Geotechnical Finite Element Analysis

Geotechnical Finite Element Analysis

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Geotechnical Finite Element Analysis A practical guide

Andrew Lees BEng PhD CEng MICE


Published by ICE Publishing, One Publishing, One Great George Street, Westminster, London SW1P 3AA Full details of ICE Publishing sales representatives and distributors can be found at: www.icebookshop.com/bookshop_contact.asp Other titles by ICE Publishing:

Finite Element Analysis in Geotechnical Engineering: Volume two – Application D. Potts and L. Zdravkovic´ ´ .. ISBN 978-0-7277-2783-1 Structural Analysis with Finite Elements P. Rugarli. ISBN 978-0-7277-4093-9 Finite Element Design of Concrete Structures G. Rombach. ISBN 978-0-7277-3274-3 www.icebookshop.com A catalogue record for this book is available from the British Library. ISBN 978-0-7277-6087-6 #

Thomas Telford Limited 2016

ICE Publishing is a division of Thomas Telford Ltd, a wholly-owned subsidiary of the Institution of Civil Engineers (ICE). All rights, including translation, reserved. Except as permitted by the Copyright, Copyr ight, Designs Designs and Patents Patents Act 1988, no part of this publication publication mayy be re ma repr prod oduc uced ed,, st stor ored ed in a re retr trie ieva vall sy syste stem m or tr tran ansm smit itte ted d in an anyy form or by any means, electronic, mechanical, photocopying or otherwise, without the prior written permission of the publisher, ICE Publishing, One Great George Street, Westminster, London SW1P 3AA. This book is published on the understanding that the author is solely responsible respo nsible for the statem statements ents made and opini opinions ons expressed expressed in it and that its publication does not necessarily imply that such stat st atem emen ents ts an and/ d/or or op opin inio ions ns ar are e or re refle flect ct th the e vi view ewss or op opin inio ions ns of th the e publishers. While every effort has been made to ensure that the statements made and the opinions expressed in this publication prov pr ovid ide e a sa safe fe an and d ac accu cura rate te gu guid ide, e, no liliab abililit ityy or re resp spon onsi sibi bilility ty ca can n be accepted acce pted in this respect by the author or publi publishers. shers. While every reasonable effort has been undertaken by the author and the publishers to acknowledge copyright on material reproduced, if there has been an oversight please contact the publishers and we will endeavour to correct this upon a reprint. Commissioning Editor: Laura Balchin Development Editor: Maria Ineˆ s Pinheiro Production Editor: Rebecca Norris Market Development Executive: Elizabeth Hobson

Typeset by Academic + Technical, Bristol Index created by Simon Yapp Printed and bound in Great Britain by TJ International Ltd, Padstow


Contents

Preface About the author

vii ix

01 . . . . . . . . . . . . . . . . . . . . . . . . . . . .

How is a geotechnical geotechnical finite element analysis analysis set up? 1.1. Analysis planning 1.2. Geometry 1.3. Meshing 1.4. Analysis stages 1.5. Constitutive models 1.6. Groundwater and drainage References

1 1 7 17 18 26 27 27

How are constitutive models selected? 2.1. Introduction 2.2. Aspects of ground behaviour 2.3. Co Common constitutive model types 2.4. Typical applications References

29 29 31 36 48 52

02 . . . . . . . . . . . . . . . . . . . . . . . . . . . .

How 03 . . . . . . . . . . . . . . . . . . . . . . . . . . . . How

are so are soil il an and d ro rock ck pa para rame mete ters rs ob obta tain ined ed? ? 3.1. Introduction 3.2. Soil and rock sampl sampling ing and and groun groundwat dwater er measurement 3.3. Parameter testing 3.4. Pa Parameter derivation and validation Appendix 3.1 – Useful equations in the validation of model or initial state parameters References

04 . . . . . . . . . . . . . . . . . . . . . . . . . . . .

05 . . . . . . . . . . . . . . . . . . . . . . . . . . . .

55 55 59 64 85 97 99

How are groundwater groundwater effects taken into account? 4.1. Introduction 4.2. Drained and undrained analyses 4.3. Groundwater flow analyses 4.4. Consolidation analysis References

105 105 109 118 120 123

How ar How are e ge geot otec echn hnic ical al st stru ruct ctur ures es mo mode dell lled ed? ? 5.1. Structural geometry 5.2. Structural materials 5.3. Soil–structure interaction References

125 125 125 152 156 160 v


Can 06 . . . . . . . . . . . . . . . . . . . . . . . . . . . . Can

FE an anal alys ysis is be us used ed wi with th de desi sign gn co code des? s? 6.1. Introduction 6.2. Serviceability limit state (SLS) 6.3. Geotechnical ultimate limit state (ULS) 6.4. Structural limit states References

163 163 163 167 168 180 181

07 . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Ho H ow is th the e acc ccu ura racy cy of outp tpu uts ass sse ess sse ed? 7.1. Introduction 7.2. Assessing accuracy 7.3. Managing errors References

183 183 188 192 197

08 . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Examples 8.1. Introduction 8.2. Raft foun foundatio dation n with with settlem settlementent-redu reducing cing piles example 8.3. Shaft excavation example 8.4. Em Embankment construction example References

199 199 199 225 243 261

Index

263

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Preface

It soon became clear to me while coord coordinati inating ng the European Commission Lifelong Learning COGAN Project on improving competency in geotechnical numerical analysis that finite element (FE) analysis is now widely used in geotechnical engineering but, in contrast to other fields of engineering, there are few fulltime users of such software. Geotechnical FE analysis places heavy demands on the competency of engineers but it is difficult to gain sufficient competency when applying appl ying such softwa software re part-ti part-time me betwe between en other engineering tasks. There was an obvious need for a ready reference for users of geotechnical FE analysis software to learn about and refresh their knowledge on applying the technique in practice. This book is intended primarily to address that need. Before using this book, it may also be useful to know the following: g

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The book book is strictly software software neutral. neutral. I did not not want to appear to be favouring any particular software. I have not endeavou endeavoured red to cover the essentia essentiall background soil mechanics, rock mechanics and geotechnical engineering knowledge needed to perform FE analysis since this can be found readily from other sources. Worked Work ed examples examples in FE analysis analysis are complica complicated ted to present and explain. So that readers can access information quickly, I have avoided putting examples within the topics in Chapters 1 to 7. Rather, three examples illustrating application of many of the topics are presented and described separately in Chapter 8. Some parts parts of the NAFEMS NAFEMS guidebo guidebook ok Obtaining Obtaining Parameters for Geotechnical Analysis which Analysis which I authored have been reproduced in this book, particularly in Chapter 3, with the kind permission of NAFEMS. This book provides the background information information covering about 160 competence statements from the COGAN Competency Tracker maintained by NAFEMS. NAFEM S. This Competency Tracker is avail available able online to individuals free of charge for monitoring and recording competency in geotechnical numerical analysis. Andrew Lees Nicosia May 2016 vii



About the author

Andrew Lees graduated

with a BEng in Civil Engineering at the University of Southampton in 1996, where he also obtained a PhD in the fields of centrifuge modelling and FE analysis of soil–structure interaction in 2000. He was then a geotechnical engineer at a major UK consultancy until 2004 when he took up a lectureship at Frederick University, Cyprus where he taught geotechnical engineering until 2015. In 2007, he also set up and continues to run the successful consultancy Geofem, specialising in geotechnical FE analysis. In 2016, he was also appointed Senior Application Technology Manager at Tensar International, where one of his tasks is to improve techniques of modelling geogrid-stabilised soils by FE analysis. He is a member of the NAFEMS Geotechnic Geotec hnical al Work Working ing Group and authored their first guidebook on obtaining parameters for numerical analysis and is a founding member of the Professional Simulation Engineer scheme administered by NAFEMS. He coordinated the European Commission Lifelong Learning project COGAN on improving competency in geotechnical numerical analysis. He was convener of the evolution group advising the Eurocode 7 committee on the use of numerical methods in accordance with the design code and has since been involved in the redrafting of Eurocode 7. He is a member of the British Geotechnic Geotec hnical al Assoc Associatio iation n and the Institu Institution tion of Civi Civill Engineers.

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Geotechnical Finite Element Analysis ISBN 978-0-727 978-0-7277-6087-6 7-6087-6 ICE Publishing: All rights reserved http://dx.doi.org/10.1680/g http://dx.doi.or g/10.1680/gfea.60876.001 fea.60876.001

Chapter 1

How is a geotechnical finite element analysis set up? The following sections in this chapter describe the steps taken and decisions to be made when wh en set settin ting g up a geo geotec techni hnical cal fini finite te ele elemen mentt (FE (FE)) an analy alysis sis mod model. el. In ma many ny cas cases, es, readers are referred to sections in subsequent chapters where more detail is provided. The implementation of these steps is demonstrated in the examples in Chapter 8.

1.1 .1.. 1.1. 1. 1.1 1

Analy Anal ysis pl pla ann nnin ing g Does Do es FE an anal alys ysis is ne need ed to be us used? ed?

This is an important question because FE analysis usually involves a lot more time and expense than conventional design methods, so choosing this method needs to be justified. The mere use of FE methods does not guarantee accurate predictions. Arguably there is greater scope for error due to the power and complexity of such software. Non-numerical, or conventional, methods of design are usually quicker and cheaper, but they the y hav havee majo majorr assum assumptio ptions ns (e.g. linea linearr elas elasticity ticity,, unifo uniform rm gro ground und pro propertie perties) s) and provide limited information (e.g. average settlement of a foundation, limit states). Nevertheless, in spite of the assumptions and probable conservatism, they are often sufficient to demonstrate a satisfactory design without significant loss of economy. In such cases FE analysis cannot normally be justified. However, in other instances there may simply be no conventional method to calculate the required output, or the greater precision and detail offered by FE analysis at the design stage could bring significant economies during construction. For example, FE analysis rather than conventional analysis methods might be required when any of the following need to be considered: g

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complex ground ground behaviour behaviour (e.g. non-linear stiffness, hardening soil, soil, anisotropy, anisotropy, creep), more realistic ground behaviour or changing ground behaviour (e.g. ground improvement or treatment, consolidation) complex hydraulic conditions unusual geometry soil–structure interaction interaction and internal internal structural forces in complex complex structures, and and interactions with adjacent structures complex loadings the effects of the construction construction sequence sequence and constructi construction on method method 1



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applying observational observational approa approaches ches to design design time effects effects (e.g. creep, creep, consolida consolidation) tion) back-analysis back-ana lysis of field field trials or monitored monitored structures. structures.

To help decide whether the use of FE analysis can be justified, a preliminary analysis can be performed with rudimentary project information and the outputs compared with appropriate conventional methods to assess the potential economic benefit of investing more time and money at the design stage in FE analysis.

1.1. 1. 1.2 2

What Wh at ar are e the the ai aims ms of th the e FE FE ana analy lysi sis? s?

Before thinking about building an FE model, the aims of the FE analysis need to be defined. For example, it may need to be demonstrated that a geotechnical structure has adequate safety against failure, or that the movement of an adjacent building is small enough not to cause damage, or to predict the flow of water into a cofferdam. Each requires a different approach, so the aims need to be defined at the start so that the decision-making throughout the preparation of the model helps to ensure that the model prov pr ovide idess suf suffici ficient ently ly ac accur curat atee pr predi edicti ctions ons.. If one of the aim aimss we were re the pr predi edicti ction on of ground gro und defor deforma mations tions,, for exa example, mple, then softw software are and cons constitut titutive ive mode models ls tha thatt wer weree known to produce accurate predictions of ground deformation for the site conditions would wo uld be cho chosen sen an and d pa para ramet meter er tes testin ting g wou would ld foc focus us on obt obtain aining ing ac accur curat atee sti stiffn ffness ess parameters for the ground. From the start, the analysis’ aims should be discussed with other stakeholders in the project to help ensure that the FE analysis meets their needs. FE models can take a long time to prepare and it is frustrating to learn of a new issue near the end of the process that could have been addressed by the FE model if it had been included in the aims of the analysis at the start. Some stakeholders will be third parties, particularly if ground movements might affect adjacent structures, services and infrastructure. So, as part of the site investigation, check with neighbouring property owners, utility companies and infrastructure agencies (e.g. highways, railways, metro lines) that their requirements are covered by the aims of the FE analysis. Document the aims of the analysis clearly and have them checked by the project stakeholders so that everyone knows what to expect from the analysis model and to avoid any misunderstandings. Once agreed, the written aims should be kept close at hand and refer re ferre red d to wh when enev ever er de deci cisi sion onss ar aree ma made de re rega gard rdin ing g th thee FE mo mode dell an and d ob obta tain inin ing g parameters. The outputs from the FE analysis that will be used to meet the specified aims are the key outputs. outputs. Clearly, it is vitally important for these outputs to have sufficient accuracy because they will influence the design of the project. Every decision during the design of the FE model should be made considering its effect on the key outputs.

1.1.3 1.1 .3

Whatt inf Wha inform ormati ation on nee needs ds to be ga gathe there red? d?

To pr prod oduce uce an ac accur curat atee geo geotech technic nical al FE mo model del,, com compr prehe ehensi nsive ve inf inform ormat ation ion on the historical, present day and proposed conditions at the site is needed. This requires an 2 Downloaded by [ University College London] on [14/11/16]. Copyright © ICE Publishing, all rights reserved.

How is a geotechnical finite element analysis set up?

extensive search of information sources, largely as part of the site investigation, as well as regul re gular ar com commu munic nicat ation ion wit with h mem member berss of the pr proje oject ct tea team m and thi third rd par partie ties. s. Ev Every ery project is different but the types of information gathering often include the following broad categories:

Ground information Careful planning of the ground investigation is needed to obtain the information necessary to for form m a suf suffici ficient ently ly re repr presen esenta tativ tivee sim simula ulatio tion n of gr groun ound d beh behav aviou iourr in the FE model, and this stage is covered in detail in Chapter 3. Essentially, sample descriptions and characterisation tests are used to form a ground model representative of site conditions. Then, by referring to the aims of the analysis, the required geotechnical parameters can be obtained by dedicated parameter testing. When interpreting the findings of the ground investigation and parameter testing results, it is important to understand the geological history of the site and the mechanisms of strata formation. The uncertainty in the interpretation of the ground conditions and parameters needs to be judged in order to select appropriate characteristic values, and sensitivity analyses are necessary to assess the potential effects of the uncertainties on the FE model outputs. Regular communication with those undertaking the ground investigation will help in judging the uncertainties.

Historical information During the desk study stage of a site investigation, information on historical land uses on and around the site is gathered, but how is this relevant to an FE analysis of today’s situation? Stress history and stress path have significant effects on the behaviour of the grou gr ound nd and therefor thereforee infl influen uence ce the inp input ut pa para ramet meters ers to a mod model. el. Als Also, o, in ord order er to recreate the stress path and current stress state in the model accurately, it may be necessary to simulate historical construction stages in the FE model leading up to the present day situation. Therefore, Therefore, the gathered historical information information should be used to build up a timeline of significant loadings (e.g. foundations), unloadings (e.g. excavations), tunnelling and other structures that may exist in the ground (e.g. unused piles or foundations). When preparing the FE model some of these historical activities may be important enough to be simulated in the construction stages or may influence the input parameters and in situ stresses.

Existing structures and infrastructure information If the site has existing structures or infrastructure, details of the existing geotechnical structures (e.g. foundations, retaining walls, slope supports, tunnels, buried services) and loads from the existing structures and infrastructure will need to be obtained. This may include structures and infrastructure adjacent to the site where they influence ground behaviour or feature in the aims of the FE model. Ideally, as-built drawings will be available together with designs and load schedules, and these can be sought from owners of the existing structures and infrastructure. Often such comprehensive information is not available, particularly for older structures, and some intrusive investigation of existing geotechnical structures will need to be included in the 3 Downloaded by [ University College London] on [14/11/16]. Copyright © ICE Publishing, all rights reserved.


site investigation. Even with intrusive investigation, assumptions will probably have to be made regarding existing geotechnical structures, so their type and geometry will need to be estimated based on experience of similar structures of the same age and by using design methods appropriate for the period of construction, and different options studied where there is uncertainty. Regarding existing loadings, rarely will these be available from the original design of older structures, so they will have to be estimated based on typical loadings for the type of structure and its use. Remember that existing loadings can often be favourable: for instance, an existing structure on a site to be demolished will have pre-loaded the ground such that settlement of the subsequent structure’s foundations will be reduced. In such a case it would be appropriate in an FE model taking account of pre-loading effects to apply the estimated actual loading rather than an upper bound value typically adopted for the design of new structures. Where the aims of the FE analysis include verifying that the settlement or distortion of adj adjac acent ent st struc ructur tures es and inf infra rast struc ructur turee ar aree wit within hin ac accep ceptab table le lim limits its,, the ga gathe there red d information could be used to set these limits. Sometimes, particularly for infrastructure, the owner will provide acceptable deformation limits. On other occasions, the limits may need to be judged to help ensure that existing structures do not suffer an unacceptable level of damage resulting from construction-induced ground movements on the site.

Proposed structures and infrastructure information Naturally, information on what is proposed to be constructed on the site will need to obtained in order to simulate its construction. Consequently, at least the following will normally be required: g

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drawingss and building informat drawing information ion modelling modelling (BIM) files for the pro proposed posed substructure in order to define the geometry of the FE model sufficient information information (drawings (drawings and BIM) regarding the superstructure superstructure in order to assess whether this will affect the behaviour of the ground proposed pro posed loading loadingss on the substructur substructuree and foundat foundations ions and the different different load cases that need to be considered limits on acceptable movement movement and distortion distortion of the substructure and foundatio foundations ns proposed pro posed constructi construction on sequence sequence in order to prepare prepare constructio construction n stages in the FE model proposed construction construction programm programmee in order to estimate estimate time intervals between construction stages, which will be important for deciding whether to simulate lowpermeability strata as drained, undrained or with consolidation in the various construction stages, or other temporal effects such as creep.

1.1.4 1.1 .4

Which Whi ch FE ana analys lysis is sof softw tware are sho should uld be use used? d?

Some software will perform certain tasks better than others, so try to choose the software most suited to the task. In every case the user needs to know the software very well, including its strengths and limitations. 4 Downloaded by [ University College London] on [14/11/16]. Copyright © ICE Publishing, all rights reserved.


Look for case studies including FE analysis of similar problems to see which software was used by others and how well the software performed. Most programs have user groups providing forums for the exchange of ideas and experiences, as well as ready access to case studies. Verify that the software works properly on the computer being used. The range of computers, devices and operating system versions continues to grow and there is always the potential that one combination of these may not be fully compatible with all features of a particular FE analysis program (see Section 7.2.1).

1.1.5 1.1 .5

How Ho w will will the FE ana analy lysis sis fit into into the des design ign pr proc ocess ess? ?

It is likely that the FE analysis will form one part of larger design process, so be aware of the ‘bigger picture’ to ensure that the FE analysis work fits within the design process as seamlessly as possible. A lot of the information gathered for the FE analysis inputs will be sourced from the main design process while the outputs and recommendations will be fed back into the subsequent design stages. The FE analysis outputs must meet the needs of the follow-on designers. For instance, consider the design of a raft foundation where an FE model of the ground volume and foundation has been used to predict settlement and deformation of the raft. The structural design of the raft may need to consider multiple load combinations subject to frequent revision so the structural engineer would prefer to calculate structural forces (bending moment and shear force) in the raft using his/her own, simpler soil–structure interaction analysis software. Some of the FE analysis outputs would need to be presented in a form that could provide input to the structural engineer’s simpler model (e.g. coefficients of subgrade reaction for a beam-spring model – see Section 5.3.2). Also, outputs of structural forces from the FE model could be provided to help the structural engineer to validate the simpler model. In more straightforward cases the structural engineer may use the FE analysis outputs of structural forces directly in his/her design. Consequently, knowledge of the wider design process is needed in order to meet the needs of other designers using the outputs. Any outputs provided to other designers must be clearly explained to avoid misunderstandingss and delays or errors in the ongoing design process. For example, standing example, be clear cle ar about units un its,, axi axiss dir direct ection ionss (glob (global al an and d loc local), al), sig sign n con conve venti ntion on,, con const struc ructio tion n st stage age,, da datum tum values for outputs, load case and any factors applied to inputs or outputs. Also show a clear legend for contour and vector plots. Regular communication among the design team is key to help avoid misunderstandings. Keep up to date on the wider design process through document management systems and regular communication to ensure that the FE model stays up to date and that the outputs and recommendations are relevant to the latest design. It is common for FE models to be revisited long after they were completed due to delays, changes in design or is issu sues es en enco coun unte tere red d du duri ring ng co cons nstr truc ucti tion on.. Th This is is on onee of th thee re reas ason onss wh why y a go good od write-up of the analysis work is essential (see later in this section) so that engineers can get up to speed when revisiting an analysis model with minimum delay and without misunderstandings. 5 Downloaded by [ University College London] on [14/11/16]. Copyright © ICE Publishing, all rights reserved.


