.! G.1 G.1 G. i G. 1 G.2 G.2 G.3 G.5 Sum.! Sam.! Per.I

Chapter 1

Introduction

1.1 Foreword

In recent years, the structural design and analysis of self-elevating jack-up units has been the subject of various research programs of marine structures. Due to the complexity of the type of the structure and the environmental conditions, no complete agreement has been achieved as the best choice for the structural model and the analysis method. Presently there exist two

methods for the design of jack-up structures. In the first method, the designer is usinga pseudo-static type of analysis with reference to structural nonlinearities. In the second method, he/she focuses on the dynamic analysis without structural non-linearity such as

implied to the tension leg platforms. In contrast to the nonlinear static or linear dynamic

analysis, the exact magnitude and probability distribution of the nonlinear dynamic response

of the jack-up structuresis seldom found in the state of art studies.

In this thesis, the philosophy of structural analysis of jack-up platforms is addressed from probabilistic point of view. An overview of limit state analysis and reliability based design aspects for various failure elements are given in elevated condition. Although the dynamic response analysis of jack-up is apparently used in this thesis, it is acknowledged that the

dynamic simulation of complex structural models are practically impossible with the present

computer capacities. Reférence is made to simplified structural models and a methodology

is developed to correlate the dynamic response of structure with the response of design wave

method. Implying the technique of structural analogy, the stochastic procedure is extended to the complex structure of jack-ups considering the post failure behaviour of structural

components.

Much progress has been made, particularly in the last two decades, in the development of both the theories of structural non-linearity and in probability computation methods for

structural and mechanical reliability analysis. The standard method of evaluating the safety

of offshore platforms in extreme environmentäl conditions is the so-called pseudo-static method of analysis. In this method, the effects of stochastic wave, wind and current

conditions on a jack-up mass are represented by an equivalent dynamic force determined as the product of a dynamic amplification factor and the response of platform from single design wave analysis. The design environmental conditions are also limited to extreme sea-state for the operational period in which the contribution of noh-severe sea-states is neglected. For the

probabilistic analysis, the failure criterion is expressed in term of external loads and their

internal load effects, whichever is most convenient.

To ensure a rational design of deep water platforms, the dynamic response of offshore structures to environmental loading has been receiving significant attention. In this scope, from the practical point of view, various aspects including linear dynamics, nonlinear hydrodynamic forces, free surface effects, P-ô and P- effects (second order effects), mechanisms of jacking system and the soil-structure interaction are encountered. Including all of these features, the main train is concentrated on implied safety levels of jack-up

structures in harsh environmental conditions. Since the emphasis is on the jack-up structure itself, the soil-structure interaction is excluded from the present study.

A literature survey on the state-art research programs on the structural and reliability analysis

of jack-up platforms is shown in Table 1.1.

Table 1.1 Comi,arison of recent studies on the structural behaviour of jack-up platforms

X Include - Not inciuue pprox

-.) ) H Q I.., )

-Q

--

C Q . -.) . U

-_ > _{- -}n H

-n .

--, n . n -U Nonlinear hydrodynamic force X O ? X X X X X x

x .x

x P-z and P-ô effects X O - + + ? ? X + + X X Soil-structure interaction - O X X X - X X - X X -Mechanisms of jacking system - X + - + X - - - X Linear dynamics; 3-D stick model or discrete elements X O - X X - X - - X - X Nonlinear dynamics SDOF model - O - - - X Nonlinear dynamics stick model in structural state [1] - O - - -

x

Detail hull model X O O - - X - - - X

Nonlinear static analysis - O - - - X - - X X Nested technique in reliability analysis O O - X O O X - X X - X System reliability -O . O - -. - -- -O . O -n X t ._-_.-_-.-... -

-n

.T.X X

Introduction 1.3

1.2 Objectives and plan of work

The main objective of this research is to develop a system reliability approach to be used in an engineering environment for complex, dynamic sensitive structures. The method is applied

to jack-up structures in deep water. This type of structure consists of various types of

components (lattice structure for the legs, stiffened panels for the hull, mechanical structures

for the jacking mechanism) and is subjected to nonlinear and dynamic behaviour. The

combination of a methodology to relate the structural response distribution of jack-ups using advanced and simplified models, with a system reliability assessment based on the integrated system of jack-up structure in an early stage of the limit state design procedure is considered as a new contribution to jack-up analysis.

To achieve the purpose of this study, one has to select rational methods for determining

wave-induced design loads for jack-up structures. In particular, the probabilistic distribution

of wave-induced bending moment and axial force for the jack-up legs have been used as

examples for cases where a design wave analysis would be insufficient. Since the distribution

of response characteristics may differ for each sea state characteristic, a response surface

model is developed for the distribution parameters of short term responses. With this track, the number of time domain simulations and the total computational time is reduced compared to the previous investigations in dissertation of Karunakaran [1993]. Despite of the reduction

in the simulation time for short term sea-state analysis, some general points have been

investigated in the parametric study:

The extreme response of jack-up platforms in deep water is affected by contribution of

different sea-states. Consequently, the system reliability of structure is evaluated such as the

convolution integral of sea-state parameters gives the system reliability for long term

analysis.

The evaluation of stochastic responses is extended to structural non-linearity using absolute

velocity formulation in Morison equation. The extreme response of jack-up structure with

relative velocity fOrmulation is compared to the extreme response with absolute velocity for

higher sea-states with significant wave heights H> 10 m. Considering the structural

non-linearity, the dynamic equation of motion has to be solved for the external forces which are

related to the structural velocity (or the strain rates). However, the current finite element codes does not cover this particular aspect [see for instance ANSYS manuals] and in this study, keeping the absolute velocity formulation for nonlinear structural analysis, we

employed different damping constants depending on the severity of sea-state parameters.