The results of the FE analysis may help to make important decisions in the design process, particularly when considering different design options in the analysis. In order to present a clear case on the advantages and disadvantages of each option, which of the many possible outputs should be presented? Being aware of the wider design and financial issues through discussions with other members of the project team will help in understanding the key outputs that need to be presented. For example, in an FE analysis of a basement construction adjacent to other properties, party wall negotiations may be a critical element to the whole project, so the presentation of FE analysis results could focus on the predicted foundation movements at the party walls with potential options to overcome any unacceptable movement, while not forgetting to present any other outputs considered to be important or flag up potential issues or cost savings to the rest of the project team. When ad When adop opti ting ng an ob obse serv rvat atio iona nall ap appr proa oach ch to de desig sign n to he help lp ma mana nage ge err error orss (s (see ee Section 7.3), the FE analysis outputs are compared with site monitoring data during cons co nstr truc ucti tion on.. In su such ch ca case ses, s, th thee exp xpec ecte ted d mo moni nito tori ring ng re resu sult ltss ba base sed d on th thee FE analysis outputs should be clearly documented together with ranges of values outside of wh whic ich h act ctio ion n sh shou ould ld be ta tak ken on si site te to mo modi dify fy th thee co cons nstru truct ctio ion n pr proc ocess ess.. Th Thee project team should be made aware of the importance of the monitoring data both to the FE analysis model and the project as a whole. Ensure that clear responsibilities have been assigned for regular viewing and interpretation of the data and that the data will be fed back into the FE analysis work for validation of the output, as described in Chapter 7. As wi with th al alll en engi gine neer erin ing g de desi sign gn,, it is ver ery y im impo port rtan antt to wr writ itee up ca calc lcul ula ati tion onss in a clear cle ar wa way y so tha thatt use users rs of the res result ultss can un under derst stand and the ass assump umptio tions ns ado adopte pted, d, to facilitate checking of work to help avoid errors, to satisfy any approval or licensing processes and ensure those who revisit the FE model at a later date can get up to speed quick qu ickly ly.. Ho Howe weve ver, r, wr writ ite-u e-ups ps of de desig signs ns by FE an anal aly ysis ar aree no nott st stra raig ight htfo forw rwar ard d beca be caus usee th thee ca calc lcul ula ati tion onss ar aree to too o co comp mpli lica cate ted d to pr prese esent nt an and d ar aree pe perf rfor orme med d by a computer. Consequently, the requirements for documenting design by FE analysis may differ dif fer fro from m an org organ anisa isatio tion’s n’s pr prac actic ticee for con conve venti ntiona onall des design ign.. NAF NAFEMS EMS pr prov ovide ide usefu us efull gu guid idan ance ce on qu qual ality ity as assu sura ranc ncee pr proc oced edur ures es fo forr en engi gine neer erin ing g an anal alys ysis is,, e. e.g. g. Chillery (2014). (2014) . A write-up should include at least the following information. As much of the information as possible should be obtained from direct reporting features in the software to minimise the chance of errors in transferring analysis data to the report: g

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background backgr ound informa information tion to the pro project ject and how the FE anal analysis ysis is related related to this summary of information gathered for the FE analysis any previou previouss FE analyses analyses superseded superseded by this one aims of the the FE anal analysis ysis software version version and any add-ons plus verification verification reports reports geometrica geom etricall assumption assumption (3D, 2D plan planee str strain, ain, 2D axisymmetr axisymmetric) ic) and any axes of symmetry

6 Downloaded by [ University College London] on [14/11/16]. Copyright © ICE Publishing, all rights reserved.


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plots showing showing geometry, scale, axis axis directions, strata, strata, structures, structures, boundary boundary conditions tables showing showing constitutive constitutive models and input input parameters for all materials derivation of material parameters and and their validation validation description descri ption of each constituti constitutive ve model and justifica justification tion for its selecti selection on plots of initial stresses, stresses, pore water water pres pressur suree and state state variables variables (elas (elastic, tic, yielding, yielding, etc.) tables and geometry plots showing showing each construction construction stage, material material models assigned to each element, assumptions (drained/undrained, displacements reset, etc.), loadings, time (for consolidation analyses), calculation method and convergence criteria as well as presentin presenting g the outputs outputs required required to meet the aims of the analysis, analysis, as a minimum plots of the deformed mesh, stresses, strains and state variables at key construction stages should be presented to show satisfactory completion of calculations validation validatio n of analysis model interpretations, interpretation s, discussions and recommenda recommendations tions based on the analysis analysis results results any recommended recommended site monitoring, expected expected values, trigger values values and remedial remedial measures.

1.2. 1.2.1

Geometry 2D or 3D?

Whether to build the FE analysis model in three dimensions (3D) or in two dimensions (2D) using a geometrical assumption (plane strain or axisymmetric) is an important decision because there can be an enormous difference in the workload between the two options. Setting up the geometry for a typical 2D analysis model of one section may take about a day, for instance, while to set up a 3D model of the same structure may take a whole week due to all the additional geometrical information that must be specified. So, perform 2D analysis when possible to save time and resources, but only when the assumptions required to perform 2D analysis will not have a detrimental effect on the accuracy of the model. The following paragraphs describe some of the effects of the 2D assumptions to assist readers in making the right decision about whether to build a 2D or 3D FE model.

2D plane strain assumption A 2D plane strain model model involves involves the analysis of a plane, vertical section through through the site. The strain and displacement in the ‘third dimension’ (i.e. perpendicular to the plane) is assumed to be zero, hence strains can only occur in directions within the plane and they aree ind ar indepe epend ndent ent of the out out-of -of-pl -plan anee dir directi ection on.. Co Conse nsequ quent ently ly,, she shear ar str stress ess and she shear ar strain can be non-zero only in the plane of the analysis, although normal stress perpendicular to the plane is calculated and can be non-zero. This assumption is suited to sites with a uniform cross-section (including ground conditions) and stress state/loading for a sufficiently long straight dimension for virtually zero strain to be expected in the long dimension dime nsion (e.g. str straigh aightt tunn tunnels, els, emba embankme nkments, nts, long exc excava avations tions,, strip foun founda dations tions), ), as shown in Figure 1.1. 1.1. It is not suited to sites with foundation piling, ground anchors or similar structural geometries. 7 Downloaded by [ University College London] on [14/11/16]. Copyright © ICE Publishing, all rights reserved.


Figure Figur e 1.1 Suitab Suitable le geomet geometries ries for the plane str strain ain assumption

Plane strain section

Raft foundation

Tunnel

Embankment Plane strain section

Plane strain section

Basement excavation

Raft foundation



It is im impo porta rtant nt to vi visu sual alis isee wh wha at a 2D pl plan anee st stra rain in mo mode dell is actu ctual ally ly sim simul ula ati ting ng in 3D in or orde derr to understand the limitations limitations of the model. To do this, simply extrude a plane strain strain model in the direction direction perpendicular perpendicular to the plane. This is illus illustra trated ted in in Figure Figure 1.2 for 1.2 for a basement example that is probably unsuitable for a plane strain assumption. A section was taken throug thr ough h the true true 3D geometry geometry as sho shown, wn, which which formed formed the geome geometry try of the 2D pla plane ne strain strain model. Extruding the section in the out-of-plane direction as shown demonstrates clearly the geometrical assumptions of of the plane strain model and that that they are very different from the true 3D geometry. The excavation is modelled as a long trench instead of the true boxshape sha pe and the str strat ata a ar aree ass assume umed d ho horizo rizonta ntall in the out out-of-of-pla plane ne dir directi ection. on. Str Structu uctura rall elements are also heavily influenced by the plane strain assumption, with the struts in the original geometry now being modelled as continuous slabs. For this reason, linear structures, such as struts, ground anchors, piles, nails, etc. are not suited to the plane strain assumption, as described further in Section 5.1.5. Furthermore, the apparent point load in the plane strain model actually acts as an infinite line load in the out-of-plane direction.

2D axisymmetric assumption A 2D axisymmetric model also involves the analysis of a plane, vertical section through the site except that one vertical side of the plane (the left hand side usually) is the axis about which the site has rotational symmetry. The horizontal axis is the radius from the axis of symmetry, and the strain perpendicular to the plane and in the circumferential or hoop direction is assumed to be zero; hence displacement, strain and shear stress can only occur in the analysis plane. All stresses and strains perpendicular to the plane are zero except for the normal stress in the hoop direction. This assumption is suited to sites with a vertical structure in the ground with a uniform, radial cross-section (e.g. vertical shaft, circular cofferdam, single vertical pile, circular spread foundation) and vertical loading which is uniform around the central axis, as shown in Figure 1.3. 1.3. If there are any other features nearby that are not symmetrical about the axis, these cannot be simulated. Note that torsional loadings (e.g. to simulate pile boring) also cannot be simulated in an axisymmetric analysis. To visualise the geometric assumption of a 2D axisymmetric model, extrude the model through 360 about the axis of symmetry. The strata and the ground surface can be horizontal or slope only toward or away from the axis of symmetry. Any structure becomes circular in plan, centred about the axis of symmetry. Point loads applied in axisymmetric models are treated as circular line loads centred about the axis of symmetry, while line loads are treated as distributed loads over areas of circles centred about the axis of symmetry. Care should be taken when specifying the input parameters for and interpreting the outputs from structures in axisymmetric models (see Section 5.1.5). 8

1.2. 1. 2.2 2

How Ho w det detai aile led d doe doess the the ge geom ometr etry y nee need d to be be? ?

To save time in setting up and running an analysis, the geometry of the FE model needs to be as simple as possible but without compromising too much on accuracy. As with many of the decisions to be taken when setting up a geotechnical FE model, it comes down to a compromise between detail and efficiency. Enough detail is required in order to obtain reasonably accurate key outputs, but not excessive detail such that the task becomes unnecessarily time-consuming and expensive. 9 Downloaded by [ University College London] on [14/11/16]. Copyright © ICE Publishing, all rights reserved.


Figure 1.2 Visualising the plane strain assumption: (a) true 3D geometry; (b) 2D plane strain Figure model; (c) 3D geometry assumed by plane strain model

(a)

(b)

Extrusion

(c)



Figure 1.3 Suitab Suitable le geomet geometries ries for the axisymmetric axisymmetric assumption

Circular spread foundation Axisymmetric section

Axisymmetric section

Axisymmetric section

These decisions are taken using judgement and experience, as well as with the help of test runs when there is any doubt. For example, regarding geometrical detail, an FE analysis can be run with and without a particular geometrical detail and then the key outputs compared to see whether that detail had a significant effect and needed to be included. 11 Downloaded by [ University College London] on [14/11/16]. Copyright © ICE Publishing, all rights reserved.


For exa xamp mple le,, im imag agin inee a str tra atu tum m bo boun und dar ary y at a de dep pth of 9. 9.8 8 m an and d th thee ba base se of an exc xca avati tion on at 10.0 m depth. To avoid the detail in the mesh required to include the 20 cm difference in eleva ele vatio tion n be betw twee een n th thee st strratu tum m bo boun unda dary ry an and d ex exca cava vati tion on flo floor or,, it ma may y be acce ccept ptab able le to mo move ve the stratum boundary to the same depth of 10.0 m in the model, particularly given the uncertainty in ground conditions, without a significant effect on the outputs. Further away from the area of interest, less detail is required. In a typical city centre site, there will be many features in the surrounding ground such as basements, piled foundations, metro tunnels, etc., as illustrated in Figure 1.4, 1.4, while even in greenfield sites there may be topographical or geological features nearby that could potentially influence ground behaviour in the area of interest. Again, it comes down to judgement which features around the area of interest need to be included, and if there is any doubt, try running the analysis with and without certain features to see if they influence the key outputs significantly and need to be included. Taking advantage of axes of symmetry can also simplify the geometry significantly by allowing half, or even more, of the geometry to be omitted. Axisymmetry allows 3D geometry to be simplified to a 2D plane, as described above, while planes of symmetry may permit only half or a quarter of the geometry to be modelled in, for example, a rectangular piled raft, as shown by example in Section 8.2. Similarly, Similarly, a 2D plane strain model may may be simplified further by omitting half the geometry on one side of a vertical axis of symmetry. Bear in mind, however, that not only should the geometry be symmetrical about the axis or plane of symmetry, the construction methods, timing and ground conditions must be symmetrical too. If construction on one side of a geometrical plane of symmetry follows a different sequence or timing to the other, then that should not be considered a plane of symmetry in the FE model and the different sequences should be fully simulated in a model of the whole geometry. This is due to non-linearities in ground modelling and soil–structure interaction that do not follow the principle of superposition.

1.2.3 1.2 .3

Where Whe re sho should uld mod model el bou bounda ndaries ries be loc locate ated? d?

The FE mesh needs to be fixed in space in order to establish equilibrium and solve the global glo bal st stiffn iffness ess equ equat ation ion to det determ ermine ine dis displa placem cement ent.. Th Thee fixi fixitie tiess ar aree app applie lied d at the boundaries to the model, but in field problems there is often no obvious boundary for the FE model because the ground extends indefinitely. Therefore, some judgement is required when deciding where to place the model boundaries. The boundaries should not be placed too close to the area of interest because that would be unrealistic and introduce a significant boundary effect, effect , i.e. the fixities imposed at the boundary would start to influence the key outputs. The only common situation in the field where a model boundary would correctly impose a significant boundary effect on the area of interest is where a relatively soft soil overlies a strong or hard layer (e.g. rock or very dense soil) at shallow depth. The top of the hard layer could form the bottom boundary to the FE model, as shown in Figure 1.5, 1.5, provided that the layer is of substantial extent and deformations in the real layer due to the imposed loads would be insignificant compared with the deformations in the upper layers. 12 Downloaded by [ University College London] on [14/11/16]. Copyright © ICE Publishing, all rights reserved.


t s e r e t n i f o a e r A

t s e r e t n i f o a e r a e h t d n u o r a y r t e m o e g e h t g n i y f i l p m i S

t s e r e t n i f o a e r A

4 . 1 e r u g i F



Figure Figur e 1.5 Example of a clearly defined boundary

Softer soil

Hard soil or rock undergoing insignificant strain

Hard soil or rock represented by fixed bottom boundary

In other cases, the boundaries to the FE mesh need to be placed sufficiently far away from the area of interest for the fixities to be imposed without causing significant boundary effects on the area of interest. But how far away should this be? This clearly depends on the particular characteristics of each model, so there are no concrete rules on model boundary locations. It is best to experiment with different locations, unless the most appropriate locations are already known from previous experience of similar problems. By placing the model boundaries progressively nearer or further from the area of interest in preliminary analysis runs and plotting plotting the key outputs, outputs, it should be possible to identify a threshold boundary position at which boundary effects are no longer significant, as illustrated in Figure in Figure 1.6. 1.6. Model bou Model bounda ndarie riess wit within hin the thr thresh eshold old wil willl re resul sultt in pr progr ogress essiv ively ely lar larger ger bo bound undary ary effects on the area of interest as they are located nearer to the area of interest, but anywhere beyond the threshold should result in insignificant boundary effects on the key outputs. Therefore, the final model boundary locations should be set anywhere outside the threshold position. Another method to test whether the model boundaries are located sufficiently distant from the area of interest is to change the fixities (e.g. add and remove vertical fixity at the vertical boundaries) to see if this affects the key outputs. If no significant effect is observed, then the boundaries are located sufficiently far away.

Figure Figur e 1.6 Threshold boundary location Threshold boundary location Area of interest

Increasingly significant boundary effects

No significant boundary effects Increasingly significant boundary effects

No significant boundary effects

Generic output in area of interest



Also, when outputs are checked, the stress state should not be on the failure envelope to a significant extent at any model boundary, except perhaps on axes of symmetry. As a general rule, stress changes should be less than 5% at model boundaries, and ideally less than 1%. The bottom boundary can usually be placed closer to the area of interest because the grou gr ound’ nd’ss st stiffn iffness ess an and d str streng ength th inc increa rease se wit with h dep depth, th, and ev even en clo closer ser wh when en a str strain ain-dependent stiffness is adopted in the constitutive model. The vertical boundaries usually need to be located further from the area of interest, particularly for ground models with linear stiffness because these tend to exaggerate the deformation further away from the area of interest. Figure 1.7 shows 1.7 shows some rules of thumb that can be used as a first-guess for model boundary locations when starting to investigate the most appropriate locations. The distances shown often need to be increased for sloping ground, undrained behaviour and for groundwa ground water ter flow analyses. The appropria appropriate te distances for ground groundwater water flow analyses can be estimated from Sichardt’s empirical formula (Equation 1.1) providing the approximate radius of influence R in metres of a well, as described in, for example, Cashman and Preene (2012). R

√ 

= Cs

k

(1.1)

where s drawdown in borehole (m), k permeability (m/s) and C where s and C metric conditions and 1500 to 2000 for plane strain conditions.

=

=

= 3000 for axisym-

Analysis models of deep tunnels need not include the ground surface if it is sufficiently remote from the area of interest not to influence the key outputs. A pressure should be applied to the top surface of the model to represent the total stress from the overlying ground. There are similar situations where a small detail of a larger analysis model may need to be stud studied, ied, in which case the mode modell boun boundarie dariess can be loca located ted closer to the area of interest than usual, with the total stresses obtained from the larger model applied at the boundaries of the smaller model.

1.2.4 1.2 .4

Whatt fixit Wha fixities ies are app applie lied d at at the the mod model el bou bounda ndaries ries? ?

Note that the term ‘boundary conditions’ refers to all conditions imposed on a model in order to define a particular problem (e.g. loads, pore pressures, prescribed displacements, accelerations, etc.) and not just the conditions at the outer boundaries to the FE mesh. As mentioned in Section 1.2.3, the model needs to be fixed in space. As shown in Figure 1.7, the standard fixities applied at the model boundaries are zero displacement in all directions at the bottom boundary and zero displacement on the vertical sides in the horizontal direction perpendicular to those boundaries, including on axes of symmetry met ry.. Th Thee top sur surfa face ce has no fix fixiti ities es imp impose osed. d. Str Struct uctur ural al ele elemen ments ts wit with h ro rota tatio tiona nall degrees of freedom, e.g. beams and shells (see Section 5.1.1) that extend to vertical boundaries must also be fixed rotationally to simulate the restraint from the structure beyond the boundary. This is particularly important at axes of symmetry. 15 Downloaded by [ University College London] on [14/11/16]. Copyright © ICE Publishing, all rights reserved.


Figure Figur e 1.7 Appropriate first-guess FE mesh boundary locations B

~3B

B

Largest of 3B or 2D

B

Largest of 3B or 4H

~3B

D

~2B

H

~2B

D

~5D ~3D

The standard fixities should be used at axes of symmetry, but at remote model boundaries located sufficiently far from the area of interest, the nature of the fixities is less important. The vertical boundaries could be fixed in the vertical direction, for instance, and the bottom boundary allowed to move freely in the horizontal direction. Indeed, varying these fixities provides a means of checking the sensitivity of the key outputs to boundary effects. 16 Downloaded by [ University College London] on [14/11/16]. Copyright © ICE Publishing, all rights reserved.