For jack-up structures in deep water depths and/or harsh environment, the dynamic amplification factor is influenced by the response sensitivities. This condition requires that

different dynamic amplification factors are applied within the design wave analysis. Since the

dynamic effects are mainly influenced by the sea-state characteristics, the concepts of

Calibration factor and Dynamic amplifi cation factor are introduced for definition of relevant responses of the dynamic time domain responses and the quasi-static time domain responses

from the static design wave analysis. Employing these factors in the structural respoñses of detailed structural model with static design wave analysis, accurate extreme responses are

4- In the case of linear systems, the dynamic amplification factors are found proportional to

the amplitude of excitations, from which essential differences are observed for resonance

response of nonlinear systems. and consequently the nonlinear dynamic responses. One other feature is also identified for nonlinear elastic models compared to elastic-plastic systems. In

both of systems, the nonlinear dynamic amplifications are dependent on the load factors. 1.3 Organization of work

A literature review in the structural analysis of jack-up platforms is presented in chapter 2. The most versatile tool of finite element technique in structural dynamics is used with

particular attention on the jack-up structural elements, their nonlinear stiffness and damping

models. The Neka jack-up platform has been chosen for the parametric study with the

prototype design for operation in the South coasts of Caspian Sea, Iran. Since it was virtually

impossible to obtain reliable data for the operational condition of Caspian Sea, a North Sea location (Hutton area, Marex report [1979]) is chosen for the parametric study. For the structural analysis of Neka jack-up platform, the overall detail model of three legs jack-up

structure is introduced adopting the finite element codes of SAP9O and ANSYS. The models are generated using plate elements for the hull, interface elements for jacking system and 3-D

beam elements for the legs. Reference is also made to the numerical test with the proposed

jacking model in compound action of bending moment and leg reaction.

Fundamental differences in three different design approaches, the Allowable Stress Design

(ASD), Plastic Design (PD) with load factor and Load and Resistance Factor Design (LRFD)

are discussed in chapter 3. The failure criteria of jack-up components in the elastic and plastic regimes are discussed for all three types of components in legs and jacking system.

A distinction is made for tubular and non tubular sections with different limit state functions provided by design codes. Apart from beam-column elements, the ultimate strength equations of jacking pinions, shock-pads and fixation systems have been modelled using the Ramberg-Osgood model.

Structural reliability methods in element level and system level are presented in chapter 4.

The proposed methodology for system reliability has been selected by simple test examples. The accuracy of proposed system reliability formulation in component reliability and system reliability has been verified for linear and hyper-spherical limit state functions for structural components. By comparison of the reliability methods (nested routine, numerical integration),

a system reliability approach for complex structural systems has beenintroduced using the combined numerical integration and FORM (First Order Reliability Method) technique.

In chapter 5, a simplified response surface technique for reliability analysis of jack-up legs

is introduced with stick structural models using stochastic dynamic analysis of jack-ups. The type III extreme value distribution (Weibull distribution) is matched to the extrapolated time

series of maxima response histories. The short term and long term reliability analysis are presented with various type of hydrodynamic damping models and different statistical

estimators for extreme values. Adopting the same strategy for quasi-static analysis of jack-up

structure, the extreme responses of jackup are compared with the static design wave analysis. The dynamic amplification factors and calibration factors of elastic structure are

Introduction _{1. 5}

Sensitivities of dynamic effects for different responsequantities and sea-state parameters are set up to the single design wave responses of the detailed structural model in chapter 6. The

uncertainties of dynamic effects and calibration factors are also introduced with respect to

their variation with the design wave height distribution The long term reliability analysis of the jack-up structure is formulated both with conventional design wave disthbution arid nested technique. The procedure for the reliability analysis of the detáiled structure is also extended

to the reliability analysis of a jack-up structure including nonlinear structural and nonlinear dynamic effects. The reserve strength and complexity measure of jack-up structure are

verified for different structural models of jacking system with and without fixation system.

The main conclusions from the present study and suggestions for the future studies are

discussed in chapter 7. The Appendices and the References belonging to Chapters 1 to 7 are

attended to the end of the dissertation; A list of symbols for all Chapters is provided in

Nomenclature.

1.4 Jack-ups risk versus fixed platforms

Statistics of structural failure for jack-up platforms is presented in this section. The accident

statistics of jack-up units is extracted from the Worldwide Offshore Accident Databank (WOAD) and is compared to those of fixed offshore platforms. Accidents in the WOAD

databank are defined as those events or conditions which have caused damage to supporting

structures or equipment and environment causing death or injury to individuals. Bearing in mind that the jack-up structures are inherently different structures than thefixed platforms, the integrity assessment of jack-up platforms should confirm that the target probability of failure can be chosen based on the experience of fixed offshore platforms or not.

For the case of a constant failure rate, i.e. X(t) independent of time, the time t between

successive failures of a system has an exponential distribution. The relation between failure

rate and probability of failure is then obtained by the natural logarithmic of the probability of survival or Xt=-Ln[1-PF(t)]; in which the PF(t) is the ratio of the total number of lost platforms to the total number of units in the period of consideration.

With the frequency rate of accidents, we are able to fmd the (instantaneous) failure rate for the total losses of two kinds of platforms for two types of failures, i.e. either for the total

loss or for losses resulting from environmental loading or foundation failure. The philosophy

for the integrity assessment of offshore structures can be organized with consideration of different uncertainties in the design parameters and assumed models. From different types of fixed and mobile offshore platforms, the relative reliability of jack-ups with the fixed

platforms are compared in the period of 1970-1991. For this purpose, the exposure statistics of losses for the jack-ups and fixed platforms has been extracted in the period 1970-'91 from the' WOAD [19921.

For the fixed platforms 83700 is the total number of platform-years exposure while for, the jack-ups the corresponding exposure is equal to 5420 rig-year in the period 1970-'91.