1.3. 1.3. 1. 3.1 1

Meshing Whic Wh ich h el elem emen entt typ types es sh shou ould ld be us used ed? ?

Having defined the geometry of the model, this is then replaced by an equivalent FE mesh, with continuum elements used for the ground. The mesh is formed of elements according to the degree of precision required in the model (a greater number of smaller elements gives more precision). The elements are connected together at their nodes. The nodes are the discrete points where the primary unknowns (displacement or excess pore pressure) are calculated. Nodal displacements are then interpolated by shape functions or interpolation functions for all locations in each element to obtain the secondary or derived quantities of strains or strain rates and, via constitutive relationships, stresses or stress rates. The stresses and strains are calculated at Gauss, stress or integration points located across the element. The hierarchy of element types is shown in Table 1.1. 1.1. The higher order elements have more nodes and Gauss points so they produce more accurate calculations of stress, particularly for stiff behaviour. Linear and cubic strain element types are commonly used in geotechnical FE analysis. The linear strain elements have fast computation times and are adequate for typical deformation analyses provided a sufficient number are used, but they may not be suitable for 2D axisymmetric models and they may over-predict failure loads in all models (although this tendency is reduced when adopting reduced integration). To predict failure states and for any axisymmetric models, the cubic strain elemen ele ments ts (e. (e.g. g. 1515-nod noded ed tri triang angle) le) ar aree pr prefe eferre rred, d, in spi spite te of the their ir slo slowe werr com compu puta tatio tion n times. In groundwater flow analyses, lower order elements are adequate, or even preferable in some programs. The advantages of triangular (2D) and tetrahedral (3D) elements over quadrilateral (2D) and hexahedral/brick (3D) elements are that they fit into awkward shapes more easily so are more suited to automatic mesh generators and they are less susceptible to distortion errors (see Section 1.3.2). Rock discontinuities, if modelled explicitly, require interface elements with appropriate material laws to allow slippage and separation along the discontinuity surface. Interface Table 1.1 Hierarchy of element types Shap ape e function

1st order 2nd order 3rd order 4th order TRI triangle, QUAD per element

=

Var ariiation across element Displacement

Strain

Linear Quadratic Cubic Quartic

Constant Linear Quadratic Cubic

Example elements for continua

TRI3, QUAD4, TET4, HEX8 TRI6, QUAD9, TET10, HEX20 TRI10, QUAD16 TRI15

= quadrilateral, TET = tetrahedron, HEX = hexahedron. Number refers to number of nodes 17



elements are also used between structures and the ground for the same reasons. Interface elements and structural elements (beams, shells, etc.) are covered in Section 5.1.

1.3. 1. 3.2 2

What Wh at ma mak kes a go good od FE me mesh sh? ?

The size and arrangement of elements in a mesh can have a critical effect on the accuracy of an FE analysis. A poorly formed mesh is a common source of error, so a lot of attention needs to be paid to mesh quality and automatic mesh generators cannot be relied on to produce good meshes on their own. Essentially, large stress concentrations and zones of rapid stress (including pore pressure) or strain change need smaller elements. These locations typically occur, for example, at large stiffness changes, discontinuities, foundation corners and pile bases. A very fine mesh with small elements everywhere would be the most accurate but this would need long computation times. A good FE mesh is graded with small elements where they are needed and larger elements remote from the area of interest and where stresses and strains are more uniform. Thus faster computation times can be achieved without a significant loss of accuracy. Examples of graded meshes are shown in Chapter 8. To check whether the mesh is adversely affecting outputs, try running the analysis with a finer mesh and compare the key outputs. If the outputs are essentially the same, then the mesh is not affecting the outputs. If the outputs are different, then theoretically the finer mesh is closer to the true solution. Experiment with different meshes to determine the coarsest and hence most computationally efficient mesh that does not influence significantly the key outputs. Meshes formed of higher order elements can be coarser because of the higher number of nodes per element. Note that the prediction of collapse loads is heavily influenced by mesh geometry and elemen ele mentt typ type, e, par partic ticula ularly rly whe when n coa coarse rse mes meshes hes an and d lo lowe werr ord order er ele elemen ments ts ar aree use used. d. Higher order elements should be used and meshes made progressively finer until collapse loads appear uninfluenced by mesh geometry. Some programs have adaptive mesh refinement where, based on the outputs from an initial mesh, more elements are added automatically where the greatest changes in stress and strain occur. Subsequent analyses and refinement are continued until no further refinement is necessary (refer to Sloan, to Sloan, 2013, 2013 , for example). The distributions of displacement and stress calculated by the interpolation functions are only reliable if the element shapes are not excessively distorted. Where the calculated variables varia bles chan change ge rap rapidly idly,, e.g. at str stress ess conce concentra ntrations tions,, the dist distribut ribution ion is eve even n mor moree sensitive to element shape. Automatic mesh generators cannot control distortion, so this needs to be checked manually. Distortion is less of a problem for triangular and tetrahedral elements provided that the sides of each element are about the same length.

1.4 .4.. 1.4. 1. 4.1 1

Analy Anal ysis sta tage ges s How Ho w ar are e the the in init itia iall str stres esses ses se sett up? up?

Soil and, to a certain extent, rock are frictional materials so their strength and stiffness are heavily dependent on internal stresses. In terms of FE modelling, the stress–strain 18 Downloaded by [ University College London] on [14/11/16]. Copyright © ICE Publishing, all rights reserved.


behaviour predicted by all non-linear constitutive models depends on the current stress state. Unless the entire geological history is simulated, which is rare, analyses of field problems require the direct establishment of the initial stresses in the ground. These are usually by far the largest stresses in the model so they are important and this is one aspect that sets geotechnical FE analysis apart from other sectors of engineering analysis. The first stage of any geotechnical FE analysis involves setting up the initial stresses. For field problems, these are the stresses in ‘greenfield’ conditions, i.e. before any significant man-made stress changes occurred, which should be relatively homogeneous across the model. Then significant stress changes caused by historical constructions or groundwater changes are simulated in subsequent stages in order to establish the present day stress state and recent stress history along the correct stress path. When sim When simula ulatin ting g lab labor orat atory ory tes tests, ts, on ma many ny occ occasi asion onss the sel self-w f-weig eight ht st stres resses ses of the specimen are insignificant compared with the applied stress throughout the specimen. In such cases, the specimen can be assumed to have zero density and the initial stresses set to zero. The stresses applied to the specimen in the real test would then be applied in the simulated test. For the remainder of this section, field-type situations where the initial stresses need to be established will be considered. Except for cases where undrained conditions are simulated in terms of total stress, the pore water pressure is clearly an important variable in the setting up of initial effective stresses. In hydrostatic cases and in relatively simple steady-state flow conditions, the pore pressures can be specified directly in the input data to the FE analysis. For more comple com plex x gr groun oundw dwat ater er flo flow w con condit dition ions, s, a sep separ arat atee gr grou oundw ndwat ater er flo flow w ana analy lysis sis (see Section 4.3) may be required whose output of pore pressure can form some of the input for the ini initia tiall str stress ess in the st stres ress–s s–str train ain FE an analy alysis sis.. Th Thee gr grou oundw ndwat ater er lev level el sho should uld coincide with element boundaries in cases where material properties (e.g. saturated and unsa un satur turat ated ed we weigh ightt den densit sity) y) dep depend end on the ma materi terial’ al’ss po positi sition on abo above ve or bel below ow the groundwa ground water ter level. The ve The vertic rtical al effe effecti ctive ve st stres resss is re rela lativ tively ely st stra raigh ightfo tforw rward ard to cal calcul culat atee onc oncee the por poree pressure profile and ground densities are known. But the horizontal effective stress, as calculated from the vertical stress using the stress ratio K ratio K 0 , is heavily dependent on stress histor his tory, y, str stress ess pa path, th, top topogr ograp aphy hy and oth other er geo geolog logica icall pr proce ocesse ssess ex exper perien ienced ced by the ground. Do not underestimate the importance of this stress. There are two horizontal directions and only one vertical, so it has the strongest influence on the overall stress state. It also has a major influence on the outputs of some FE analyses, e.g. for retaining walls, cut slopes and piled foundations. Unfortunately, in situ horizontal stress is difficult to measure accurately (see Section 3.3) and careful judgement is needed before using measured values. To estimate K estimate K 0 , or otherwise to help validate measured values, a number of approximate equations are available which are given in Appendix 3.1. Note that these equations are 19 Downloaded by [ University College London] on [14/11/16]. Copyright © ICE Publishing, all rights reserved.


intended for homogeneous ground with horizontal ground surface and layers and in greenfield conditions where it is reasonable to assume the same in situ stress in both horizontal directions. With an inclined ground surface or strata and in the vicinity of man-made structures, such an assumption should not be made. The equations are also approximate and empirical, so the influence of K of K 0 values on the key outputs would need to be considered carefully in a parametric study. Very high (approaching the passive limit) and very low (approaching the active limit) K 0 should be avoided otherwise the initial stress in the FE model may be in a state of failure. There are two methods of establishing initial stress in an FE analysis:

Direct specification (K 0 method) This is intended for homogeneous stress profiles with horizontal ground surface, strata and groundwater levels, otherwise equilibrium may not be obtained since the FE analysis achieves vertical equilibrium while the horizontal stress is based only on the specified K specified K 0 or horizontal stress values. Small equilibrium errors may be acceptable, perhaps due to a small inclination in the layers or ground surface, in which case a plastic nil-step should be performed following the establishment of initial stress (a plastic nil-step is an additional analysis stage, with no change in load, intended to restore equilibrium and allow stresses to return within failure limits). Note that initial stresses for soil layers simulated as undrained in terms of total stress should be specified in terms of total stress also and the K 0 value would be the stress ratio for total stresses. Initial stresses for cases with a sloping ground surface but horizontal strata can still be established with direct specification. This is performed by having a horizontal ground surfa sur face ce in the ini initia tiall st stage age an and d the then n cr crea eatin ting g the slo slope pe by ac activ tivat ating ing or dea deacti ctiva vatin ting g elements to create the slope in a subsequent analysis stage.

Gravity switch-on In cases of non-homogeneous stress profiles, such as with sloping strata, the initial stress is generated by activating the self-weight of the ground and by specifying the initial pore pressures in the model. This is the same method used for activating new volumes of ground during subsequent analysis stages (even if direct specification was adopted in the initia ini tiall sta stage) ge).. A bas basic ic con const stitu itutiv tivee mo model del,, suc such h as the lin linear ear ela elast stic ic per perfec fectly tly pla plasti sticc (LEPP) Mohr–Coulomb model, can be used in the initial stage with the appropriate parameters to establish the required stress state before changing to an advanced model with wit h ap appr propr opria iate te ma materi terial al par param ameter eterss for sub subseq sequen uentt sta stages ges if nec necessa essary ry.. Ad Advan vanced ced models may establish horizontal stresses in a complex way, whereas with LEPP models, K 0 can can be ma mani nipu pula late ted d mo more re st stra raig ight htfo forw rwar ardl dly y fr from om th thee eq equa uati tion on (for ela elast stic ic conditions): K 0

n

= 1 −

n

(1.2)

However, K 0 values in excess of 1.0 require a Poisson’s ratio above 0.5 which is not possib po ssible le num numeri erical cally ly.. In suc such h cas cases, es, mor moree of the loa loadin ding g his histor tory y wou would ld nee need d to be 20 Downloaded by [ University College London] on [14/11/16]. Copyright © ICE Publishing, all rights reserved.


simulated, experimenting with different Poisson’s ratios for loading and unloading to achieve the required stress state. Alternatively, increased self-weight can be imposed in the first stage before reducing the ground weight in a subsequent stage in order to create an over-consolidated state. All gravity switch-on stages must be performed with drained conditions, even when undrained conditions will be simulated in subsequent stages. It is important to check the outputs from the initial stress analysis stage to ascertain whether the initial stresses have been established correctly. Also, the model must be in equilibrium with an error of less than 1% and no significant plastic yielding. Where there is a small equilibrium error or a few Gauss points yielding, performing a plastic nil-step should restore equilibrium and return all stresses within failure limits. As we well ll as est establ ablish ishing ing the ini initia tiall st stres resss st stat ate, e, ad advan vanced ced con const stitu itutiv tivee mo model delss re requi quire re certain state parameters that define, for instance, the initial location and size of the yield surface. Examples include the initial void ratio and pre-consolidation stress. The former should be relatively straightforward to measure while the latter often requires a degree of interpretation of test data (see Section 3.4.1). Application of the gravity switch-on method to establish the initial stress state is shown in the example in Section 8.4.

1.4.2 1.4 .2

How Ho w ar are e the con const struc ructio tion n st stage agess set up up? ?

Any geotechnical FE analysis of new or existing structures must consider explicitly how the structures were constructed because this affects stress paths and ground behaviour. Thee tim Th timee per period iodss for constru constructi ction on ar aree als also o imp import ortant ant for tem tempor poral al eff effects ects suc such h as groundwater flow and excess pore pressure dissipation in low-permeability soils or creep effects. Furthermore, if outputs only for permanent works were required, the temporary works stages taken to get there cannot be ignored because of non-linear effects. The principle of superposition cannot be applied in geotechnical FE analyses. Construction activities can be complex, with many processes occurring simultaneously and in different phases across the site. Rather like the creation of the analysis geometry, it is not possible to simulate every detail of the construction activities. For example, the placement of each 0.3 m-thick layer of fill in the embankment construction example in Section 8.4 was not simulated. It was found that 2 m-thick layers could be installed without a significant loss of accuracy. Judgement is needed to identify the essential elements of the construction activities that need to be included in the analysis model and which are likely to have a significant effect on the key outputs. The most reliable way to test whether a feature of the construction activities needs to be included in the model is to run the analysis with and without the feature included, and to check whether the key outputs change significantly. If the key outputs are not affected, then the unnecessary detail could be omitted so that the analysis could run more efficiently. If the key outputs are affected, then that feature would need to be included in the analysis model. One of the most common activities in construction stages is the deactivation (to simulate excavation of ground or removal of structural components) and activation (to simulate 21 Downloaded by [ University College London] on [14/11/16]. Copyright © ICE Publishing, all rights reserved.


placement of fill or installation of structural components) of elements. All elements must be present in the FE mesh but may be deactivated at the start where necessary. On deactivating elements in a construction stage, immediately their material properties are ignored, stresses and nodal displacements are set to zero, any model boundaries formed by the deactivated elements become free and permeable and any external loads applied to those elements are ignored. However, removal of the weight of the deactivated elements can cause a large out-of-balance force so the equivalent weight of the deactivated elements is applied automatically by most software at the newly exposed ground surface and reduced in a stepwise fashion in non-linear analyses. Some simpler software, particularly non-geotechnical-focused programs, may just assume a very low stiffness for inactive elements, but this could lead to inaccurate predictions. On activating elements in a construction stage, the material properties of these elements are taken into account from the start of the construction stage, while the weight of the activated elements is introduced in a stepwise fashion due to the large out-of-balance force and the activated element stresses begin to grow. The new nodes also immediately become active but their initial displacement is set to match the already deformed mesh to which they are being added. The disadvantage of this is that a false impression of the deflection of newly placed layers can be formed. The deflection of nodes occurring prior to activation of the element needs to be subtracted from the output in order to obtain the deflection of the elements since activation, as demonstrated in the embankment construction example in Section 8.4.4. The ac The activ tivat ation ion of gr groun ound d ele elemen ments ts is the sam samee as the gr grav avity ity swi switch tch-on -on pr proce ocedu dure re described in Section 1.4.1 except that a specific area or volume is activated during the analysis rather than the whole ground mass at the first analysis stage. So, in the same way, it is often necessary to use a different constitutive model or parameters for the ground during element activation to simulate behaviour during construction and obtain an appropriate stress state. For example, Poisson’s ratio may be manipulated with an LEPP model to obtain an elevated K 0 value, perhaps resulting from compaction, or different drainage conditions may be used for low-permeability fills during construction. In a subsequent construction stage, the constitutive model may be changed to simulate the post-placement behaviour more accurately.

Ground impro improvement vement The actual processes of ground improvement, such as compaction and treatment, are too complex to simulate in routine FE analyses. Therefore, the strength, stiffness and density properties are usually changed in a construction stage to reflect the improvement of the ground, with the new input parameters obtained from site trials or previous experience on similar sites.

Installation effects The term ‘installation effects’ refers to the effect on the ground around new structures such as piles and diaphragm walls during their installation. If installation effects are 22 Downloaded by [ University College London] on [14/11/16]. Copyright © ICE Publishing, all rights reserved.


likely to influence key outputs from an FE analysis, then they must be considered in some way. Unfortunately, simulating installation effects can be complex and, at best, probably only provides qualitative assessments of installation effects rather than accurate predictions. In some cases the installation effects may be less significant, for example during CFA (conti (co ntinu nuous ous flig flight ht aug auger) er) or cas cased ed bo bored red pil piling ing,, whe where re the sup suppor portt pr prov ovide ided d sho should uld prevent significant stress and strain changes in the soil. In other cases, installation effects may be more significant. Most commonly, structures are introduced in the ground in FE analyses as ‘wished in place’. This means that line or surface elements are simply activated in the FE mesh, while area (in 2D meshes) or volume (in 3D meshes) elements have their material models changed from a soil or rock material to the new structural material. At the same time, it is possible to alter the properties of the surrounding ground to attempt att empt to tak takee acco account unt of inst installa allation tion effects effects.. Alter Alternat nativel ively, y, addi additiona tionall cons construct truction ion stages can be added to try to simulate the installation process. Some examples of these techniques are described in the following: g

g

Fluid support to pile bores and and diaphragm diaphragm wall excavations: excavations: fluid provides provides less support than casing and results in more ground deformation and stress relief. Consequently, the wished in place option is likely to be conservative in terms of earth pressures applied to the structure but not so in terms of ground deformation. For an approximate assessment of installation effects, fluid support can be simulated with its hydrostatic pressure applied to soil surfaces in contact with the fluid. Fluid concrete can be simulated in a similar way, but note that pressures increase hydrostatically initially but at a critical depth a maximum is reached below which the concrete fluid pressure stays approximately constant. This is due to increases in effective stress as its behaviour changes from a fluid (aggregates in suspension) to a granular medium (aggregates in contact) with pore pressure. The critical depth depends on many characteristics of the concrete, as described by Clear by Clear and Harrison (1985). (1985) . Pouring concrete under water or a support fluid further complicates the pressure distribution because it is dependent on both the effective concrete weight and the initial fluid pressure (refer to Lings to Lings et ., 1994). 1994). Three dimensional analysis et al ., is required for diaphragm wall installation simulations due to the complex stress redistributions during construction. The approximate nature of these assessments does not usually warrant their inclusion in the main FE model of the entire construction sequence. It is usually more appropriate to conduct a separate, detailed study of installation effects in order to assess the approximate error in the outputs resulting from adopting the wished in place option in the main analysis model. Driven/displacement Driven/displa cement piles: the installation installation effects of displacement displacement piles depend on the in situ ground density, geological history, installation method and any installation aids, and include settlement or heave and changes to ground density. The installation process is too complex to be simulated in routine FE analyses, so, on pile or wall activation, the adjacent soil parameters need to be modified appropriately (e.g. Engin (e.g. Engin et ., 2015). 2015). et al ., 23



g

g

Grouting: compaction compaction and compensation compensation grouting grouting cause displacement displacement of soil, but in a less dynamic way than for driven piles, so the grouting can be simulated by widening elements with careful validation using field tests and monitoring (see Section 5.1.4). Casting thick concrete concrete slabs: some raft raft foundations foundations and basement slabs have have a large thickness and therefore a significant self-weight. On casting, the fluid concrete self-weight interacts with the ground in a different way to hardened concrete since it applies its self-weight without any stiffness so is more inclined to sag. Therefore, in some situations it may be necessary to consider the installation effects of a thick concrete slab by applying its self-weight only without structure before substituting the load for the structural elements of the slab (with self-weight included) in a subsequent stage to simulate the hardened concrete slab’s weight and stiffness.

Further examples of the simulation of installation effects can be found in Hicks Hicks et et al . (2013).

1.4.3 1.4 .3

Which Whi ch cal calcu culat lation ion opt option ionss sho should uld be cho chosen sen? ?

Linear elastic FE analysis is more straightforward and computationally simple than nonlinear analysis. Unfortunately, such analyses are inadequate for geotechnical problems, except perhaps for intact rock. Geotechnical modelling generally requires the introduction of plasticity, non-linear elasticity, frictional contact, large displacements or creep, or a combination of them. Each of these introduces non-linearities to an FE analysis which require more complex solution methods. In particular, applied loads or displacements must be divided into increments or steps and equations solved iteratively, ensuring that equilibrium is satisfied before moving on to the next iteration or load step to prevent the solution drifting from the correct equilibrium value. This lengthens computation times and increases the probability of failing to obtain a final convergent solution or, worse still, obtaining an inaccurate solution. Therefore, it is important to exercise caution and engineering judgement when interpreting outputs from non-linear FE analyses. This section covers a few issues that are common to most programs, but always consult the software manuals carefully to learn about the calculation options available and seek guidance on appropriate selections.

Step size Non-linear equations need to be solved in calculation steps, but what step size should be used? Too small and many steps will be required leading to a slow solution. Too big and a high number of iterations will be required, also leading to a slow solution or no solution at all. Therefore, the right balance needs to be found for step size and many prog pr ogra rams ms de dete term rmin inee th thee st step ep si size ze au auto toma matic tical ally ly.. If th thee st step ep si size ze is set ma manu nual ally ly,, experimentation will be required to find the most efficient step size.

Solution scheme A widely used method of solving non-linear equations is the Newton–Raphson iterative method. It establishes the load–displacement curve for every degree of freedom using an initial guess or trial solution for each load increment based on the slope of the curve. It 24 Downloaded by [ University College London] on [14/11/16]. Copyright © ICE Publishing, all rights reserved.


calculates the out-of-balance or residual force vector which is the difference between the external load increment and the corresponding resisting internal force computed using the strain–displacement and stress–strain relationships. If the residual force exceeds a particular tolerance, then the displacement is corrected back to the equilibrium solution and the process repeated successively until the residual force is within the tolerance. The trial solution must be reasonably close to the true solution in order to achieve convergence, and the slope of the load–displacement curve should not change sign. At or near maxima or minima where the slope changes sign, the arc length method (or Riks method) can be used to obtain more reliable solutions. There are various stress point algorithms used to integrate the constitutive equations to obtain stress change (and hence internal force) and each must make additional assumptions since the constitutive behaviour is changing in each increment. Users should verify that their software uses an appropriate stres st resss poi point nt alg algori orith thm m in the their ir FE ana analy lysis. sis. Th Thee Ne Newto wton– n–Rap Raphso hson n met metho hod d use usess the current slope of the load–displacement curve, which is the (tangent) stiffness matrix, at every iteration. In large FE meshes, calculation of the stiffness matrix is computationally demanding and can slow down the calculation. An alternative is the Modified Newton– Raphson method where the same slope is used in successive iterations – although convergence will be slower the overall calculation may be faster, provided the behaviour is not overly non-linear, since the stiffness matrix is not re-calculated for every iteration.

Equilibrium error At the end of each load increment, an equilibrium check is performed by converting the externally applied loads and internal stresses into equivalent nodal loads and calculating the difference or out-of-balance load between the external and internal values at each node. The maximum difference expressed as a ratio or percentage of the out-of-balance load to external load is termed the maximum equilibrium error. It should be less than 1% preferably, particularly at the initial stage, but definitely less than 5%. While achieving a low value is a requirement, it eliminates only one of the many potential sources of error in a non-linear FE analysis so should not be viewed as a guarantee of accuracy. Increasing the allowable value in the calculation options above this level to achieve convergence merely obtains a false equilibrium and certainly an inaccurate result. Do not be tempted to do this.

Large deformations In conventional small deformation analysis, the external loads and internal stresses are assumed in equilibrium in the original mesh geometry (which is called Total Lagrangian formulation). So, while nodal deflections are calculated, the actual coordinates of the nodes in the calculations do not change. This is a good approximation for small deformations in most cases, but in some cases the changing geometry of the mesh needs to be taken into account in the calculation. Such cases include the analysis of soil reinforcement (e.g. geotextile) where membrane action can help support loads perpendicular to the reinforcement plane at large strains. If only the original mesh geometry is considered, then the tension mobilised in the reinforcement would only act in the original horizontal orientation of the reinforcement where wh ereas, as, in re reali ality ty,, the rei reinfo nforce rcemen mentt ben bends ds an and d so its ten tensio sion n dev develo elops ps a ve verti rtical cal 25 Downloaded by [ University College London] on [14/11/16]. Copyright © ICE Publishing, all rights reserved.


component to help support the load. There are also more general cases where large defo de form rma ati tion onss ne need ed to be pr pred edic icted ted ac accu cura rate tely ly,, of ofte ten n in so soft ft so soil ils, s, as sh sho own in th thee emban emb ankme kment nt con constr struct uction ion ex exam ample ple in Sec Sectio tion n 8.4 8.4.. Not Notee tha thatt for the pr predi edicti ction on of collapse loads (and not for the prediction of collapse deformations), usually the conventional tio nal met method hod agr agrees ees bet better ter wit with h an analy alytic tical al app appro roac aches hes.. Usi Using ng lar large ge def deform ormat ation ion methods can lead to stiffer behaviour near failure and a higher predicted failure load. There are two common options to account for large deformations in the mesh geometry: g

g

Updated coordinat Updated coordinates es (only): here the nodal coordina coordinates tes are updated updated to matc match h the calculated deflection and equilibrium is satisfied in the final deformed geometry. But this is not a rigorous treatment of large deformation behaviour because no account is taken of stress changes due to rotation and straining. Updated Lagrangian Lagrangian formulation: formulation: here, here, as well as updating coordinates, coordinates, the stress stress changes due to rotation and straining are taken into account.

These methods are slower and less robust, so only use them when necessary. Perform a conventional analysis first and check whether deformations are large enough possibly to require large deformation methods. If so, try using these methods and check whether the key outputs have changed enough to justify their use. Groundwater pressures (for pore pressure at Gauss points and external water pressures) may also be updated due to the changing geometry of the ground under a constant groundwater level. Check how the software handles distributed loads – usually the equivalent nodal loads remain unchanged in spite of the changing nodal coordinates, and the loads will either follow the initial direction or rotate with the deformed shape. Note tha Note thatt ev even en the these se lar large ge def deform ormat ation ion met metho hods ds ha have ve the their ir lim limits its.. Oth Other er met metho hods, ds, beyond the scope of this book, are under development to predict larger deformations and material flow, including a number of Eulerian methods, such as the material point method, usefully summarised by Soga Soga et et al . (2016). (2016).

1.5. 1.5. 1.5.1 1.5 .1

Con ons sti titu tuti tiv ve mo mode dels ls Which Whi ch co cons nstitu titutiv tive e mod model el sho should uld be use used? d?

Constitutive models should be selected that simulate each soil or rock stratum’s behaviour with suf suffici ficient ent ac accur curac acy y under under all the loadin loading g con condit dition ionss to be imp imposed osed.. To avo avoid id unnecess unnecess-ary complexity, the simplest constitutive model that satisfies this requirement should be selected. Consequently, some compromise compromise is needed and a complex model that that recreates all aspects of ground behaviour behaviour may not be necessary. Identify which regions regions of the FE model aree of gr ar grea eates testt int inter eres estt (wh (wher eree mor moree pr precis ecisee con consti stituti tutive ve mod modelli elling ng ma may y be need needed) ed) and whi which ch aspe as pects cts of gr grou ound nd be beha havi viou ourr ar aree th thee mo mosst cr criti itica cal. l. Th Then en sel select ect th thee co cons nsti titu tuti tiv ve mo mode dell acco ccord rd-ingly. Note that the same soil or rock with different structures (e.g. an embankment or an exca ex cava vatio tion) n) ma may y beh behav avee ve very ry dif differ ferentl ently y beca because use beh beha avio viour ur is str stressess-pa path th dep depend endent. ent. So, one model cannot be said to be suitable for a particular soil or rock in all situations. More guidance on constitutive models is provided in Chapter 2. 26 Downloaded by [ University College London] on [14/11/16]. Copyright © ICE Publishing, all rights reserved.


1.6. 1.6. 1.6.1 1.6 .1

Ground Grou ndwa wate terr an and d dr drai aina nage ge How Ho w are are the the effects effects of gro ground undwa water ter incl include uded d in the the anal analys ysis? is?

Groundwater is very important in any FE model because it has a direct influence on effective stress, time-dependent soil response and the forces acting on structures (e.g. retaining walls). Sometimes groundwater flow can be neglected and hydrostatic conditions assumed, or the steady-state pore pressure distribution is simple enough to be specified directly. On other occasions, a groundwater flow calculation is required to generate the steady-state or transient pore pressure distributions in the ground, either in a sep separ arat atee an analy alysis sis or ful fully ly cou couple pled d wit with h the st stres ress–s s–stra train in cal calcul culat ation ions. s. Inc Includ luding ing groundwater effects in an FE model is covered in Sections 4.1 and 4.3. Submerged surfaces require the application of perpendicular, external water pressures corresponding with the water level. Some programs create these pressures automatically once the external water level has been specified. If excavating or filling under water, the external pressures need to be changed in the same construction stage.

1.6.2 1.6 .2

Should a drain Should drained, ed, und undra raine ined d or con consol solida idatio tion n analy analysis sis be performed?

The dissi dissipa pation tion of exc excess ess por poree pr pressu essures res is a timetime-dep depend endent ent phen phenomen omenon on req requirin uiring g equations of consolidation (usually Biot’s equations) for its simulation in a consolidation analysis. Alternatively, it may be acceptable to simplify the analysis of a soil layer to wholly drained (when the rate of loading is slower than the rate of drainage) or wholly undrained (for short-term periods during which no significant dissipation of excess pore pressu pr essure re has occu occurre rred). d). A con consoli solida dation tion ana analy lysis sis is req requir uired ed to dissi dissipa pate te exc excess ess por poree pressure either to change from undrained to drained conditions or to obtain temporal outputs of deformations and structural forces, etc. during consolidation. Drained, undrained and consolidation analyses are described in more detail in Chapter 4 (Sectio (Se ctions ns 4.2 and 4.4 4.4), ), in pa partic rticula ularr the iss issues ues ass associ ociat ated ed wit with h mod modell elling ing und undra raine ined d behaviour. REFERENCES

Cashma Cash man n PM an and d Pr Pree eene ne M (2 (201 012) 2) Groundwa Groundwater ter Low Lowering ering in Cons Construct truction ion,, A Pra Practica ctical l Guide to Dewatering, Dewatering, 2nd edn. CRC Press, Boca Raton, FL. Chille Chi llery ry M (20 (2014) 14) NAFEMS NAFEMS Sim Simula ulatio tion n Ha Handb ndbook ook – Qu Quali ality ty Man Manage agemen mentt . NAF NAFEMS EMS,, Hamilton. Clear CA and Harrison RA (1985) Concrete Pressure on Formwork. Formwork. CIRIA Report R108. CIRIA, London, UK. Engin Eng in HK, Bri Brinkg nkgre reve ve RBJ and Va Van n To Toll AF (20 (2015) 15) Approxima Approximation tion of pile inst installa allation tion effects: a practical tool. Proceedings of the Institution of Civil Engineers – – Geotechnical Geotechnical Engineering 168(4): 319–334. Hicks Hic ks MA, Dijkstr Dijkstra a J, Llo Llore ret-Ca t-Cabot bot M and Kar Karstu stunen nen M (20 (2013) 13) Installa Installation tion Effec Effects ts in Geotechnical Engineering. Engineering. CRC Press, Leiden, Netherlands. Ling Li ngss ML ML,, Ng CW CWW W an and d Na Nash sh DF DFT T (1 (199 994) 4) The The la late tera rall pr pres essu sure re of we wett co conc ncre rete te in diaphr dia phragm agm wa wall ll pan panels els cas castt und under er ben benton tonite ite.. Proce Proceed edin ings gs of th thee In Inst stit itut utio ion n of Ci Civi vil l Engineers – Geotechnical Geotechnical Engineering 107(3): 163–172. 27 Downloaded by [ University College London] on [14/11/16]. Copyright © ICE Publishing, all rights reserved.


Sloan SW (2013) Geotechnical stability analysis. Ge Ge´ ´ otechnique otechnique 63(7): 531–571. Soga K, Alonso E, Yerro A, Kumar K and Bandara S (2016) Trends in large-deformation analy ana lysis sis of lan landsl dslide ide mas masss mov moveme ements nts wit with h par partic ticula ularr emp emphas hasis is on the ma materi terial al poi point nt method. Ge Ge´ ´ otechnique otechnique 66(3): 248–273.



Chapter 2

How are constitutive models selected? 2.1. 2.1. 2. 1.1 1

Introduction Int What Wh at is a co cons nsti titu tuti tive ve mo mode del? l?

A constitutive model is a series of mathematical expressions relating stresses and strains (or stress rates and strain rates) that are used to model material behaviour in an element. When implemented into an FE analysis that also ensures equilibrium and compatibility between each element, the constitutive model allows complex problems to be analysed and displacement and stress to be calculated everywhere in the model at every construction stage. All constitutive models are an approximation of material behaviour. Advanced models may recreate several aspects of material behaviour but none can recreate all aspects and knowing what they do not model is just as important as knowing what they do model. Many constitutive models have been proposed and published in the literature to recreate particular aspects of soil and rock behaviour. Most are rarely used. This is often due to their complexity and the high number of input parameters, some of which may be hard to obtain. Furthermore, to enter practical application in FE analyses, a constitutive model needs to work in general stress space whereas during development the model may have been tested on only limited stress space, stress range and stress paths. Implementation of constitutive models in FE analysis is not straightforward and requires rigorous testing to help ensure that the implementation is robust. This leaves a relatively small number of constitutive models that have been implemented successfully and applied widely in simulations of actual construction projects. Yet, even the selection of appropriate models from this short list is not easy. This chapter provides the background knowledge required to make informed selections of appropriate constitutive models for a particular analysis task.

2.1.2 2.1 .2

Why is it it impo importan rtantt to use use appr appropr opria iate te cons constit tituti utive ve mode models? ls?

While FE analysis has allowed the simulation of complex ground behaviour and soil– structure interaction problems, its accuracy depends heavily on the constitutive models adopted for each material. The constitutive model needs to account for all the important aspects of material behaviour for a particular problem otherwise the outputs from the FE analysis will be erroneous. 29 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.


Soils are usually the softest and weakest material in any soil–structure interaction analysis, so their behaviour governs the deformations and probability of failure. Therefore, it is important to simulate their behaviour accurately over the range of stresses and strains they the y will exp experien erience ce with an appr appropria opriate te cons constituti titutive ve mode model. l. Co Const nstructio ruction n ma material terials, s, such as concrete and steel, are stiff in comparison with soils so it is often sufficient to model these with simple linear elastic constitutive models (see Section 5.2).

2.1.3 2.1 .3

How Ho w is the app appro ropri priate atenes nesss of a cons constitu titutiv tive e model model jud judged ged? ?

An appropriate constitutive model is one that recreates the important aspects of stress– stra st rain in beh behav aviou iourr for the ra range nge of st stre ress ss and strain strain con condit dition ionss in the pr probl oblem em to be analysed. At the same time, it should be economical, i.e. not include other aspects of material behaviour that are not important, to avoid unnecessary complexity. Furthermore, the model parameters need to be obtainable from the tests that can be (or were) performed in the site investigation. The selection of appropriate constitutive models depends primarily on three aspects of the project:

1

2

3

Aims of the FE analysis analysis and required required outputs: outputs: above above all, a constitutiv constitutivee model needs to be selected that will provide the required outputs accurately, but there is no need for the model to provide accurate outputs that are not required. For instance, an FE analysis of a deep excavation may be required to obtain the structural forces in a retaining wall for its design. In this case, a linear stiffness varying with depth or a stress-dependent stiffness may be sufficient to model the elastic behaviour of the soil. There would be no need, in this case, to use a straindependent stiffness because the additional complexity would probably result in only a marginal increase in the accuracy of the required outputs of retaining wall forces. On the other hand, if outputs of excavation-induced settlement were also required from the FE analysis, then a constitutive model with strain-dependent stiffness would be required in order to obtain that output accurately. Structure Struc ture type and expected expected stress stress path: the type of stru structur cturee being simulate simulated d influences the stress path in the ground around the structure. Broadly speaking, foundations and embankments cause increased loading in the ground while excavations and tunnelling cause unloading. Of course, many projects will be more complicated than this with combinations of structure types and a construction sequence that may result in load reversals (e.g. demolition and reconstruction of a building) rather than just monotonic loading. The expected stress path is important because some constitutive models are more suited to particular stress paths than others. For instance, the Modified Cam Clay (MCC) model provides realistic predictions of deformation for the compression of soft clays, but less so for unloading. More guidance on the influence of structure type on constitutive model selection is provided in Section 2.4. Soil and rock rock types: it would be wrong wrong to say that a constitutiv constitutivee model is suited to a particular soil type in all applications because there are required outputs and expected stress paths to consider. If a constitutive model was used successfully to simulate a particular soil on one project, do not expect the same model to


How are constitutive models selected?

simulate the same soil successfully on another project with a different stress path and/or different required outputs. Yet, clearly, the soil or rock type is an important consideration. Certain soil and rock types will exhibit behaviour less prevalent in other types, and if this behaviour influences the required outputs then a constitutive model that recreates this behaviour needs to be selected. For instance, soft clays and silts often experience time-dependent creep or stress relaxation not seen in other soils, while other soils may exhibit particularly anisotropic properties that could influence the required outputs. The different aspects of ground behaviour that need to be considered are covered in Section 2.2. Further project-specific guidance on the selection of appropriate constitutive models can be found in published case studies of similar projects in similar ground conditions using FE analysis where justification of the selection of constitutive models may be provided.

2.2. 2.2. 2.2.1 2.2 .1

Aspect Aspe cts s of gr grou ound nd be beha havi viou ourr Which Whi ch aspe aspects cts of soil soil beh behavi aviour our ma may y need need to to be be consi consider dered? ed?

Soil is a complex material consisting of a skeleton of soil grains in frictional contact and voids filled with air and/or water water (or other fluids). Forces are transmitted through through the soil skeleton via normal and shear forces at grain contact points. This is a behaviour actually more suited to discrete element modelling where each contact is modelled. However, due to limitations on the size of discrete element models and the time taken to set up and run such analyses, most practical problems are modelled using the principles of continuum mechanics with, for example, FE analysis. This leaves the engineer with the challenge of characterising chara cterising a mixture of solid grains, grains, water water and/or air as a continuum. continuum. For For this reason, compared with other engineering materials, soil is among the most difficult to model. The following are some of the aspects of soil behaviour that need to be considered when selecting a constitutive model:

Soil type Soft clay and dense gravel, for example, each display very different behaviours with wide ranges of strength and stiffness. Many constitutive models are more suited to either finegrained or granular soils. Each soil type will respond in different ways to most of the further aspects of soil behaviour that follow.

No tensile strength Soils have little or no tensile strength, so constitutive models need to include this important aspect of behaviour.

Strength changes during shear Loose soils compress while dense soils dilate during shear, both toward the critical state, and these changes in density cause changes in the current shear strength of the soil, i.e. shear hardening in the loose soil and shear softening in the dense soil. Continued shear can lead to a further reduction in strength toward a residual shear strength. Softening is a particular issue with stiff plastic clays which are rather brittle and progressive failure is a common phenomenon in slopes in such clays. 31 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.