Therefore a relative exposure ratio of about (83700/5420 15) is obtained for this period.

The relative exposure ratio for the fixed platforms was fifteen times higher than that of

fixed platforms in comparison to the jack-up structures. Note that the frequency rate of accidents is determined by the total losses during the period of investigation (years) per 10,000 unit-years. For example the total losses of fixed platforms during 1970-'79 was 17

and the number of unit-years of exposure was 23,204 in the same period, thus the frequency

rate of total losses is 7.3.

Focusing on the most severe accidents, i.e. those found with the total loss of the unit, two results can be drawn. Firstly the total loss of integrity for the jacket type structures is

considerably less than for the jack-ups. Secondly, whereas the frequency rate of accidents for

the jacket structures shows a significant improvement with time (from 7.3 to 1.5), such a

trend cannot be observed for the jack-ups (134 vs. 110). On the other hand, if the same issue

is used for the total losses due to the environmental loading or foundation failure, i.e. excluding total losses due to blowout, fire, collision or during transportation, then the

frequency rate of accidents for the fixed platfórms will be (1.7 vs. 0.5) for the period of 1970-'79 and 1980-'91, while for jack-ups the frequency rate of accidents will be equal to (35 vs. 32) for the period of 1970-'79 and 1980-'91 respectively.

The relative failure rates for total losses (the ratio of failure rateof jack-ups to fixed platforms) are extracted from Table 1.2. These are 18 and 73 in the periodof 1970'79 and 1980-'91 respectively. The relative failure rates for losses from environmentalloading and

foundation failure are also estimated as 21 and 64 in the period of 1970-'79 and 1980-'91

respectively. From the statistics of failures it can be concluded that regardless of type of failurê (total failure or structural failure), the failure rate of jack-ups has beenconsiderably increased during the period of 1980-'91.

Table 1.2 Failure rates for fixed platforms and jack-ups in the period 1970-'91 (Numbers

in the table should be multiplied to 1/10,000)

The failure rate of jack-up platforms is at least times more than the fixed platforms during the period of 1970-'79 and surprisingly with the recent statistics an increase of relative failure

rate to is proven in the period of 1980-'91. Since the jack-upsafety is considered due to

the losses from environmental and foundation failure, this comparison could not be directly used for the purpose of the study in which the interest is only on the structural failure. While

the new jack-up designs is often used in deep water conditions, the integrity assessment of jack-up platforms will give valuable informations on the failure mechanisms and their

interactions in the extreme environmental condition. Therefore, the contribution of structural failure on the failure rates of jack-up structures can be identified by the structural reliability analysis methods in which the foundation failure is excluded from other failure modes.

Total losses consist of environmental Losses from environmental loading, fire, transportation, foundation

loading and failure

1970- 1979 1980- 1991. 1970- 1979 1980- 1991

Fixed Jackup Fixed Jackup Fixed Jackup Fixed Jackup

Chapter 2

Finite element analysis of structural dynamics

in jack-ups

2.1 Introduction

The use of adequate structural models is an important tool in the prediction of the extreme response and of the failure of the offshore platforms. This chapter deals with numerical simulation of jack-up units in order to conduct an assessment of system reliability of these

structures.

Methodologies for the dynamic response of jack-up structures are discussed. The finite

element theory is implemented to the integrated system of the jack-up structure consisting of

the hull (frame, plating), the space frame of the leg structure and the jacking system with

different types of load transfer models. The finite element formulation of undamped structural

behaviour has been discussed in literature and for the damped structural behaviour, two damping models, the linear viscous damping and the hysteretic damping, are derived in

appendix F.

The detail structural models of a jack-up structure consist of the orthotropic model of the hull module, the slender beam-columns for chords and interface elements in the jacking system.

Simplified and advanced solution methods for the dynamic equations in the elasto-plastic regions are discussed with emphasis on the damping models of jack-up structures. The equivalent leg model is verified with the original space frame in order to provide a suitable model for hydrodynamics and structural behaviour of the equivalent leg. The definition of the equivalent leg model is different for the hydrodynamic leg model and the structural leg

model. A numerical experiment on the jacking system is also used to check the effectiveness of the proposed model. Mechanisms of failure for the fixed and floating jacking systems are explained considering the three-dimensional behaviour of the jacking interface.

A coherent picture of the safety of jack-up structures is obtained by converting the

consequences of failure to the reliability scale: To calculate the ultimate strength, the reserve strength of different structural components has to be taken into account while the simplified

structural models usually under/óverestimätes the beariñg capacity of structure. To demonstrate the importance of structural modelling, Moan [1994] showed that in certain situations, the use of different ship design rules for a merchant ship implied a difference in steel weight of order of 20-30%. An example of the dynamic analysis of jack-up platforms has been given by Kjeoy et al. [1989], in which dynamic responses of the SDOF in single

design sea-state has been compared to the stochastic dynamic analysis whithin the irregular

sea-state. They concluded that for the jack-up in moderate conditions, the regular wave

analysis method underestimates the design responses by approximately 25-35% compared to

the time-domáin simulatiôn of design sea-states. Occasionally it is desirable to check the validity of structural simplifications, their individual and combined effects in partial and overall structural failure. In this chapter we focus on the structural design and analysis

methods of dynamically sensitive jack-up platforms.

2:2 Developments in tructura1 analysis of jackuP platforms

There are several

more or

less Table 2.1 Dynamic amplification ofthe extreme

advanced methods to analyse jack-up structural wave induced response;

platfOrms and the results are sensitive Excerpted from [Carlson et al. 1986]

to the method used. The initial static

deterministic analysis method is based

on the

full size modelling of the

structural framework and its supports. A dynamic response analysis has been

concerned out for the self-contained units which were moving to deeper waters and harsher environments. In 1982 Hattori et al. suggested that an equivalent single degree of freedom

(SDOF) would be suitable for the

dynamic analysis of jack-up rigs. They

calculated the natural frequencies of jack-up units considering the virtual

mass of the vibrating water, and the supporting condition of the sea bed. When the first natural period of the

jack-up rig is below approximately three seconds then a static deterministic "design wave" analysis is considered adequate for the structural assessment of the rig [SIPM 19891 sincethe

period of the sea-state is 10 18 s. The extreme natural period for a deterministic analysis is equal to five seconds inDNV classification notes [DNV 1992]. According to the simplified dynamic analysis procedure, a typical dynamic amplification factor of the extreme structural

response may be obtàined from Table 2.1 (Dynamic Amplification Factor for static design wave response). The natural period of the jack-up in the first mode of vibration is an important indicator of the degree of dynamic response to be expected. This leads tothe pseudo-static response analysis for typical jack-up drilling rigs in the elevated condition.

To simulate the structural behaviour with a realistic model, additional members were assumed to model the connection of hull and legs. Nagy [1986] defined the equivalent

stiffness of jack-up leg and the linear behaviourofjacking system located at the lower guide level. By decoupling two translation modes and one rotational mode, three uncoupled single degree of freedom systems designate the important modes of the structure. This model takes

into account the leg stiffness between lower and upper guide level. Development in the

jacking system can be found in the work of Sakata et al. [1985] and Gründlehner [1989]. For

a zero clearance jacking system, Sakata provides a linear model of the jacking system with

inclusion of the stiffness of the leg section between the guides concerning the guide position,

the guide stiffness itself and the stiffness of jacking system For non zero clearance, a three

-dimensional model has been developed by Gründlehner which takes into account the stiffness of the jackhouse structure and the backlash in the jacking system.

The second order moment due to the deck weight, the so-called P-ô andP- effects, was formulated in the structural model of the jack-up (idealized leg, beam model of hull). This

3-legged jack-up with lattice type legs 2 Dynamic amplification is only applied to the

amplitude value Water depth (m) Typical natural period (s)' Typical dynamic amplification2 70 5 1.1 90 7 1.2 110 9 1.4 130 11 1.8

Finite element analysis of structural dynamics in jack-ups 2.3

geometrical non-linearity has been, implemented bythe. equivalent sway stiffness in the dynamic analysis of jack-ups by Liii [1989]. An equivalent method has been suggested by Barltorp and Adams [1991] with the calculation of average leg compressive force and the corresponding reduction of the sway stiffness of the leg. To simp1if' the structural model, an idealized leg model for the lattice leg with equivalent hydrodynamic properties was

developed. Nonlinear problems (physical and geometrical) in the idealized leg model can be

implemented using approximate 'models. A reduced stiffñess of the legs was used to count for the geometric reduction in the stiffness due to the presence of axial load.

Today, the structural trends are directed to use superelements for structural modeling of jack-ups [Maison et al. 1991 and Lewis et al. 19921. These models present a better insight in the

stress analysis of the total structure. For nonlinear behaviour of structures, the present

methods of superelements can not used and new development may be needed in superelement techniques for the physical and geometrical nonlinear analysis of jack-up structures. Bearing all aspects in mind, one may conclude that there is a relation between the accuracy

of the method and the time needed for computer simulation. While for static analysis, a

detailed finite element model can be used, time domain dynamic simulation is often implied by the simplified stick structural models.

In this thesis, two finite element models for three leg jack-up structure are adopted. A rather detailed finite element model of the jack-up structure with limited design wave simulation and a three-dimensional stick model for the complete time history analysis.

2.3 Damping models for jack-up structures

Real structures that are subject to vibration are lightly damped and hence part of the energy is dissipated during vibration. For a jack-up structure, the damping terms are generated from the structure, soil and hydrodynamic damping. If the standard deviátion of a response gives sufficient information on the response of the structure, the viscous damping model provides

accurate estimates in a frequency domain analysis. For ultimate analysis of dynamically

loaded structures, an appropriate damping rnòdel should be substituted for each of the three terms mentioned previously.

The structural damping is described by a linear and viscous damping model. This is modeled by a proportional or Rayleigh damping model.

C=aK+ M

(2.1)

In which K and M are the stiffñess and mass matrices and proportional damping means that

it is proportional to the stiffness and mass matrix.

The soil-structure interaction is modelled by the linear spring models with the soil damping specified in the viscous damping models. In the soil-structure interaction, the characteristics of the viscous damping is difficult to obtain and the soil-structure interactiôn is modelled with

the hysteretic damping model [Karunakaran 1993]. For the present study, the soil-structure interaction is excluded from the investigation since our interest is on the behavibur of

structure itself. Derivation of the finite element formulations with viscous and hysteretic

damping mödels will be discussed in appendix F.

The most important damping source for jack-up structures is the hydrodynamic damping.

According to design codes, for robust jacket structures the hydrodynamic damping (potential and drag damping) is normally modelled by a linear viscous model while for slender jack-up

platforms, the generalized Morison equation is usually adopted [DNV 1984, 1992]. For drag-dominated structures, the dominating source of hydrodynamic damping is drag

dämping. This hydrodynamic damping is normally modelled implicitly with the

hydrodynamic forces by the extended Morison equation. For a flexible slender structure, the

hydrodynamic wave load may be expressed by the generalized Morison equation including the current velocity. According to this force model, the external force per unit length, ft+M at time t+&, for a vibrating system may be written as.

ft=KM1 MaÜ +KD.v -û .(v-ü)

(2.2)

in which y = v,, + v, vs,, denotes the water particle velocity induced by waves, VC denotes

the current velocity, and KM,KD, and M3 are constants related to the inertia, drag and added

nass.