Stress-dependency of stiffness and strength As the confining stress stress on a soil increases, the volume of voids decreases, the soil densifies and so it becomes stronger and stiffer. The stress-dependency is non-linear, although a linear relationship is often assumed particularly for soil shear strength. It is possible to specify the changing stiffness and strength with depth in the model parameters to take account of increasing in situ stress, while more advanced models take account of stress changes during the analysis, which is important when the stress changes are large, e.g. in embankment construction or excavations.

Stress-path dependency of stiffness On primary loading (i.e. when loaded to a particular level for the first time), soil shows a highly non-linear stiffness (Figure (Figure 2.1). 2.1). When unloaded or reloaded it shows a higher, more mo re lin linear ear st stiff iffnes ness. s. Whe When n re reloa loadin ding g cha change ngess to pri primar mary y loa loadin ding g as the pr previ eviou ouss maximum stress is exceeded, then there will be a sudden reduction in stiffness. Therefore, stress history as well as stress path is important. Furthermore, primary loading behaviour differs significantly between deviatoric and compressive loading, as illustrated by comparing the typical graphs from triaxial and oedometer tests in Figure 2.1. Whereas stiffness decreases with deviatoric load, it increases under compressive load due to the increasing density of the soil.

Permanent deformations Many materials have a significant elastic stress range within which reversible, elastic deformations occur. Soils, however, have a narrow elastic stress range such that, even at quite low stress levels well below the failure stress, permanent deformations occur, particularly in soft normally consolidated and lightly over-consolidated clays.

Bonding and structure Natural soils develop a fabric and inter-particle bonding called structure structure that that gives soil additional strength and stiffness that cannot be explained by void ratio and stress history alone. Significant straining causes loss of structure (destructuration) and a change in behaviour. Yet, many constitutive models are based on the results of laboratory tests

Figure Figur e 2.1 Typical deformation behaviour of soils q

, s s e r t s c i r o t a i v e D

′

p

Primary loading stress path

Unload/reload stress path

Axial strain, εa Triaxial compression

, s s e r t s e v i t c e f f e n a e M

Primary loading stress path Unload/reload stress path

Axial strain, εa Oedometric (K 0) compression



on reco econs nstit titut uted ed sa samp mples les wi with thou outt st stru ruct ctur uree an and d so do no nott in incl clud udee th thee eff effec ects ts of destructuration.

Intermediate principal stress Many constitutive models are based on the results of laboratory triaxial tests where the intermediate principal stress s 2 equals either the minor principal stress s 3 (compression tests) or the major principal stress s 1 (extension tests). However, when implemented into FE analyses, constitutive models operate in general stress space and s 2 may vary between s 3 and s 1 – stress states that may not have been tested in the original conthe values of s stitutive model. This variation is defined by the ratio b where b = 0 corresponds with s 2 = s 3 and b = 1 with s 2 = s 1 (as shown in Figure in Figure 2.2). 2.2).

Anisotropy Most soils are anisotropic to some extent. Assuming isotropy can over- or under-estimate the strength and stiffness of the soil, it is not necessarily a conservative assumption. Fabric and stress history can give an element of (inherent) anisotropy to the strength and stiffness of soils, while stress and strain changes can increase the anisotropy (induced) or reduce it. Soil properties generally do not vary in the plane of deposition (which is often horizontal, but not always) but only between the plane of deposition and direction of deposition (often vertical). Consequently, cross-anisotropic (also called transverse anisotrop tr opic ic and ort orthot hotro ropic pic)) con condit dition ionss can usu usuall ally y be ass assume umed d for soi soils. ls. An Aniso isotr tropy opy is expressed in terms of the angle a between the major principal stress direction and the direction of soil deposition, as shown in Figure 2.3. 2.3.

Strain-dependent stiffness In addition to stress-dependency, soil stiffness is also strain-dependent. At small strains, soil stiffness is high and it decays to a lower value as strains increase, as illustrated in Figure 2.4. 2.4. Note that the ra rate te of deca decay y is particularly particularly high in the typical ranges ranges of str strain ain occurring around geotechnical structures, so it is highly relevant to the modelling of most structures. The stiffness also returns to higher small-strain values on stress reversal and, to a lesser extent, after stress rotation before decaying again with increasing strain. Figure 2.2 Example principal stress orientations b = 0

b = 1

σ 1

σ 3

σ 3

σ 2 σ 2 = σ 3

σ 2

σ 3

σ 1

Triaxial compression

σ 1

0 ≤ b ≤ 1

σ 2

σ 1

σ 3

σ 2 = σ 1

σ 2

σ 1

σ 3

Triaxial extension

σ 1

σ 3

σ 3

σ 2 σ 2 σ 3

σ 1

σ 1

σ 1 > σ 2 > σ 3

Typical application in general stress space



Figure Figur e 2.3 Expressing anisotropy of soils α =

0º

α =

σ 1

σ 3

90º σ 3

σ 3

90º

σ 1

σ 1

bedding planes

σ 3

σ 3

σ 1

σ 3

σ 3

σ 1

active state α = 0º (when bedding planes are horizontal)

Triaxial extension

bedding planes

σ 3

σ 1

σ 1

Triaxial compression

σ 1

passive state α = 90º (when bedding planes are horizontal)

Spread foundation

When using linear elasticity, a lot of care is needed to ensure that the stiffness value selected is appropriate for the strain level.

Creep Creep (or secondary compression) is deformation that continues even after excess pore pressures have dissipated and under constant pore pressure and effective stress. Alternatively, if deformations are constrained, stress relaxation will occur over time. It is a major contributor to the deformation of soft clays, silts and peat. No constitutive model can take account of all of these aspects of soil behaviour. Even if one could, it would probably be too complex to implement into an FE analysis and the parameters would be too difficult to determine. Many of these aspects of behaviour are still to be researched in detail and understood before accurate constitutive models can be produced and rigorously tested. Nevertheless, continuous developments are leading to more unified models that incorporate more of the aspects described above. from Mair (1993)) (1993)) Figure Figur e 2.4 Typical decay of soil stiffness with strain (redrawn from Mair Typical strain range around geotechnical structures Linear elastic s s e n f f i t S

LEPP model

Perfectly plastic

0.0001

0 .0 0 1

0.01

0.1

1

10

Strain: %


Q1


2.2.2 2.2 .2

Which Whi ch aspe aspects cts of of rock rock beha behavio viour ur may may need need to be be consi consider dered? ed?

While the aspects of behaviour in soft rocks and hard soils, e.g. mudstone and hard clay, are quite similar, in harder rocks the behaviour becomes rather different and constitutive models intended specifically for rocks are usually more appropriate. The following are some of the aspects of rock behaviour that need to be considered when selecting a constitutive model: g

g

g

g

Stress-dependency of strength: strength: highly jointed or weathered weathered rock is a frictional frictional material and the stress-dependency of shear strength is significant, and is also highly non-linear. Stiffness, on the other hand, is not very stress-dependent and a constant stiffness can usually be assumed. Influence Influ ence of discontinuit discontinuities: ies: discontinu discontinuities ities are surfaces surfaces in a rock mass where there is a sudden change in physical and chemical characteristics. In this book, discontinuities will be considered as the physical type where relative movement can occur between the rock on each side of the discontinuity surface and they include joints, fractu fractures, res, cleavage and faults. They gov govern ern rock mass behavio behaviour ur in lowstress conditions because they are significantly weaker than the intact rock. They introduce strength and stiffness anisotropy and pre-determined failure planes. Somehow the mechanical properties, spacing, orientation and persistence of the discontinuities need to be incorporated into a constitutive model of a continuum. Strength Str ength changes changes during failure: failure: the stress–str stress–strain ain response response of roc rock k shows a relatively linear elastic response initially, compared with soils, until failure occurs. Rock failure can be brittle or ductile depending on the stress state, as defined by the empirical Mogi line (Mogi, ( Mogi, 1971). 1971). When the ratio of major to minor principal stress exceeds 3.4, brittle failure is predicted. Brittle failure in rock results in strain softening to a residual strength that is much lower than the peak strength. Progressive failure mechanisms can also form. Tensile Te nsile strength: strength: rock possesses possesses tensile strength, strength, but not at discontinui discontinuities, ties, which requires specific treatment in constitutive models for rocks.

2.2. 2. 2.3 3

When Wh en ca can n FE FE ana analy lysi siss be be use used d for for ro rock cks? s?

Fractured rock is a complex assembly of intact blocks between discontinuities. Rather likee soi lik soil, l, for forces ces ar aree tr trans ansmit mitted ted thr throug ough h the gr grou ound nd via no norma rmall and shear for forces ces at contact points, but behaviour can be more complex than soil due to the effect of the size of the intact blocks relative to the structure being analysed. At some point it becomes necessary to model each block explicitly because their form starts to dominate engineering behaviour and failure mechanisms. The advantage of using FE analysis is that it is quicker and more practical than modelling with discontinuous media. It is also easier to compare analyses with different discontinuity patterns and properties. Yet, on some occasions it will be necessary to model a rock mass as a discontinuous medium using, for example, discrete element modelling. Here are three different options for modelling rock masses, two as a continuum and one as a discontinuous medium, together with descriptions of when it is appropriate to adopt 35 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.


each option, as illustrated in Figure 2.5: 2.5: g

g

g

As a conti continuum nuum,, e.g. FE analysis, analysis, with implicit modelling modelling of discontinuiti discontinuities es (when present) by smearing them into a continuum with appropriately reduced strength and stiffness and, perhaps, anisotropy. This is appropriate when the rock mass is essentially free of discontinuities or when the discontinuity spacing is small relative to the structure being modelled and there is no unfavourable dip in the discontinuity that could lead to instability. As a conti continuum nuum,, e.g. FE analysis, analysis, with explicit explicit modelling modelling of disco discontinu ntinuities ities using interface elements. Interface elements have zero tensile strength, a compressive stiffness and a shear stiffness subject to a Coulomb friction criterion and are described in more detail in Section 5.1.3. This method is appropriate when the discontinuity spacing is similar to the size of the structure being modelled, provided that the discontinuity geometry remains unchanged during the construction process since nodal coordinates remain unchanged during FE analysis. The geometry of the discontinuities can be created individually or, since that level of detail is not usually available from geological field mapping, using different fracture network models to investigate the effect of different discontinuity patterns, spacings and orientations. In continuum modelling, the onset of failure is predicted but not the complete separation or rotation of blocks. As a disco discontin ntinuous uous medium, medium, e.g. discrete discrete element method, method, where complete complete separation and rotation of individual blocks are possible. This method is also appropriate when the discontinuity spacing is similar to the size of the structure being modelled, but also when large deformations are expected or when the contact points between blocks are expected to change during the analysis due to slippage, rotation or separation of blocks. This method is particularly suited to rock slope stability problems. Relatively well-defined discontinuity patterns are needed to model the locations of the discontinuities.

Hybrid numerical methods combining, for example, discrete element modelling with FE analysis may be used to combine the advantages of each method. For borderline cases, the quality of the field data plays a role in the choice of modelling method. If no discontinuity pattern emerges and spacings and orientations seem arbitrary, then continuum modelling is better because it is easier to study different discontinuity patterns. Remember that bedrock situated below a softer soil and the area of interest can be simulated as a fixed bottom boundary if strains in the bedrock are expected to be insignificant (see Section 1.2.3).

2.3. 2.3. 2.3.1 2.3 .1

Common Comm on co cons nsti titu tuti tive ve mo mode dell ty type pes s How Ho w do con const stitu itutiv tive e model modelss commo commonly nly ac accou count nt for for the ela elast stic ic behaviour of soils?

Rarely is an elastic constitutive model on its own sufficient to model soil behaviour because soil is a comparatively weak material and irreversible strains, shear-dilatancy 36 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.


Figure 2.5 Metho Methods ds of model modelling ling rock masses depending depending on disco discontinu ntinuity ity spacing Essentially free of discontinuities Model as a continuum

Slope stability analysis may require discontinuous medium due to large deflections Discontinuity spacing large relative to structure Model as continuum with interface elements or as a discontinuous medium

May require explicit modelling of discontinuities due to unfavourable dip

Discontinuity spacing small relative to structure Model as continuum with implicit discontinuities



and other behaviours associated with being near or beyond failure are not recreated by an elastic model. Perhaps only very stiff clays at low stress levels could be simulated with an elastic model. Therefore, soil constitutive models need both elastic and plastic behaviour to produce sufficiently accurate predictions. This section provides a summary of the ways that the elastic part of elastoplastic constitutive models work. For more detail on specific constitutive models, readers should refer to their software manuals and the reference papers associated with each model.

Linear elasticity Isotropic linear elasticity, also called Hooke’s law, is the most basic way to treat the elastic part of soil behaviour. It requires only two parameters (Young’s modulus E and and Poisson’s ratio n). Real soil stiffness is stress- and strain-dependent, so the parameters selected for a linear elastic model must be appropriate for the expected stress and strain level and stress path (e.g. primary loading or unload–reload path). It may not be possible to select an appropriate value for large ranges of stress and strain (see Figure (see Figure 2.6) 2.6) or for complex stress paths. An increasing stiffness with depth (a linear increase specified in the input parameters or else separate soil layers with different stiffness) can be specified to account for increasing in situ stress with depth, but subsequent stiffness changes due to stress changes are not be taken into account.

Anisotropic linear elasticity Cross-anisotropic stiffness can be added to the linear strain elastic model with five input parameters instead of two for the isotropic model (Clayton, ( Clayton, 2011). 2011 ). The more advanced non-linear elasticity models described below usually adopt isotropic elasticity because of the high number of parameters that would be required to define non-linear, anisotropic behaviour. However, some non-linear elastic, anisotropic models are used in research and it is possible that they will see more common use in practical problems in the future.

Non-linear elasticity and stress-path dependent stiffness To take account of the non-linear stiffness of soils on primary loading, hyperbolic functions are commonly used to fit observed stress–strain curves in triaxial compression tests Figure Figur e 2.6 Linear and non-linear elasticity Linear elasticity q


Hyperbolic function for primary loading Real soil behaviour

Unload/reload stiffness Axial strain, εa



up to a failure stress plateau (Figure 2.6). This was first proposed by Kondner (1963) and (1963) and is used in the Duncan the Duncan and Chang (1970) and Hardening Soil (Shanz ( Shanz et et al ., ., 1999) 1999) models. On unloading, the soil does not return along the same stress–strain path but exhibits a stiffer, linear elastic response requiring a separate unload–reload stiffness to be specified in the constitutive model. Then the model provides good predictions of displacements under deviatoric monotonic loading and load reversals.

Stress-dependent stiffness As stated previously for linear elasticity, increasing stiffness with depth can be specified to take account of its variation due to in situ stress, but this does not take account of stress str ess chan changes. ges. Ful Fulll str stress-de ess-depend pendency ency is obta obtained ined if the cons constituti titutive ve mode modell inclu includes des expressions relating stiffness with confining stress. For example, a power law between confining stress and stiffness forms the basis of the expressions used in the Duncan and Chang and Hardening Soil models. The MCC model ( Roscoe and Burland, 1968; 1968 ; Muir Wood, 1991) uses a logarithmic relationship between average effective stress p ′ and void ratio e and therefore a linear stress-dependent stiffness which is appropriate for normally or lightly over-consolidated clays. The Lade model (Lade, ( Lade, 1977) 1977) uses a logarithmic dependency of stiffness on the stress state ( p ( p′ and q). Full stress-dependency of stiffness is important for more accurate deformation prediction where stresses change sign si gnifi ifica cant ntly ly,, e. e.g. g. fo forr se sett ttle leme ment nt un unde derr a ne new w em emba bank nkme ment nt or he hea ave un unde derr a de deep ep excavation.

Strain-dependent stiffness The reduction of stiffness with strain is specified with parameters defining stiffness degradation curves. The original Jardine Jardine et et al . (1986) model (1986) model simulated undrained behaviour and the decay of shear modulus G only. Since then the model has been extended to drai dr aine ned d be beha havi viou ourr wi with th tw two o tri trigo gono nome metr tric ic fu func nctio tions ns to de defin finee th thee de deca cay y of sh shea earr modulus G and bulk modulus K requiring requiring ten input parameters and upper and lower limits to the curves. Its main disadvantage is that it does not simulate small-strain behaviour following stress reversals or stress rotations. The HS Small model (Benz ( Benz et et al ., ., 2009) 2009) is simpler in that only two parameters are required to fit a logarithmic function to the decay curve of G while the decay of K is is calculated from G using a constant Poisson’s ratio. In reality, Poisson’s ratio is not constant but varies with strain and real soil data indicates that the decay curves for these two moduli are only broadly similar, so this is a disadvantage of the model. In any case, accurate measurement of the K decay decay curve is notoriously difficult. Stress reversals cause a resetting of the small-strain stiffness to its maximum value before decaying again with strain while stress rotations of less than 180 are also reset to an intermediate, interpolated stiffness. Since these models are an ‘add-on’ to other elastoplastic models, careful attention should be paid to the way stiffness is modelled at the interface between the models. 8

2.3.2 2.3 .2

How do cons How constit tituti utive ve model modelss common commonly ly acc accoun ountt for the the plas plastic tic behaviour of soils? Yield surfaces The elastic limit in stress space is defined by a yield surface. Stresses beyond the yield surface surfa ce cause permanent permanent and irrev irreversibl ersiblee str strains ains and the cons constituti titutive ve rela relations tionship hip is 39 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.


Figure Figur e 2.7 Types of idealised plastic behaviour Perfectly plastic Yield surface maintains original position

Isotropic hardening

σ′2

Yield surface after hardening

σ′2

Stress path

σ′3

σ′1

Stress path

σ′1

σ′3

Kinematic hardening

Isotropic softening

σ′2

σ′2

Yield surface after hardening

Yield surface after softening Stress path

σ′3

σ′1

σ′3

σ′1

always non-linear (regardless of whether the elastic part of an elastoplastic model was linear). Usually, plastic analysis assumes that stress–strain behaviour is independent of strain rate, although in reality this is not quite true – high strain rates result in a slightly stiffer response – and this should be considered in parameter selection (see Section 3.4.1). Plastic behaviour can be idealised in the following ways, as also illustrated in Figure 2.7: 2.7: g

g

g

perfectly plastic (or non-work perfectly non-work hardening): hardening): the yield str stress ess remains constant constant during during yield strain str ain hardening hardening (or work hardening): hardening): the yield stress stress increases increases during yield – isotropic hardening: the yield surface increases in size with increasing plastic strain but remains the same shape and in the same position – kinematic hardening: the yield surface is translated to a new position in response to plastic strain but does not change size or shape strain str ain softening: softening: the yield stress stress decreases decreases during yield, for example example due to dila dilation tion of a dense soil during shear. Consequently, the yield surface needs to expand up to the peak stress and then contract beyond the peak. Convergence of the calculation needs to be monitored very carefully and outputs are highly meshdependent, so such models tend to be used in research rather than in practice.



Figure 2.8 Principal stress space and the stress invariants Space diagonal σ′1 = σ′2 = σ′3

Deviatoric plane

σ′2

Deviatoric plane

Space diagonal σ′1 = σ′2 = σ′3

Current stress state

σ′2

′ p

g n i s a e r c n I

Lode angle θ

Increasing q σ′1

Current stress state

σ′1

σ′3

σ′3

View 1

View 2

Yield surfaces need to be defined in general stress space in order to be adopted in constitutive models for FE analysis, and they are visualised on three axes of the principal effective (or total) stresses (s 1′ , s 2′ and s 3′ ), i.e. in principal stress space as shown in Figure 2.8, 2.8, so that the graphs are unaffected by the chosen coordinate axes directions. To describe any point in that stress space, rather than use the three values of principal stress, it is more useful in constitutive modelling to use the following three invariants (an invariant has the same magnitude and direction no matter which directions are chosen for the coordinate axes):

1

2

mean effective stress p′ which is the average of the three components of normal effective stress with no shear stress component. Changes in this stress cause volumetric strains. deviatoric stress q which is the shear component of stress remaining after subtracting p′ . Changes in this stress cause deviatoric strains.