KM=CMpA Ma=(CMl)PA ,KD=-pDCD (2.3)

where D and A are the equivalent diameter and area, CM and CD are the inertia and drag coefficients and p is the density of water. The values of KM and KD are generally influenced by the presence of current [Sarpkaya et al. 1981], but here they are considered constant and

equal to the KM and KD values for the zero current situation. Equation (2.2) posses a

nonlinear drag term which is un-symmetric in the relative velocity. Different approximations

are used for the non-linearity to simplify the generalized Morison equation. To realize the relative velocity expressions in the Morison equation, the constants in equation (2.2) are

substituted.

f

t_(c_1)pA()_ü)+pA+±CDp

_{.Iv+v-ûI.(v+v-u)}

(2.4)

From this equation, evidently the hydrodynamic damping forces are proportional to products of the following terms.

,

M. , v.IIfl

u.II

The first term is linear and time invariant as viscous damping, the second is also linear but time dependent, while the third is nonlinear with a quadratic nature. In practice, some simplified approximations of the. full interactive nonlinear Morison equation are used to

Finite element analysis of structural dynamics in jack-ups 2.5

describe the wave-current induced hydrodynamic foîáe. As proposed by Borgman [19691, the

nonlinear term is replaced by a linear one such that the mean square value of the difference is minimum. This gives the following models for the force equation.

Model i (M + Ma) ' + (C KD C )ú + Ku = KM + KD (a + by)

(2.6)

Model 2 _{(M+M)ü+(C+KDC)ü+Ku=KMI)+KDJvIv} (2.7)

where a = E(vlvI) and b = C = 2 E(IvI). If the mean Gaussian process y (t) with mean p and standard deviation r are given, the minimum mean squared error linearization gives:

a=E(vv) _(2+2) i _2(_)I +2o

exp[-2/(2o2)] (2.8)

o

b=Ca=2E(Iv)=2[l_2(_)i

(2.9)

o]

in which CF(x) denotes the cumulative normal distribution function. The first linear model can

be solved analytically because it is linear in y. The response statistical properties for the

nonlinear asymmetric Morison equations can be obtained by using numerical simulations. in chapters 5 and 6, the time domain dynamic analysis will be discussed with application of the

relative velocity formulation (eq. 2.2) and model 2 presented by eq. (2.7).

2.4 Overall Structural Modeling for a three leg platform

A jack-up platform system consists of four subsystems, namely the hull, legs, leg-hull

interface (jacking system) and leg-foundation interface. The reason behind this classification

is as follows. The jack-up have been built of a steel frame hull structure supported by three or four legs. The way of connecting deck to legs by the jacking system has great influence

on the stress distribution near the deck-leg connection due to the interaction of leg inclination and directional moment distribution. A model for the connection of deck floors to the jacking

system is obtained choosing the deck structural model in different elevations. In the first

periods of reasearch, a three dimensional fmite element model has been generated with linear

elastic analysis PEA program SAP9O [Wilson and Habibullah 1990] shown in Fig. 2.1 and for the nonlinear structural analysis, the overall structural model has been created with

ANSYS code (version 5.0 A) given in Fig. 2.2.

In the detailed analysis of the elevated condition, the deck structure transveres a combined

effects of the following forces:

- the leg reactions;

- the own structural weight plus dead load;

Figure 2.1 Overall model with SAP9O

For the purpose of this study, it

is assumed that the gravity

loads are the dominant loads for

the deck structure whereas the

variability of environmental

condition during the design life

is limited for the deck structure.

For jack-up structures, most of the work on collapse analysis has assigned to the collapse of

legs and foundation failure.

These studies indicate that in

the survival condition, structural.

failure mostly occurs in the

chords close to the leg guide and jacking unit. The collapse

mode of the legs depends on the

type of leg-hull interface and

may be

classified into two

modes,. bending and shear

collapse as suggested in

Mommaas et al. [19841 and

Rashed et al. [1987]. The

ultimate strength analysis of

jack-ups showed that the

bending collapse occurs by the application of a fixation system which leads to the failure of one chord in the vicinity of the lower guide (case b in Fig. 2.3 shows the reaction forces schematically). Furthermore the shear collapse happens in the bracing elements of the

diagonals between the leg guides since the deck-leg reactions are transferred by the horizontal

guides (case a in Fig. 2.3 indicates the fôrce distribution in guides). For different deck-leg

interface models (with or without fixation system), different fäilure paths are distinguished,

Figure

system

Figure 2.2 Overall model with ANSYS

2.3 Reaction forces with and

[Mommaas and Blankestijn 1984]

Finite element analysis of structural dynamics in jack-ups 2.7

thus a proper model for the jack-up leg to hull connection should be selected for this study. The most vulnerable parts of a jack-up, the legs, are subjected to deformations due to the combined action of the platform loads, the operational loads and the structural loads from

legs due to the wave, current, wind and possible earthquake loads. Failure of one leg means

that a three legged platform will turned with the total platform which is considered to be

catastrophic failure. A lattice leg structure collapses in a sequence of failures of members or

its joints. Member failures occur in a beam-column mode with a semi-brittle behaviour.

Tubular joints may fail due to collapse or fracture which is not studied in the present thesis.

It is acknowledged that the system effect for overload failure can be more significant for jack-up structures than for typical jacket structures, since the dynamic effects are important for jack-ups and their structural failure can be occured by failure of different types of structural elements. Due to possible random imperfections in structural behaviour, many failure sequences are expected and this is particularly important when the geometrical

non-linearities are included in the leg elements.

2.5 Hull Module modeling

From the structural point of view, the hull of a jàck-up is an all-welded structure and is

usually composed of different structural elements such as outer and inner shells, deckhouses including hatches and bulkheads. In selecting structural elements for the jack-up analysis two criteria are employed. Some elements are of imniediate interest for the design loads and their

response must be reported. Others are not interested in the global analysis and can be deferred or not reported at all. For example, most of the main deck and bottom plating of

the hull will be designed for a uniform pressure or a hydrostatic head and these locations will

not be controlled by the environment. This design strategy seems to be conservative based on the reliability analysis methods. Thus, such a detailed hull module is chosen based on a

suitable membrane element model accounting for the stiffness of the deck, sides, bottom, and main structural bulkheads.

We adopt here the hull structure as a complete, 3-dimensional structure including frame beams, plate elements and their connections'. Eàch floor of the hull module consists of stiffened panels intersected by transverse deck beams. The beams are extended to the side shells and they build a 3-dimensiOnal frame for the hull structure. All these beams are

attached to the plating and the centre line of these beams differ from the panels. Because the neutral axis of beams in the hull frame has offset from the neutral axis of plating, modeling

of both axises doubles the number of nodes. In order to build a complete model for the hull module, several approximations are proposed in literature. In most cases, the bending of

plating is considered in a local analysis of panels and in the global analysis only the in-plane

action of plating is considered. The equivalent bendiñg stiffness of plating is achieved by

either hybrid beam or eccentric beam theories (Hughes [1991]). With the hybrid beam model Hughes transfers the neutral axis of plating to the beam axis and by the eccentric beam model he uses the plating neutral axis as the reference. However, in this context, the application of offset unsymmetrical tapered beam elements will be presented which is an equivalent to the eccentric beam model.

The two

different

solutions are illustrated

schematically in Fig. 2.4.

As it can be seen from the

figure, for the hybrid

element, the structural

modeling is such that the structural nodes are located at the neutral axis

of the combined section of

the beam and the. plate (nodes 1 and 2 in hybrid beam element, Fig. 2.4). In this case, the neutral

axes

of

combined

elements are estimated

before

to

nodal

numbering.

In the

eccentric

(tapered)

elements, the element

nodes are located not

along

the

beams'

centroidal axes but rather at the toe of the web as

shown in Fig. 2.4. The

plate elements are located

at their true position or in

the plane of plating. The transformation of coordinate system for combined stiffness is done

automatically by the finite element code and the preliminary calculations for determination of the neutral axis is not needed by the application of the eccentric beam elements.

Figure 2.4 Hybrid and eccentric beam elements [Hughes 1991]

Figure 2.5 Hull module with plate modelling Figure 2.6 Frame module for the hul and leg

The hull module can be represented by either the plate elements on a hull frame (Figure 2.5)

- - -'j_

s:<P\

-__\

Finite element analysis of structural dynamics in jack-ups 2.9

or only the frame structure of the hull model without plate elements (Figure 2.6). In the

linear analysis using the SAP9O program, the additional membrane stiffness with the action of stiffeners is modelled by extra thickness of the membrane elements. Besides the bending

stiffness for the transversal beams in x direction is modelled by different thickness for the bending of plate elements than the membrane thickness. The model is rather complicated compared to the standard designs where the hull structure is normally modelled by beam elements with equivalent stiffness. Plate elements are included for two decks (main and bottom), and beam elements for vertical and horizontal bulkheads. Note that for plate

elements, 4 nodes elements in the three dimensional fmite element model are used taking into

account the flexural and membrane forces. For beam elements the constitute equations are

established based on the Timoshenko theory for beams with six degrees of freedom and the stiffness matrix method has been adopted for the evaluation of the structural displacements. The results of the stress analysis with a linear static structural program, SAP9O, showed that

the vertical bulkheads are mainly loaded by the membrane forces. Findings from the

preliminary study with the linear analysis program SAP9O have been used for generation of the hull model with ANSYS software (Figures 2.7 and 2.8).

Figure 2.7 Hull model with minimum nodes Figure 2.8 Frame model and machinery deck

In this model, the main floors have to be designed for the combined action of the membrane and bending stresses. Moreover the effect of longitudinal stiffeners on the hull structure can

be modelled by the unidirectional orthotropic plates which are also called the transversal

isotropic stresses in orthotopic materials. The orthotropic equivalent material characteristics are found by the elastic material properties and the stiffener areas directed in y direction. In

the hull frame, the deck floors (main-, machinery- and bottomdecks) can be adequately

modelled by transversal isotropic plate elements which are stiffened by beam elements along

x direction (for example at each 2.5 m).

Based on the results of parametric study, adopting a linear elastic model of the hull structure,

the detailed model of the hull structure will yield an accurate estimation of the structural

responses of the legs and the jacking systems. For jack-up structures with a fixation system,

of the hull model and the lattice leg structural model.,

2.6 The three-dimensional beam-column models for a leg module

The lattice geometry of a jack-up leg is designed with a space frame structure including chords horizontal and diagonal braces with different diameters There are two models for

the structural simulation of jack up legs namely the detailed model and the equivalent model

Clearly the detailed model is recognized to be suitable for the legs especially near the hull leg connections where the complex behaviour is not achieved by suprposition of individual effects. However, it is established that the equivalent model gives satisfactory results when

the hull-leg connection is equipped with a fixation system.

The legs are designed taking into accouñt the interaction of axial loads and bending moments

due to the combined action of gravity, wave and wind loading. The distribution of axial

forces and bending moments is influenced by the connection in the deck leg mterface Studies forjack-ups equipped with different jacking systems indicate that the most vulnerable of these

elements, the legs, may behave considerably different for the various jacking systems.

Nevertheless, the reserve strength of the structure will be important in case of application of

a strong floating jacking system without fixation system as well as fixedjacking unit with a

fixation system.

Space frame

offshore structures

are most comínotily modelled by

beam-column type

elements as shown in Fig. 2.9. The principles of a finite e

1 em e

n t representation of beam elements has

been studied by

several authors and is not repeated here

again.