The value of p p′ is a measure of the distance along the space diagonal where s ′1 = s 2′ = s 3′ (see Figure 2.8). In other fields of engineering this space diagonal is called the hydrostatic stress axis, but this term is used less often in geotechnical engineering because hydrostatic has a different meaning concerning the behaviour of bodies of water. Any pl Any plan anee pe perp rpen endi dicu cula larr to th thee sp spac acee di diag agon onal al is ca call lled ed th thee de devi via ato tori ricc pl plan anee (se (seee Figure 2.8) while q is a measure of the distance from the space diagonal to the current stress state along that plane. But in which direction is the current stress state from the space diagonal? This is defined by the third invariant:

3

Lode’s angle u which which is the angle between a chosen reference axis and the line between the space diagonal and current stress state. Due to the requirement that 41



′

′

′

s 3 , u has has a range limited to 60 with the extreme ends of the range representing triaxial compression (s ′2 = s 3′ ) and triaxial extension ( s ′1 = s 2′ ) s 1

≥

s 2

≥

8

conditions.

Flow rules When the stress state reaches the yield surface, the material undergoes plastic deformation. So an elastoplastic constitutive model needs a flow rule that defines the plastic strain increment at every stress state. There are two types of flow rules: g

g

Associated Associa ted flow: the direction direction of plastic plastic strain strain is the same as the outward outward normal normal to the yield surface. This simplifies calculations in FE analysis but is better suited to the simulation of metals. In frictional materials like soil, the dilation angle for the prediction of irreversible volume change during shear and the friction angle are assumed the same, leading to the prediction of excessively high rates of dilation, particularly for high friction angles such as with dense sand. Clays with lower friction angles may be modelled with reasonable accuracy with associated flow. Non-associated flow: flow: the direction of plastic strain strain is specified separately. separately. This adds complexity to the calculations but is required for frictional materials such as soil and concrete to avoid the excessive dilation of associated flow rules. Even though the dilation is less, it can still continue indefinitely, which is unrealistic. Dilation can be linked to plastic strain or a pre-defined cut-off to keep volumetric strains to realistic levels.

Failure surfaces Failure points in stress space are defined by a failure surface. Stresses inside the failure surface are not in a state of failure while stresses on the failure surface are in a state of failure. Stresses cannot exist outside the failure surface. The Mohr–Coulomb failure surface is the most commonly used for soils (see Figure 2.9) 2.9) and is an extension of Coulomb’s friction law (defined by the internal friction angle w ′ and cohesion c′ ) to general stress space. Its failure predictions in drained conditions are quite good, but the strength is obtained from the difference between the major and minor principal stresses so the intermediate principal stress s 2 is not taken into account in the prediction of failur fai luree in gen gener eral al str stress ess spa space. ce. The Theref refor ore, e, car carefu efull sele selecti ction on of str streng ength th par param ameter eterss is required (see Section 3.4.1). Sometimes, a c a c ′ value above zero is required to fit test data but this can give the soil an unrealistic tensile strength. In such cases a tension cut-off should be imposed allowing small or zero tension. Undrained strength predictions in effective stress analyses depend heavily on accurate excess pore water pressure predictions (see Section 4.2.5). With w ′ set to zero, Mohr– Coulomb becomes equivalent to the Tresca failure surface for undrained shear failure in terms of total stress. This also provides good predictions of failure provided that the selected undrained shear strength c strength c u value, which is not a fundamental soil parameter, is appropriate for conditions at that time. Note that Tresca takes no account of strength changes due to consolidation. 42 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.


Figure 2.9 Common failure surfaces in principal stress space: (a) Mohr–Coulomb; (b) Tresca; (c) Druck Drucker–Pr er–Prager; ager; (d) von Mises

σ′2

(a)


(b)

σ′2


σ′1

σ′3

(c)

σ′1

σ′3

σ′2


(d )

σ′2


σ′1

σ′3

σ′1

σ′3

The hexagonal shape of the Mohr–Coulomb and Tresca failure surfaces creates some difficulties in their implementation into FE analysis, so versions simplified to a circular cross-section are available in some programs, i.e. the Drucker–Prager and von Mises surfaces, surfa ces, resp respectiv ectively. ely. Note tha thatt these simpl simpler er surfa surfaces ces pro provide vide rea reasona sonably bly acc accura urate te failure predictions only for quite simple stress paths (e.g. triaxial compression or extension), but for complex stress paths they can significantly over-estimate shear strength and should not be used. While the hexagonal shape provides good predictions of failure for many stress paths, it does not match perfectly with laboratory test observations. Some rounding of the corners of the hexagon provides a more exact fit with observed data, such as in the Matsuoka and Nakai (1974) surface that is used in some constitutive models. 43 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.


Perfect plasticity This is the most simple form of plasticity because the yield surface remains unchanged during yielding and adopts the same surface as the failure surface. Consequently, it does not include aspects of behaviour such as continuous yielding from the onset of primary loading, or the effects of stress path (e.g. unload–reload versus primary loading). It is sufficient for the prediction of failure mechanisms and assessing factors of safety.

Isotropic hardening single surface plasticity Single surface plasticity marks the first level of sophistication over perfect plasticity. It allow all owss con contin tinuou uouss pla plast stic ic (i. (i.e. e. irr irrev evers ersibl ible) e) st stra rain in bel below ow a fai failur luree sur surfa face. ce. Ex Examp amples les include the cap models (DeMaggio ( DeMaggio and Sandler, 1971) 1971 ) which have a cap yield surface incorporating volumetric hardening and a fixed failure surface due to deviatoric stress, and MCC (Roscoe and Burland, 1968; Muir Wood, 1991) which has an elliptic yield surface in q – p′ stress space and is based on critical state soil mechanics. At the critical state, no more volume change on shearing occurs, deviatoric strain becomes infinitesimal and the soil has failed. The MCC model forms the basis for many other advanced constitutive models addressing different aspects of soil behaviour. There are a number of drawbacks associated with the original model, some of which have been overcome in various implementations of the model into FE analyses, so users of the model need to check carefully the particular implementation in their software. Some of the drawbacks of the original MCC model are as follows: g

g

g

g

The critical critical state line is similar to the Drucker–Pr Drucker–Prager ager failure failure surface, surface, so the same inaccurate failure predictions are possible in some stress states, but some models have been implemented with the Mohr–Coulomb failure surface for more robust strength prediction. Plastic Plas tic deviatoric deviatoric straining straining is derived derived from an associated associated flow rule so it depends on the amount of friction angle mobilised but some modifications have adopted non-associated plasticity in the deviatoric plane. Since only volumetric volumetric hardening is considered, deviatoric deviatoric loading in heavily heavily overoverconsolidated soils remains inside the yield surface leading to an overly long and linear elastic range and a high peak strength. Some modifications have introduced a cut-off surface to address the overly high peak strength. The K 0 value is determined implicitly from the soil strength parameters. Furthermore, the yield surface is based on isotropically consolidated clay whereas K 0 consolidated clays have a yield surface rotated toward the K 0 line in stress space and some models include such a modification (e.g. Sekiguchi and Ohta, 1977). Nevertheless, many field problems have been predicted well with the MCC model without this modification.

The MCC model is good for predicting the deformation behaviour of very soft soils under compression (e.g. embankment construction on very soft soil). It is not so good for unloading problems (e.g. excavations).

Isotropic hardening double surface plasticity As well as a volumetric yield surface (often called a ‘cap’) that is pushed out toward higher p′ valu values es durin during g comp compacti action on or comp compressi ression on hard hardening ening ( Figu Figure re 2.10 2.10), ), doub double le 44 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.


Figure 2.10 Compression (cap) hardening e

, o i t a r d i o V

q

Primary compression line W

X

Elastic unload/reload lines

Z

Y


Mohr–Coulomb failure surface Expanding cap yield surface Y

XZ W

p′p Mean effective stress, p′

p′p Mean effective stress, p′

p′p = isotropic pre-consolidation stress

surfa surf ace pl plas astic ticit ity y mo mode dels ls in incl clud udee a de devi via ato tori ricc yi yiel eld d su surf rfac acee to too o (Figu Figure re 2.11 2.11), ), e. e.g. g. Vermeer Ver meer (197 (1978) 8),, the ha harde rdenin ning g soi soill mo model del (Sc (Schan hanzz et al ., ., 19 1999) 99) an and d La Lade’ de’ss do doubl ubleehardening model (Lade, 1977). The deviatoric yield surface expands during shear or friction hardening toward a failure surface. Such plasticity models are often combined with the non-linear and stress-path dependent stiffness elasticity type models described earlier since they have the unload–reload and primary stiffness values to be activated depending on whether the stress state is inside or on the yield surfaces. These models provide pro vide mor moree real realistic istic disp displace lacement ment pred prediction ictions, s, parti particular cularly ly for exc excava avations tions wher wheree primary shear loading (on the yield surface) may occur even though the volumetric yield surface predicts elastic unloading. These types of models are becoming the most commonly used due to their versatility and ability to handle changes in stress-path direction.

Kinematic hardening multi-surface plasticity models or ‘bubble’ models These ar These aree the mor moree sop sophis histic ticat ated ed of the ad advan vanced ced mod models els whi which ch can des descri cribe be man many y aspects of soil behaviour including anisotropy, destructuration and small-strain stiffness.

Figure 2.11 Friction (shear) hardening q


q

Hyperbolic primary loading stiffness

C

B Linear elastic unload/reload stiffness


Mohr–Coulomb failure surface B

C

Expanding inner yield surface A

A Permanent plastic strain

Axial strain, εa

Mean effective stress, p′



Figure 2.12 Multi-surface plasticity models a → b, d

e

→

q d

Kinematic surface stationary,, elastic strains stationary

b → c, e

f

→

Kinematic surface dragged by stress point, plastic strain controlled by kinematic surface c → d Bounding surface enlarged, plastic strain controlled by bounding surface

c

Kinematic surface

Stress rotation

e

b a

f

p′

a c e f a u r f s n g d i n n u B o

Small, kin Small, kinema ematic tic (i.e (i.e.. the they y can mo move ve abo about) ut) yie yield ld surf surfac aces es or ‘bu ‘bubb bbles’ les’ ar aree with within, in, typ typica ically lly,, a critical state bounding surface based on the MCC yield surface except that within the surface, behaviour is elastoplastic rather than elastic. Within a smaller kinematic surface, elasti ela sticc beh beha avio viour ur is pr predi edicted cted but whe when n the str stress ess sta state te rea reache chess the surf surfac ace, e, the whole kinematic surface is dragged toward the bounding surface ( Figure 2.12) 2.12) and plastic strain occu oc curs rs in acco ccord rdan ance ce wi with th a flo flow w ru rule le an and d ha hard rden enin ing g la law w as asso socia ciate ted d wi with th th thee sm smal alle lerr su surf rfa ace. When Whe n the sma smalle llerr sur surfa face ce re reach aches es the bo bound unding ing surf surfac ace, e, beh behav aviou iourr is con contro trolled lled by the lar larger ger surface. Examples include Al-Tabbaa include Al-Tabbaa and Wood (1989) and and Stallebrass Stallebrass and Taylor (1997) which has a second kinemati kinematicc surface to tak takee account account of recen recentt stress history. history. Overal Overall, l, there is potential for a growing use of such model types in the simulation of clay soils.

Stress-dependent strength A curved failure envelope, particularly for granular materials, can be more appropriate than the common linear envelop e nvelopee of the Mohr–Coulomb Mohr–Coulomb failure surface. The Lade model (Lade, 1977) is an example of a constitutive model with a curved failure envelope. It is useful in simulating granular materials with large stress changes, for instance in the construction of a rockfill embankment where the first layers have very low confining stress but on completion of the embankment the stresses in these layers are significantly higher.

Destructuration Stress changes may cause loss of structure and a change in behaviour. Some models recreating this phenomenon that are used in research may see more common use in practical problems. These include a multi-laminate model (Schweiger ( Schweiger et et al ., ., 2009) 2009) and those based on modifications to the MCC model ( Kavvad Kavvadas as and Amorosi, 2000; 2000 ; Rouainia and Muir Wood, 2000; Baudet and Stallebrass, 2004). 2004 ).

Anisotropic strength Some models consider only inherent anisotropy while others consider both inherent and induced and are formulated in terms of all six components of stress and strain rather than invariants. 46 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.


Model types include a rotated yield surface based on the MCC model ( Wheeler Wheeler et et al ., ., 2003) 20 03),, or on onee ba based sed on the mul multiti-lam lamina inate te fr fram amew ework ork (Sc (Schw hweig eiger er et al ., ., 2009). The MIT-E3 model (Whittle, (Whittle, 1993; 1993; Ganendra and Potts, 1995) 1995 ) is a complex model similar to kinematic surface models in that plasticity can occur within a critical state bounding surface but the bounding surface rotates in general stress space to take account of both inherent and induced anisotropy. In the NGI-ADP model ( Grimstad et al ., ., 2012) 2012) for anisotropic undrained shear strengths of clays and silts, cu is defined for active, passive and direct simple shear stress states.

Creep Constitutive models incorporating creep behaviour are complex because viscosity and ageing effects need to be considered in combination with yielding. More research is needed before models can be applied routinely in practice. One example by Yin Yin et et al . (200 (2 002) 2) is ba base sed d on th thee MC MCC C mo mode dell wh wher eree th thee eq equi uiva valen lentt ti time me co conc ncep eptt of of Bjerrum (1967) was developed to obtain time-dependent stress–strain behaviour based on both MCC and visco-plasticity concepts. It predicts accelerated creep when the stress state is near the yield surface, unload–reload behaviour, relaxation and the effects of a change in shearing rate.

Hypoplasticity Models based on hypoplasticity (e.g. Kolymbas, 1991; 1991 ; Gudehus, 1996; 1996 ; von Wolffersdorff, 1996) may see increasing use in practical problems. They take a different approach to elastoplasticity since there is no distinction between elastic and plastic behaviour and hence there is no explicit yield surface or hardening rules. They are relatively simple since a single stress tensor equation is used to describe the mechanical behaviour of a soil and the model parameters are based on fundamental properties of the soil grains. In particular, they model pressure and density coupling, dilation, contraction and variable strength and stiffness.

2.3.3 2.3 .3

How do con How const stitu itutiv tive e model modelss commo commonly nly ac accou count nt for for rock rock behaviour?

As described in Sections 2.2.2 and 2.2.3, discontinuities have a significant effect on the engineering behaviour of rock. How a constitutive model accounts for the discontinuities is the most important consideration in the modelling of rock masses. Discontinuities cause cau se dis distri tribut bution ionss of st stres resss and st stra rain in tha thatt ar aree dif differ ferent ent to tho those se pr predi edicte cted d by the common elastic or elastoplastic models for soil. They influence strength and stiffness properties in a non-linear and anisotropic fashion. If there are no significant discontinuities in a hard rock mass, then it can be modelled as an isotropic linear elastic elastic material provided provided that the stresses to be applied are truly within the elastic range. Otherwise, discontinuities can be modelled in an implicit or explicit manner as follows:

Implicit discontinuity modelling Mohr–Coulomb failure criterion The strength parameters w ′ and and c c′ of intact rock are reduced significantly to take account of discontinuities. The strength parameters need to be selected for an appropriate stress 47 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.


range because the model has a linear stress-dependency of shear strength whereas rock mass has a highly non-linear stress-dependency. Therefore, it is not adequate for large stre st ress ss ra rang nges es in ro rock. ck. Ano Anothe therr dis disadv advant antag agee is tha thatt the det determ ermina inatio tion n of equ equiva ivalen lentt strength parameters for a discontinuous rock mass involves a lot of approximation and uncertainty. The behaviour of soft rocks is more similar to soils and so the Mohr– Coulomb failure criterion is more suited to soft rocks. Hoek–Brown model This is an isotropic LEPP model with a non-linear shear and tensile strength criterion intended specifically for isotropic, homogeneous, weathered rock. It is suited to irregular discontinuity patterns where no significant anisotropy or dominant sliding directions occur. The input parameters include an intact rock strength, a Geological Strength Index (GSI) based on rock mass descriptions and Young’s modulus. The model has evolved many times since its original version in 1980, with the Hoek Hoek et et al . (2002) version (2002) version being the first suited to implementation in FE analyses, as summarised by Hoek and Marinos (2007). Anisotropic LEPP model This is suited to regular discontinuity patterns where shear is likely to occur along predefined defi ned she shear ar pla planes nes.. Suc Such h mo model delss oft often en use a Moh Mohr–C r–Coul oulomb omb fai failur luree cri criteri terion on,, so would still have the disadvantage of a linear stress-dependency of strength. Cross-anisotropy of stiffness may also be allowed in some models in order to specify a different stiffness in one direction due to the effect of discontinuities in one plane. A disadvantage with such models is that potential failure mechanisms may be missed due to the definition of pre-defined failure surfaces.

Explicit discontinuity modelling Interface elements allow separation and relative relative sliding of elements along pre-defined surfaces where the elements are installed in the mesh. When the interface is in compression, norma no rmall an and d she shear ar st stre resse ssess ar aree tr tran ansfe sferre rred d acr cross oss the ele eleme ment nt ac accor cordi ding ng to spe speci cified fied normal and shear stiffness values. A small or zero tensile strength is specified at which point the element simulates a gap opening. A Coulomb-type friction criterion is specified at which the shear stiffness drops to a specified residual value. More detail on interface elements is provided in Section 5.1.3. Setting up interface elements in an FE mesh manually can be laborious, so some programs perform this task automatically according to various fracture network models. The intact rock between the interface elements can be modelled with a simple isotropic linear elastic constitutive model (if failure modes through intact rock need not be predicted), or more usually with an LEPP model with Mohr–Coulomb failure criterion with appropriate parameters for the intact rock.

2.4. 2. 4.