For

a

designer, it is very

important to define

the most suitable

type

of

beam

element in each,

particular frame.

For example, the lattice geometry of a jack-up leg is designed with a space frame structure including chords, horizontal and diagonal braces. The beam-column elements are modelled with the rigid joint connections and the failure of tubular joints are excluded in this case Further for a jacking system, the rigid beam element often provides a proper tool for the

finite element representation.

Figure 2.9

colurñn

beam-Finire element analysis of structural dynamics in jack-ups _2.11

For the beam-colunm element the strain-displacement rlatibnship cöntains the 'second derivatives of the lateral displacements and the first derivative of the axial displacement or twist. Hence it is necessary to choose the displacement function such that (u, y, w, O, O, O,

O') are continuous at the nodes. The contribution of O' comes from the torsion of an open

cross section. due .to the warping effect (See Chen and Atsuta [1977]). For a closed Section

the so-called St. Venant theory applies which assumes a constant specific torsional angle.

However, if we exclude the warping effect, the other displacements can be exactly described by the displacements of the shear centre of an arbitrary section. The resulting shape functions

should satisfy the. completeness criterion which means that the. rigid body motions and the constant strain state of the element are included accurately.

In the ANSYS code, both a consistent and a lumped element mass formulation are available.

For the consistent mass formulation, the mass matrix is derived by assuming the same displacement functions as those employed in the determination of the stiffness matrix.

However, the elastic stiffness concept involves derivatives of the displacement functions, and the derivatives are represented with less accuracy than the displacements. Therefore it would

be reasonable to imply lower order interpolation function in the formulation of the mass matrix or using the lumped formulation.

2.7 The selection of leg-hull connection model

In the jack house with leg locking devices (fixation system) the leg-hull interaction is modelled as a totally rigid connection because the clearance has to be eliminated by the fixation system. Usually, in other types of connection systems, the superpositiòn of the responses for the two mechanisms of guide action and jacking unit reaction is not possible because the clearance and backlash introduce nonlinear spring characteristics (the backlash zone has to be added in the nonlinear model behaviour). Thus, the deck-leg connection without fixation system introduces a non-linearity in the leg characteristics between the leg guides. The interface behaviour is dependent on the inclination orientation and the vertical leg load, while the inclination and the reaction moment have different orientation. The

contact of gùide with the leg in the model is not coinciding with the direction of loading and

thus the conservative approach in the guide reactioñ with the same orientation of leg chord is corrected by this model.

For the leg to hull interface, two dimensional and three-dimensional models have been proposed in literature which are suitable for the jack-up structures. Except in the study by Gründlehner [1989], usually, the leg-hull connection is modelled by linear springs. Gründlehner has proved that the deck-leg connection behaviour for a fixed type jacking system withoUt fixation system can differ from the simplified linear spring models. Furthermore he found that the distribution of the leg bending moment over the horizontal

guiding system and vertical jacking system is strongly dependent on their stiffnesses relative

to each other and also guide clearances. Thus he has developed a PEA model that includes the following effects in the detailed model of a jack-up leg to hull connection:

Bending, shear and torsional stiffness of the leg section between guides. Axial bending, shear and torsional stiffness of the jackhouse structure. Stiffness of the guides.

Stiffness of the jacking system.

Amount of clearance of the leg within the guides.

Amount of backlash in the jacking system.

The type of leg guide arrangement (radial, tangential or combined) and the rack arrangement (single radial or double opposed) are also included in this model. Using the proposed model, the leg-hull equivalent connection is presented by two nonlinear rotational springs at the hull

mean level. If the guide reaction is imposed halfway of a bay, it turns the exact horizontal guide stiffness that is governed by the shear deformation. However, in the proposed model the upper and lower guides are imposed halfway a bay and the effect of the position of the upper and lower guides were not included. This effect has been considered by Sakata et al.

[19851 for a two dimensional leg-hull connection and the constitute relationships have been

derived for the combined effects of items (1), (3), (4) given above. The shortcoming of this

Japanese model was the ignorance of the jackhouse stiffness, the clearance of the guides and

the backlash effect oñ the jacking system. Furthermore the larger discrepancies will notbe

found by the amount of the reaction moment but in its orientation. Thus it has been decided

to develop a model where the real conditions of the guiding and jacking systems are taking into account with a FEA method.

The model shown in Fig. (2.10)

compromises between accuracy of the

model for the three-dimensional case and

flexible leg-hull connection with location of

guides. In the elevating system, the guides

can only take compressive forces. This was

simulated by using gap elements with zero stiffness. Further the effect of clearance tolerance is also introduced by these gap

elements. The pinions were modelled by the bending beams which are cantilevered form

the rigid bar in the guiding frame. The

guides are considered as rigid elements and

their stiffness are not included in this model.

In this model,, the leg is inclined until a chord makes contact with a guide or a pinion loses contact with a rack. In this stage, the stiffness of the contact zone remains zero as the chord moves through the backlash zone and the tooth contact is changed from the top of the pinion to the

bottom of the

next tooth (Fig. 2.11).

Although in the backlash zone a pinion has

no stiffness, Grtindlehner proposed that 1

percent of the equivalent stiffness of the

pinion be assigned to the backlash interval.

HULL

--HOFJZOlTtAL

(CHORD

L

LOW

Figure 2.10 Three-dimensional jacking

structure for fixed and floating jacking system HOULE

HULL

This trick avoids the rigid body motion of the leg in the backlash zone. The backlash

displacement is considered

half of the

backlash zone in order

to model the

combined effects

of upper and lower

pinions. The upper guides can be connected

to the hull or the jackhouse top and in the opposed rack system the later is preferred. The lower guides are always connected to

the hull.

2.8 Numerical tests with proposed jacking model

Figure 2.12 Leg inclination at sea-bed

connection (t is time-step in s).

10N

FREE TRAVEL

NOLOAO____-TRANSFER

The proposed model in Fig. 2.10 can be incorporated in a systematic form for the deck-leg connection of both fixed and

floating type jacking systems. In the

floating jacking systems, the impact loads associated with the engaging bottom are absorbed by shock pads. The shock pads have been introduced in the model by four springs in the connection of jackhouse box at the upper and lower levels. In fixed type

jacking system, an infinite stiffness for the shock pad springs are chosen.

zone [Bennett W.T.,

Figure 2.13 Variation of leg reaction (W is

the weight of jacking system in kN).

Backlash

Figure 2.11 Friede Jr. 1986]

2.13 Finite element analysis of structural dynamics in jack-ups

In the initial côndition, the gaps are open and

the gap size is updated at the start of each

iteration. Therefore the gap operates

biinearly and the interface -system is nonlinear. For the normal direction of a gap element, when the normal force is negative,

the gap is closed and the element responds as

a linear spring. As the normal force becomes positive, contact is broken and no force is

transmitted in normal direction. In the

tangential direction, the tangential force is assumed to be proportional with the normal

force IF ¡. Two cases are considered;- the

rotation of jack-up leg at bottom (case i

shown in Fig. 2.12) and the variation of vertical leg reaction (case 2 shown in Fig.

2.13).

Case 1) For the leg force of 36000 kN and an increment of 10 for the leg inclination at each time step, the behaviour of the jacking system

is shown in Fig. 2.14. If the tangential force exceeds the maximum shear resistance, the

guide contact slippes and no force is

transmitted in transverse direction.

0.4 9.6 11.2

Figure 2.14 Behaviour of gap connectiön: Diagram 1, the friction force in the lower

guide; Diagram 2, the normal force in guide

Case 2) Another numerical experiment illustrates the adequacy of modelling the jacking system and the shock pads. Consider the first stage of the load reversal in the case of a floating type jacking system. The springs of the jacking system will carry noInternal force

as the chord moves through the backlash zone and the tooth contact is changed from the top

of the pinion to the bottom of the next tooth. Through this stage, the jacking system is resting on the bottom shock pads. As the leg reaction increases, the springs of the jacking system begin to work against both the leg reaction and the lowershock pads.

When the internal force in the jacking system reaches the weight of jacking unit (start of third time step), it is then discharged from contact on the bottom shock pads and falls- to contact with the upper pads (Figs. 2.15 and 2.16). In this stage, the springs of the upper

shock pads are working with the springs of the jâcking system [Bennett et al. 1986]. In other words, the jacking system is in contact with the lower shock pads in transport condition and

it is in contact with the upper shock pads in elevated condition [Joint industry jack-up

committee 1992, 1993].

-4

1.6

5.6 .2 3.2 4.8 6.4 8

uumu

...

V1UUlUU

ti ne tepe (sec.)

...

u'..

...

ti ne steps (sec.)

Figure 2.15 Reactions of lower shock pads Figure 2.16 Reactions of upper shock pads for the three chords of a jack-up leg for the three chords of a jack-up leg

Finite element analysis of structural djrnaniïcs in jack-ups _2.15

-2

-4

3.1 Introduction

One of the goals of the structural design is to ensure that the load on the structure and its resulting load effect, such as bending moment, shear force, and axial force, are sufficiently below all of the applicable limit states. This condition meets with three fundamental

approaches in use today for the design of steel structures which are known as Allowable (or

Working) Stress Design (ASD), Plastic Design (PD) with load factor, Load and Resistance

Factor Design (LRFD). Under the ASD philosophy, the design is evaluated by the so-called

Factor of Safety (F.S.) which depends on the particular limit state. The stress analysis is based on a first-order elastic analysis, and the member interaction equations are implicitly

admitted for the second order effects (geometrical non-linearity).

In an attempt to quantify the actual safety level, the PD with load factor was suggested as direct measure of ultimate strength of structural component. In the PD approach, the limit state is attained by the plastic resistance of individual members and the load factor is

determined by the ratio of plastic strength to the factored load combinations. The advantage

of the PD method is that it considers redistribution of forces in a more direct manner while

in the ASD method, the safety factor for all of load conditions (dead load, live load, drilling loads and environmental loads) are assumed to be the same. The main advantage of ASD is

that it is familiar for the designers and simple to apply. On the other hand, in the PD method, the load factor reflects the associated uncertainty in load levels (using the

probabilistic approach), however the degree of uncertainty of different loads cannot be fully accomplished by applying a single load factor.

The PD is considered as a step in the development of the present probability based design

method (LRFD) which uses separate load fâctors for each load (uncertainty of load) and the

randomness of the resistance of the material. In this design method, the designer has an option to use either the elastic or the plastic structural analysis method. By adopting an

elastic analysis, the interaction equations are proposed based on the combined effects of axial

stress and bending stresses. In lieu of a second-order analysis, the (P-ô) effects for the element displacement and the (P-i) for side-way displacements of structure may be accounted for the stability checks. If the plastic analysis is combined with the second-order

effects, then a separate specification of member capacity is not required. With the evolution

of powerful computers, the direct assessment of structural system, member strength and stability of elements is considered without the need of specification of member capacity

checks. In reality, the member interaction equations in the modern limit-states specifications were developed, in part, by curve fitting to the results from the second order plastic analysis.

3.2 Nonlinear structural analysis of offshore platforms

Analysis of nonlinear structural behaviour of steel framed platforms has been a topic of research for a long time in the offshore industry. Although many factors may cause nonlinearity there is a general agreement that for most offshore structures, the material nonlinearity is the most significant. Therefore, models for nonlinear behaviour of

beam-colúmn structures have been extended by the so-called plastic limit analysis introduced some