Typ ypic ical al ap appl plic icat atio ions ns

Table 2.1 provides 2.1 provides brief guidance on the appropriate selection of the elastic and plastic parts of elastoplastic constitutive models for common structure types in soil. Further proje pr ojectct-spe specifi cificc gu guida idance nce can be obt obtain ained ed fro from m FE ana analy lysis sis cas casee st studi udies es of sim simila ilarr projects in similar ground conditions where the selection of constitutive models may be justified and the outputs appraised. 48 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.


s n o i t a c i l p p a l a c i p y t r o f s l e d o m e v i t u t i t s n o c c i t s a l p o t s a l e e t a i r p o r p p A 1 . 2 e l b a T

t r a p c i t s a l P

l , l e t a e e a w c v r i a r f f o r e p o o h o f r n r a n p o e s s i o p i h e s a t n i c n t a i , f v d t f d m i a e t u e c r n r e s i m p m u r x e i q e r n n o i d e g m o i r r f m s o e g g r d s d n e n e i n i d n h a i r d o n e i v f g s d o e r h n r y b o a h i p c ( t h a t y . i i c r s ) t i u e s e w d c c c c i r e a n e t r . o c s c r f a f r i p s o a o l f l f u t n d s p o a a e i o t r t e u r l n v t e i c e t o b b o i c e r c p r d f u m p r g u p r e e r u o o m e P d t s s D c i p

h t i w n o i t a n i b m o c n i g . n t i n r e a p d r i c a t h s a e l c e a r f e r d u s r o e r l b e u h g o i D h

. g n y g s f l n y i t i o n n a e h i l e t g a c f i u l r l f . t o f s s a i s o y t v t c s e l l h t a t y i a l c t d l l s a i u a n u i a c w d l a i t p m s y r t s m s l t e a i i i s l o c c r o p i n i r d s o t r e o r t f f s i o r a l a d f e p l e t r s p u i r g u o t n c t n u i n t i a p r q e f e u n s y r g e r e l r a - . d u e t r r l s i p s m r c o i a a f s f o i e h d y p s l y e t s t a e e t i v u d a a i v c c n i , n r a i s y d t a e f l l s a s . s r a d e r e e a n u o l s v e r i s p i s g o n l t P u o t l a k c l o i e r a c i p n 1 r b c a - f r e c o u r f e f r e d l , o e r v e e p n t P S a A c w D o p

t r a p c i t s a l E

g n g n i t i r s s e o a m d p d e r u r p r n f c i m o s u a a o n n f e s e , n i i y c n s r , o i s i l o e s m c f - s e t c e s a c r n n i l o n e r o f d u a f f f f n i e d c r i t i r r u t s v c t a s h p a c ( o t t e i r f n r h c p o u e r i w s e t v t s l y e s t , o . d d l l a n e l a r l n e e d e v e u g a e p i o o v r s e w r e v a m a u d f e p d o e ) - a h o ) e m n i m i h o l h t n t t a n g i y t o p o c a d i e r i c . a p e i t n ) i p d p t ( c d s r s i r a i o o r d n e u s u v r c t h c e q a p t r e e i p r c r e c o x o b s w a p ( f A t I s r e

h t i w d e n t i b n e m d o n c e p s s e e d n ) f f h i t t s a p t ( n s e s d e r n t e s p r . e a e s d s n e - i n l i - n a n f f r i o t S n t s

s d l i . e t t o c s a e i d r d i l a m e l r o u s o p c n n e o a o b t r c g - s l t r t u r e o c v n f o e fi f c i m i y e d l t t c s h l a e l r g a e i a p l s r . i t s o a t d u e y n a l y p b i l l c l e t n a e c f r a i f i c c p t m , i s r fi o o s r t d n y n a i o g s n n l I a I c s

s c i t s i r e t c a r a h C

. h m t i u w i d e h t a . m p g o s n t s i e d w r a l o t s l o l e g i v n c r l i e d o a t n a i o i l a v n e r t U d S

. w o l . o t h t a w p l o s s y r e r e t v s l g e n v i e d l a i n o a l n r t U S

y l s f o l i a d o t e n h t m t i r a e g o t o i u n i c n d l d e t r d n o t s l i s e r n u . c l s i p t o fi l n c o - f i . e s e e r v d s l e f h m e T t v e c o . e c o b e i n n t a y f a n l f o l n i p t a e r t t a t s i h r c d c p s i g o i s e h d p l i l l i g e r m o o o e r i P i s t s c H

e p y t n o i t a c i l p p A

l l l a a r w u t g c n u i r n t i s a t d e r n a d n e o d i d t s e c e e b fl c m e r o E d f

g n i l l e n n u t d r n o u o s t n r g n o i t d e a e m v c e a u v c d o x n E i m

s e p o l s t u C

′

′



d e u n i t n o C 1 . 2 e l b a T

t r a p c i t s a l P

r r a h l t e u g h n y t n . a o a s e l r y r c r t g a s o l t f s c l - . r i f r i o l o t f i y o e s s g v t h s o n r i s i g t c e o i n i o n t t f n e g n e a s t h o a a r e t r i g d r l t o n a a p e v i n e r e p n h l . d - v a m e e a n w p m i o n i d u c l r a r o a o k h t a f e n l , t n t f i g a h r e f a u e r b n y d e s m o l c e l h r o a r c m e a a l l e t l t s f h r b fi g u m t t e i u c i u c f n i s o d n t a e w r p s e r e d n t a a o d f d l s , s p r e o n g s e c , t s n n n i o i u n p s o i o p t s i i e o d t t o n e i t a g b t s a a r t t c s a d d o r i n t i s e n n i s o s s d l e l . p u u u e u t o o n r r t l m f A f f A i I m p S fi

t r a p c i t s a l E

) g r n i o d f t e a o n g r l a . a t r l y r s n o o a o i p t m t i e a m r i u p d ( s s d r n e n a u o n i o e f f f t n i y l a i t a - l s d c t n n t n u o f e o N o d f s . s n d e r e n g o p a n f e l a s l d - fi r s e s s h s n s f e t e f r r i o t S b t s t s

s c i t s i r e t c a r a h C

m y n l e l i l b a y o p r m r n o n p o i o – g r a t n t g g a n i n o s s n n t i a n i o d i d a . n h o l n d n d s a s t a g u a u l a o e t l o i o l i c n f l m e g v i n n f s i e a o n f s n d l u d l . f c n e . o a i l i n fi o e n i t h fi r l s i t o t s y a i g i i a n u r w n t r d w i e n o o t g a a o i r r i l t r v d s i , d v p a o ) d s r w h o a m s s a y n e m i l p h o n i g u a r l s i n o p o o e e r e o m C ( c s r r d H A f i b

d n s s a ) e r e t i s l p g n i n o d i . a s n s o s l e t e n s r s o s t i e c s l s n i r e r ( u o p t a i h m t o a v e C p d

t n n e i o m t k c n u t a r b s n m o E c

n o i t a d n u o f d e l i P

e p y t n o i t a c i l p p A

l c e i s i d e r t l o g e i y a t r o i m s i e m c v a o t n s t a r a o a i p e e t l l s r a p b u e m d i s r h s n w e t o o o o s f u l a r p o f i n h c e e i t h p f y b o l c d r t w i s a o n h n s l a . e u n i o w s s c n t i s s y I f t f a a n e e r i . n f n n s t i s a t f p m f f b i i t c , i t t i m n e s s t p t n u o m c n o a b c e i r o t t r , i r p n i o o y s r s o i t d y i s u p q u n l a o m c h s t A i t e r i s

o s , r o l n f i o i t o s t a n – l e l i e l c i a t p s fi f n u n i s o l e y i y l l i p b v g a a b n e i r h w o s l o p d l y t n f o i c e i p s t e e s a . d i t l i p l r r t o u e p s o o c i e r g v f n a p r i h e e d p n e b c a l u f l r e a e p o r i x t i m r n A i s u s t n a t d r o e t o t a s p . t l t s d i m u n l i e c o d m s a fi s n d s f m i e a t n e l o t d n f n c - t e f g e r n i e r i t s l a e s l m v t fi k o p t r n y e u n i e b e o a l b t r t d v h c n n r , m g s a e e i e s l c p e r n fi e r o o i o i d g f t t n y a g h n n l l d i t i s t a a w a n . l p o m u e t - u d o m i r f c s s i m w o d e n y o s a e r a c r e t f l n c b r S i d I p

d n r a o n f o i s s t c n e e i n fl f d f i e e r t s d i u r l a q e r d e t r r e a o l g r r n e o i n h f e i g s d r h n . o s a i t t h h t i c c e w i e f c d f e a r n e f r i o p p u t s a d u n e e i v o r l g b b o r u m p e l o o i m p D c i

e l i p y g r l n a b a i n w e b o o n o l r y l i p l i l o c ) l . v f i i a s p h o e e o t p s r h i t t e s r o d g n s d e i i n p e h d t n e o l i p r p w u p e m g o r d e i r s i n r c u a s s u f o r a s o i e , e r v t r o a i n n c f o n t h l i e i t i ( n e a c i b o l s l l – a i c t a e t s fi i f l s l a x i n e u A p i s

d a o l l t a n r e e t a d l n e e p t e a r d u ) c c . h t a s a r n p f o o ( i t s c s d i e e d r r i t r s u e p , q r e a r n o e s i t n s i l c - e n e n f i fl o f e N t s d

. s l e v e l n i a r t s w o L


t d s e n r i e u m q e e r c a . l s s s p s p e i n d u f f r f o i t g o s e t n l o i n i e t p c n d i n d i e e s n p r o e p i t d - e t c n a i r a r a e u r t t s c n c i a o s r d l o n A f a


r o f e t a r u c c a y l t n e i c fi f u s . y n l b i o a t b i c o r d p e r p y t i e c r i t u s l i a a l f p t g c i n e r f r a e e P b

) l e s s v e t e l n f n n f i e i t m r s a t e s l h t t i d t e w n ( a s r c s o i s f t s e r a t t l n e s i e r r o c a f fi e f n e u i s . l t a y n i o c i r l i t p p b c a i o o r r d t p b o o r e r s p I a p p s s c e i r r t s o t a g i n i v e d d a o e l n m o o i s s s d e r n . p a s s h m t e o a r C p t s

n o i t a d n u o f d a e r p S

n o i t a g a p o r p e v a w c i s a b r o f d e r i u q e r t o n y . t s i c i e i t s d a u l P t s

, r e o n a c l s n i i n a o t s s h i s i n t g e n r a t e g r t e r n c s i r n r a i e l a e b e h s d s e o e c u m g a d i c r e r p e v a o h r t t f g o o s n i l . e n n e r o i d t s a t c o c n e m i a l p g e s c o r n l i t i r f u p i o o r s u i q e t r n e a o s A r c i

r a e n i l i l - . o s s n s – o e t n f n f a f i r h t t i d w s t n a s n e t n d o n i n e t c e i m p e d e l e t d t r p ) e s n h t o a d i e t ( p v c o r a r s s p e t e r m n t I i s

. s l e v e l n i a r t s m u i d e M

n o i t a t r s n t e h c t a i j d w a d s t e n v e o r m p e i . c m s a r s l e e p n s h i t f f d r i u t f f o n s t n i o n o t e i t a d c d n i d n e p e e r u o d P f

, d e s e e i d n y u t s a e m k g d a i e u n i s l q d a a h t o i c r l a c e i e l p s d c n y c g n a r i r i e c i d u m n q . a u e s l n s r , e y n d d i e d a r o n r i t e m s d a r i e t t v s n s i n t o h e u n c t g a i i h m e t s r r b n e o o F p o t c

d l u o s c l n s e e o e b n d i f . d o t l s i a f u e m g t s o g c a n i p i w n t a a s o r r s r a p t s l s n e e e l n i a r v l f r f a a a t i s e w m t s r n s i l i t e c n g c s h t i i e r a a p b d l w o n r r t r e o t o u p f o f b s e d i , d s d - e e e r l s i e . i n i p u d ) u a 0 r m e u i t q t G S b s ( S e r

e k s a i e u c t s b q i r o h t t d r e a t e c d s e a i l r e e a d a i n h n c a c y e c e a p i s g m n m g . a a g i n l s n h i n i r e y c d d e a u d q o n m o e i l r m a u i c , r l t l d e v s o c i e v t r h , y c e u s g i i i d t e r e i h d s t s r u d n n n o o o F t s u c c

r c l e e i b t l i a c s e . r y c s s l g , o s p e : l v i s e d e s v n l y l e a n l a i n s a n i t r a a c . t s e r e n t l f i s f l k o a a f e t u o g c m a s q e i n f h e y g t d r r n a u e a q o i V E a r l l

s i s y l a n ) a i n c a i r m t s a n w y o D l (



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Chapter 3

How are soil and rock parameters obtained? 3.1 .1.. 3.1.1 3.1 .1

Introd Int oduc ucti tion on Why is it it diffic difficult ult to to obtain obtain ac accur curate ate geo geotech technic nical al par parame ameters ters? ?

Soil and rock are complex materials, often requiring complex constitutive models, a high number of parameters and advanced testing methods. Obtaining the parameters would be difficult enough on specified, quality-controlled, manufactured engineering materials, such as steel or concrete, but soil and rock are not manufactured. They are already in place under the site and have been subjected to natural and largely unknown geological processes for, typically, millions of years. It is usually possible to access only a tiny fraction of the ground volume influenced by a structure, so the vast majority of it is never seen. see n. Yet Yet,, finit finitee elem element ent (FE) anal analys ysis is and adva advanced nced cons constit tituti utive ve mode models ls all allow ow the ground to be simulated in precise detail. Bridging the gap between the uncertainty of real site conditions and the idealised world of constitutive models and FE analysis is the main diffi di fficul culty ty in ob obtai taini ning ng ge geote otech chnic nical al par parame amete ters. rs. It re requi quire ress ex excel celle lent nt sk skill illss in the interpretation of site information and thorough background knowledge of geology and soil/rock mechanics. Failure to obtain parameters to a sufficiently high degree of accuracy will render the subsequent FE analyses of little value, potentially potentially leading to incorr incorrect ect design decisions. This chapter provides guidance on the stages of obtaining geotechnical parameters that are part pa rticu icula larl rly y re rele levan vantt to FE an analy alysi sis. s. Fo Forr fu furth rther er gu guida idanc nce, e, re reade aders rs sh shoul ould d re refer fer to specialised publications in the particular area of testing and site investigation.

3.1.2 3.1 .2

How Ho w is par parame ameter ter test testing ing of soil soil plann planned ed in in a site site inves investig tigati ation on? ?

In order to obtain soil parameters for FE analysis, they must be measured in some way in tests. Tests can be performed either on samples in the laboratory, in which case the parameters are usually measured directly, or in situ, in which case the parameters are usually determined empirically or analytically from the test results. Both testing techniques niq ues ar aree used toge together ther,, wher wheree pos possibl sible, e, to ach achiev ievee mor moree re relia liable ble par parame ameter ter dete deterrminat min ations ions.. But befo before re sele selecti cting ng and unde underta rtaking king exp expensi ensive ve par paramet ameter er tes testing ting,, it is important to form an overview of soil conditions at a site and to characterise the soil into zones (usually layers) – a process called site called site characterisation (see Figure 3.1). 3.1). This may characterisation (see Figure appear more laborious than doing all the parameter testing in one go during the initial site investigation, but can be more cost-effective, particularly when advanced testing methods are employed, because fewer, targeted parameter tests need to be carried out. 55 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.


Figure Figur e 3.1 Steps to obtaining soil parameters for FE analysis Site characterisation Classification tests e.g. CPT, SPT, Atterberg limits, sieve analysis

Logging of samples (visual description)

Form analysis model and assess reliability

Parameter derivation, selection and validation

Form ground model

Advanced in situ and laboratory testing on representative ground or samples Obtaining parameters

Site characterisation At an early stage, the ground must be categorised into a number of discrete zones or, usually, layers within which the soil is expected to have similar engineering behaviour. Site characterisation is performed using both visual description of samples (ISO, (ISO, 2002 and 2004a and 2004a)) and a large number of inexpensive index and classification tests (e.g. moisture content, Atterberg limits, particle size distribution, standard penetration test (SPT), cone penetration test (CPT)). It should then be possible to identify soil types with certain visual characteristics and which fall within certain test result value ranges. Once the soil types have been identified, the geometry of the layers can be estimated by interpolating between the sub-surface investigation (e.g. borehole, trial pit) locations, thereby producing a ground model. During subsequent parameter testing, soil samples and in situ test locations are selected that are representative of each soil layer based on visual description and the same index and classification tests. The many published correlations between index test results and engineering parameters also allow the likely range of measured parameters to be assessed and an d pr pro ovi vide de a pl plau ausi sibi bili lity ty ch chec eck k on th thee va valu lues es of th thee me meas asur ured ed pa parram amet eter erss (s (see ee Section 3.4.2).

How many parameter tests and where? The required number of tests for each parameter for each soil layer depends on the degr de gree ee of unc uncer ertai taint nty y in ea each ch par param amete eterr an and d the sen sensi siti tivit vity y of th thee ke key y FE an analy alysis sis outputs to each parameter. Annexes P, Q and S of Eurocode 7 Part 2 (CEN, (CEN, 2007) 2007) provide guidance on the number of tests to perform depending on parameter variability and experience. Sampling and in situ test locations should be selected where the soil will influence significantly the behaviour of the structure being studied.

In situ or laboratory testing? In soils that are not conducive to high-quality sampling (e.g. sand, very soft clay, seabed sediments), parameter tests on the soil in its natural state can only be performed in situ. 56 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.

How are soil and rock parameters obtained?

Table 3.1 Exampl Examples es of dominant stress paths and test procedur procedures es Construction activity

Dominant stress path

Example test procedure

Sett Se ttle leme ment nt du due e to la land ndfil filll

Com ompr pres essi sivve st strres esss pa path th

K 0 consolidation in triaxial test

Embankment construction on soft clay

Undrai Und rained ned (de (devia viator toric) ic) loa loadin ding g

Consolid Conso lidat ated ed und undrai rained ned triaxial test

Excavation

Unloading in mean stress, loading in shear

Triaxial extension test

In soils conducive to high-quality sampling (e.g. clay), laboratory testing is preferable but, funds permitting, a combination of the two is certainly desirable. The advantages and an d dis disad advan vanta tages ges of ea each ch typ typee ar aree sum summa maris rised ed in ma many ny te texts xts (e (e.g .g.. Clayton et al ., ., 1995). The principal advantages of in situ testing are that larger volumes of soil are tested, their results can be obtained earlier in the site investigation and the soil is less disturbed. As opposed to this, the principal advantages of laboratory testing are that boundary conditions are more precisely controlled and stresses and strains are uniform throughout the specimen which allows non-linear soil parameters to be determined more precisely.

Which test methods? The critical parameters should be determi determinable nable with sufficie sufficient nt accur accuracy acy from the chosen test method(s) – see Section 3.3. If further selection is required, this will probably be based on economy and the availability availability of the methods. Then consider the str stress ess level and dominant stress path in the construction sequence to be analysed, in order to specify the correct test procedure. Some examples are given in Table in Table 3.1. 3.1.

3.1.3 3.1 .3

How is the site How site info informa rmatio tion n neede needed d to perfo perform rm FE ana analy lysis sis of rock rock masses obtained?

For massive rock, only the intact rock parameters are needed, but for fractured rock, parameters para meters for both the intac intactt rock and the discontinuities discontinuities are required. required. The parame parameters ters are then combined to form a mechanical picture of the whole fractured rock mass. Intact rock parameters are commonly obtained from laboratory tests on carefully prepared core samples, or sometimes from in situ tests provided that the volume of rock tested is free of discontinuities. Testing discontinuities is more difficult, so parameters are more commonly estimated from the observed characteristics of the rock mass and the discontinuities. Consequently, the process of site characterisation plays a more direct role in the derivation of rock mass parameters than for soil parameters. It is also possible to test the mechanical properties of the rock mass as a whole by in situ testing, provided that the discontinuity spacing is small compared with the dimensions of the test. Due to th Due thee var varia iati tion on of the ge geome ometr try y and cha chara ract cteri erist stic icss of dis disco conti ntinu nuiti ities es and the difficulty of accessing them in site investigations, it is not possible to be precise in the 57 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.


definition of fractured rock parameters for FE analyses, so parametric studies play an even greater role in rock mass modelling than in soil modelling.

Site characterisation Requirements for the characterisation of rock are described in in ISO (2003). (2003). Index and classification tests include porosity, density, hardness and abrasiveness, as well as the point load strength test which is used to derive a value of the tensile strength of intact rock. In the visual descriptions, a lot of emphasis is placed on describing the discontinuities. Ideally, large exposures can be examined, but this is not always possible and then the descriptions would need to be based on cores from drillholes from which it is more difficult to obtain a comprehensive picture of discontinuity patterns. Not only are the character char acteristic isticss of the discont discontinuity inuity itself describ described ed (e.g. colour, texture, texture, wea weathered thered state, estimated strength) but also their geometry (e.g. orientation, spacing, persistence). Often they are found in parallel sets, otherwise they need to be considered individually, which is a painstaking task.

How many parameter tests and where? As with soil, the required number of tests for each parameter for each rock type depends on the degree of uncertainty in each parameter and the sensitivity of the key FE analysis outputs to each parameter. Due to the stress-dependency of rock strength, Hoek strength, Hoek (2000) recommends about five strength tests at different confining stresses to obtain the strength parameters for the Hoek–Brown model. Coring and in situ test locations should be selected where the rock will influence significantly the behaviour of the structure being studied.

In situ or laboratory testing? Laboratory testing Laboratory testing of intact rock on core samples is more common than in situ testing. testing. In highly fractured rock, in situ testing may be more appropriate due to the difficulty of obtaini obta ining ng a suffi sufficie ciently ntly lar large ge tes testt spec specimen imen for lab labora oratory tory tes testin ting, g, pr provi ovided ded tha thatt the dimensions of the in situ test are at least six times bigger than the discontinuity spacing in order to obtain sufficiently representative parameters for the whole rock mass.

Which test methods? The triaxial compression test is preferred for obtaining the compressive strength of intact rock at different confining stresses, as well as Young’s modulus. Unconfined compression tests do not obtain the non-linear strength of rock due to stress-dependency. For detailed studies of discontinuity behaviour, triaxial testing of jointed rock is possible, as is direct shear testing on discontinuities, but more usually laboratory tests are performed on intact rock specimens. In highly fractured fractured rock where the blocks of intact rock are too small to form laboratory laboratory test specimens, in situ testing is more appropriate. Such tests obtain the strength and stiffness parameters of the fractured rock mass rather than the intact rock blocks, so are usefu us efull in val valid idat ating ing con const stit ituti utive ve mod model el par parame amete ters rs by tes testt sim simul ulat ation ion.. In si situ tu te test st methods used for rock include the pressuremeter and plate load test, as well as the SPT for index testing. Seismic testing testing is also avail available able to obtain the very small str strain ain stiffness. stiffness. 58 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.


3.2. 3.2. 3.2. 3. 2.1 1

Soil and ro Soil rock ck sam sampli pling ng and gr groun oundw dwat ater er mea measur sureme ement nt Why Wh y is sa samp mple le qu qual ality ity im impo porta rtant nt? ?

If realistic parameters are to be obtained from laboratory tests on soil, the sampling must be of the highest quality in order to ensure that the sample in the test, on reconsolidation, represents as closely as possible its behaviour in the field. This is particularly so for advanced laboratory testing where the measurements are particularly sensitive to sample disturb dis turbance ance.. Jus Justt a mino minorr dis disturb turbance ance coul could d re result sult in unr unrepr eprese esenta ntativ tivee st stre ress–s ss–str train ain behaviour, even with proper reconsolidation techniques. Advanced sampling and testing techniques still give data scatter and one important contributory factor is sample quality. The goal is to minimise the causes of disturbance as far as is reasonably possible. Similarly, when obtaining rock cores, drilling and coring methods should be selected and executed carefully to ensure that the features observed in the core are inherent rather than induced by the coring process itself.

3.2.2 3.2 .2

How Ho w ar are e hig high-q h-quali uality ty sam sample pless obt obtain ained? ed?

The least sample disturbance is achieved by block sampling from trial pits or from large diameter boreholes (using the Sherbrooke sampler in soft clays). These methods are usually too impractical and time-consuming for commercial use but are available to obtain the highest quality samples (refer to Clayton et Clayton et al ., ., 1995). The more common sampling methods to obtain the highest quality Class 1 soil samples (as defined in Eurocode 7 (CEN, 2007)) required for the measurement of soil shear strength and stiffness are listed in ISO in ISO (2006) and (2006) and are among the Category A denoted sampling methods. In essence, the triple-tube corebarrel for rotary drilling in (stiff to hard) clayey or clay soils, or thin-walled open tubes or fixed piston tubes in soft to stiff cohesive soils are the only commonly used sampling methods that can obtain Class 1 samples in fine-grained soils. In rock, the Category A methods that can obtain samples with no or only slight disturbance of the rock structure structure in all rock types, as indica indicated ted in ISO (2006), are the triple triple-tube -tube corebarrel and wireline (double or triple) corebarrel. The porosity of granular soils is too high for samples to remain intact when the stresses around them are released. released. Specialised sampling methods do exist, e.g. pore water freezing or resin or grout injection, but resulting volume changes may still prevent a Class 1 sample being obtained. Class 1 samples may be obtainable from mixed (clayey) soils (i.e. silts, sands or gravels with greater than 5% clay), provided they are homogeneous rather than layered, otherwise lower quality samples would be obtained.

Tube sampling High-quality samples can be obtaine High-quality obtained d in soft to stiff soils with hydraulically hydraulically jacked thinwalled open tubes (Shelby tubes) and thin-walled pistons. They involve pushing a tube into the ground and then retr retractin acting g it with the sample inside. The thin-w thin-walled alled open tubes can be used in clays up to stiff consistency. In very stiff clays the jacking forces become high and tubes often buckle or samples break up in the tube, so rotary coring is more 59 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.


common in stiff to hard clays. For more details on these sampling methods, refer to Clayton et Clayton et al . (1995).

Rotary coring Drilling and sampling are combined because the drilling tool is the corebarrel itself. Double corebarrels have an outer barrel with a corebit at its end that grinds away an annulus of soil or rock while the inner barrel is connected via a swive swivell to the outer barrel, barrel, so that it does not rotate as it slides over the core. The drilling fluid flows down between the inner and outer barrels, thereby avoiding the core. Triple corebarrels have a thin wall split tube or plastic liner inside the inner barrel that provides further protection to the core from the drilling fluid, reduces inside clearance and allows easier entry and withdrawal of the core. In wireline drilling the outer barrel forms a continuous rotating casing for the full depth of the hole and the inner barrel is raised and lowered through the outer barrel on a wire line. For more guidance on rotary coring, refer to Clayton et Clayton et al . (1995) and Binns and Binns (1998). (1998).

3.2.3 3.2 .3

What are What are the com common mon cau causes ses of soil soil sam sample ple dis distur turban bance ce and and how how can they be minimised? Tube sampling Reduction in effective stress Stress relief during sampling and extrusion, pore water migration and air entry all cause s ′ red reducti uction, on, part particu icularl larly y in sof soft, t, lo low-pl w-plas astici ticity ty cla clays. ys. The cons conseque equences nces of thi thiss are volume changes that will be difficult to reverse during reconsolidation without further disturbance and destructuration. To help minimise s ′ reduction, weighted drilling mud can be used in open boreholes in soft clay (Ladd (Ladd and DeGroot, 2003) 2003) while boreholes in stiff clay should be kept dry (if the clay is strong enough to resist bottom heave) to reduce swelling. Sampling operations and the preparation of laboratory specimens should be performed with minimum delay. Loss of structure Loss of structure (see Section 2.2.1) results in a shrinking of the soil’s yield surface and a reduction in stiffness. As a soil is approached by a sampler tube, it undergoes compressive strain in front of the cutting shoe, followed by extensile strain as it enters the tube. Such strains can damage the sample and cause loss of structure. The magnitude of the strains can be reduced by using the appropriate cutting shoe geometry (refer to Hight, to Hight, 2003; Clayton 2003; Clayton and Siddique, 1999). 1999). Water content changes Water Shear distortions during tube penetration create suction gradients between the periphery and centre of the samples. Moisture will therefore migrate toward the centre in soft clays resulting in consolidation at the periphery and swelling in the centre, while the opposite will occur in stiff clays. 60 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.


To mitigate this, samples should be extruded and the remoulded soil on the periphery removed as soon as possible before sealing and storing the sub-samples for later testing.

Rotary coring Provided the drilling is performed in a skilled manner, the main source of disturbance in cored samples of clay is reduction of effective stress s ′ , which is greater than with thinwalled tube samples. This occurs because of the complete stress relief around the core and the presence of drilling fluid that softens the outside of the core and fissure surfaces. Use of a triple corebarrel system helps to alleviate this. The driller should also select drilling fluids and corebits that help to avoid s ′ reduction, and undertake careful drilling.

3.2. 3. 2.4 4

Can Ca n so soil il sa samp mple le qu qual alit ity y be be me measu asure red? d?

An evaluation of strength and stiffness testing data must include an assessment of sample quality because it may explain anomalous data or indicate which are the more reliable results. Destructive techniques exist, which can only indicate sample quality during or after af ter a tes test, t, whil whilee mor moree usef useful ul nonnon-des destru tructiv ctivee tec techniq hniques ues ar aree st still ill in dev develo elopmen pment. t. Non-destructive techniques would allow the selection of the highest quality samples for testing while portable methods would even allow sample quality to be assessed on site to aid the selection of sub-samples. This section contains a summary of examples of both types of technique. De / e0 measurement

(destructive technique)

The ratio of the change in void ratio De over the initial void ratio e0 of a sample is measured for reconsolidation of a sample in a triaxial cell or oedometer to its in situ vertical effective stress s ′v . A theoretically perfect sample would behave in an undrained way (zero volume change) on sampling and reconsolidation and the ratio De/e0 would be zero. In reality, some disturbance always occurs and the ratio De/e0 is above zero. The lower the value, the higher the quality of the sample and Lunne and Lunne et et al . (1997) assigned (1997) assigned quality ratings from very poor to excellent to De/e0 ratios depending on the over-consolidation ratio (OCR) (for example, less than 0.03 for excellent and greater than 0.10 for very poor for OCR of 2 to 4). Lo 4). Lo Presti et Presti et al . (2001) stated (2001) stated that the De/e0 measurement alone is not sufficient to take account of soil type and noted a clear decrease in D e/e0 with increase in the plasticity index of Pisa clay, so they proposed lower limits than those of Lunne et Lunne et al . (1997) for high plasticity clays.

Suction measurement (triaxial) (destructive technique) ′ Prior to sampling, a soil element is subjected to in situ effective stresses s ′v0 and s h0 . On sampling, total stresses drop to zero and in a perfect sample the suctions, or residual ′ effective stress p′r , would equal the mean in situ effective stress p′m = (s ′v0 + 2s h0 )/3. In ′ ′ reality, however, p however, p r is much less than p than p m due to sample disturbance. The sample quality ′ can be assessed by comparing the values of p of pr′ a and nd pm . The drawback of this method is the need to estimate or measure K 0 . Indeed, this method is also used to estimate K 0 , so one can be left with a single measurement and two unknowns, unless an alternative method of determining K 0 is used. pr′ is also affected by pore water migration within samples and moisture moisture loss, so, on its own, it is an insuffic insufficient ient indicator indicator of sample quality. quality.



To perform the measurement, the confining stress in the triaxial cell is increased isotropically during the saturation stage of a test. The value of cell pressure when the pore pressure readings become positive corresponds with the suction pr′ in the sample. The De/e0 measurement procedure described above could then follow during reconsolidation of the sample.

Suction measurement (direct) (non-destructive technique) Direct measurements of suction pr′ can be made on unconfined samples using tensiometers (low suctions), pressure plates or with a portable suction probe (Ridley (Ridley and Burland, 1993). The disturbed outer zone of the sample should be trimmed away before taking readings.

Shear wave velocity (non-destructive technique) Shear wave velocity vs can be measured both in situ (see Section 3.3.5) and in the laboratory using bender elements (see Section 3.3.1). Differences between vs in the in situ state and in the sample will depend on changes in effective stress stress s ′ and in the soil skeleton, so they provide an indication of sample quality. vs can be measured on unconfined samples to allow sample selection, or can be measured on reconsolidated samples where the stress state sta te match matches es appro approximat ximately ely with the in situ stress state. Recent research research sugges suggests ts that a ratio of unconfined vs over in situ vs in excess of about 0.6–0.8 is indicative of a very good sample. The disadvantage with vs measurement on reconsolidated samples is that it does not allow sample selection because the laboratory test has already started. To overcome this problem, vs measurement on unconfined samples can be combined with residual effective stress pr′ measurement using a suction probe in order to apply a correction to the vs value for comparison with the in situ vs value. This allows more informed sample selection but vs measurement should still be performed subsequently on reconsolidated samples to provide the most accurate comparison with the in situ vs value and assessment of sample quality. quality. For more guidance on the shear wave wave velocity method of sample quality assessment refer to Hight et Hight et al . (2003), Landon (2003), Landon et et al . (2007) and and Sukolrat Sukolrat et et al . (2008).

3.2.5 3.2 .5

How Ho w is is gro ground undwa water ter pr press essur ure e measu measured red in the fiel field? d?

Piezometric Piezome tric pr pressu essure re is meas measur ured ed wit with h piez piezomet ometers ers and this sec section tion inc include ludess a bri brief ef description of suitable types and installation methods. Comprehensive guidance in this area is provided by Dunnicliff by Dunnicliff (1993). (1993). Piezometers can be constructed as either open or closed types, as shown in Figure in Figure 3.2. 3.2.

‘Open’ piezometers These are the traditional standpipe or Casagrande piezometers composed of a plastic pipe pi pe th thro rough ugh wh whic ich h wa water ter ri rises ses vi via a a fil filte ter. r. Th Thee pi piezo ezome metri tricc le leve vell is det deter ermi mined ned by measuring the depth to the water level in the plastic pipe. They are cheap and reliable but suffer from slow response times so are less suited to rapid changes in groundwater pressu pr essure re (e. (e.g. g. tida tidall vari variati ation, on, res reserv ervoir oir fill filling/ ing/dra drawdo wdown, wn, pump pumping ing tes tests) ts) and cann cannot ot measuree suction measur suctions. s. 62 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.


Figure 3.2 Common piezometer types

Grout seal Standpipe

Bentonite seal Diaphragm transducer, e.g. vibrating wire

Filter Sand Open type piezometer

Closed type piezometer

Multiple piezometers in a fully grouted borehole

‘Closed’ (or diaphragm) piezometers These use a transducer for direct measurement of water pressure via a diaphragm in contact with the pore water. Measurement methods include pneumatic, electrical resistance and vibrating wire strain gauge transducers. They have rapid response times and can measure suctions but are more expensive and more likely to malfunction than the open type.

Installation A porous filter element (or well point) is installed in a sand zone in a borehole, sealed from the rest of the borehole using bentonite clay or a bentoni bentonite-ceme te-cement nt grout, such that the groundwater pressure is measured in the sand zone only. The filter element can be connected to a plastic pipe (open system) or to a pressure measuring device (closed system). A more recent alternative installation method for closed type piezometers involves backfilling the borehole entirely with bentonite-cement grout, without a sand zone since diaphragm type piezometers require only a small change in water volume to equalise pressures. This is called a fully grouted piezometer and has the advantages of economy and simplicity, particularly for multiple piezometers, as well as allowing other monitoring instruments to be placed in the same borehole. Further guidance is available in Contreras et al . (2008), (2008), Dunnicliff (2009), (2009), McKenna (1995) and and Mikkelsen Mikkelsen and Green (2003). 63 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.


Multipoint piezometers It can be wrong to assume that groundwater pressures above and below a single point reading are hydrostatic, so piezometers should be installed at different depths, either in individual boreholes or in the same borehole. Multiple piezometers in single boreholes can be cost-effective and may be achieved with fully grouted piezometers, by careful installation of a pair of sand zones separated by a bentonite seal or by dedicated multipoint piezometers (Dunnicliff, 1993).

3.3. 3. 3.

Par aram amet eter er te tes sti ting ng

This section provides a summary of the laboratory and in situ test methods commonly used to obtain soil and rock parameters.

3.3.1 3.3 .1

How Ho w is the tri triaxi axial al cell cell use used d to obta obtain in soil soil and ro rock ck par paramet ameters ers? ?

The conventional triaxial cell apparatus is described widely in textbooks and test standards (e.g. ISO, (e.g. ISO, 2004b). 2004b). It is suitable for measuring the shear strength and large strain stiffness of soil and soft rock specimens. Hard rock specimens may require a different set-up with steel cell walls instead of perspex and hydraulic oil instead of cell water in order to apply the high confining stresses required to obtain the non-linear stressdependent failure envelopes (e.g. Hoek (e.g. Hoek and Franklin, 1968). 1968). For improved accuracy, the ram force should be measured inside the cell with a waterproof load cell and pore pressure should be measured at both the base pedestal and at mid-height with a pore pressure probe in effective stress triaxial tests to monitor the equalisation of pore pressures in the specimen. Also, the most accurate volume change measurements measur ements are obtaine obtained d using electr electronic, onic, scre screw-driv w-driven en combined press pressure/ ure/volume volume 3 controllers where volume is measured to a precision of 1 mm by the stepper motor. Larger diameter (75–100 mm) specimens are preferable in order to include more of the fabric of a soil and to reduce the effects of disturbance during specimen preparation.

Stress path triaxial cell with local strain measurement Stress path triaxial cells (or hydraulic triaxial cells) with computer control of cell, ram and pore pressure allow the stress state in a soil specimen to be manipulated along any stress path. Local strain measurement is performed with transducers attached to the specimen itself inside the cell of the triaxial apparatus, usually on the middle third of the specimen, remote from its ends. Local axial strain measurement eliminates the compliance err errors ors ass associa ociated ted wit with h ex exter ternal nal ins instru trument mentati ation on whil whilee an addi additio tional nal tr transd ansduce ucerr attached to a radial belt around the specimen allows radial strains to be measured, as illustrated in Figure in Figure 3.3. 3.3. Local strain measurement on hard rock specimens is achieved by fixing strain gauges directly to the specimen at mid-height for strain measurement in the axial and radial directions.

Triaxial cell with bender elements Bender elements Bender elements are piezoelect piezoelectric ric pla plates tes that can eith either er pro produc ducee a ben bending ding motion in res espo pons nsee to an in inpu putt vol olta tage ge or pr prod oduc ucee an ou outp tput ut vol olta tage ge in res espo pons nsee to a be bend ndin ing g mo moti tion on.. 64 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.


Figure 3.3 Common triaxial cell apparatus variations External vertical strain measurement

Internal load cell

Local vertical and radial strain measurement

Mid-height pore pressure probe

Conventional triaxial cell

Bender element pairs on up to 3 orientations (for assessment of anisotropy of very small strain stiffness)

Computer control

Stress path triaxial cell with local strain measurement

With bender elements

When pairs of bender elements are placed on opposite sides of a soil or rock specimen (Figure 3.3), one transmits a shear wave while the other detects its arrival and the velocity vs of the shear wave propagated through the specimen is determined from the time interval between transmission and reception. Shear wave velocity vs is converted to the very small strain shear modulus G0 using the equation G0 = r v2s

(3.1)

where r is is the bulk mass density of the soil. There are several benefits to the use of bender elements in a triaxial test: g

g

g

sample quality quality can can be assessed assessed by comparing comparing vs in the specimen with vs in situ (see Section 3.2.4) soil stiffness stiffness at at very small strain strainss (, 0.002%) can be obtained directly from vs in order to determine the elastic plateau (G (G0 ) at the upper end of the decay curve of stiffness with strain (Figure (Figure 3.4) 3.4) anisotropy anisotr opy of stiffness stiffness can be assessed from from the measurement measurement of vs between bender element pairs mounted in three orientations on the specimen (as shown in Figure 3.3) and using polarised wave sources. Note that this will still not provide enough parameters even for a cross-anisotropic stiffness constitutive model (refer to Lings to Lings et ., 2000, 2000, for further guidance). et al ., 65



Figure 3.4 Typical approximate decay of soil secant stiffness with strain and instrument Figure measurement ranges Typical strains around geotechnical structures

E , G or K ′

E 0, G0 or K ′0

Large strain Very small strain

Small strain

Bender elements

Local strain measurement

~0.002%

External strain measurement

~ 0 .1 %

εf

log ε

More guidance on the use of local strain measurement and bender elements in triaxial testing is provided in Clayton in Clayton (2011). (2011).

Reconsolidation stage It is well known that stress history, stress state and stress path all influence the engineering behaviour of soil. For the accurate measurement of soil parameters, particularly stiffness, these must be recreated in the triaxial test as accurately as possible, recognising that the void ratio will differ slightly from the in situ state (depending on sample quality). In conventional triaxial cells, only isotropic reconsolidation is possible. Soils in the field overlain by flat ground experience an anisotropic (K (K 0 ) stress history of consolidation and swelling (‘K (‘K 0 conditions’ means vertical strain with zero horizontal strain). To recreate field conditions accurately, the soil specimen must be reconsolidated along the same stress path as in the field and this requires requires anisotropic anisotropic (or K (or K 0 ) consolid consolidation. ation. This can be achieved in stress path triaxial cells by changing the ram pressure while maintaining the celll pr cel press essur uree con consstan tantt (s (str tres esss con contr trol) ol) or pr prefe efera rably bly wi with th zer zero o ra radia diall st stra rain in as mea measur sured ed by the radial belt around the specimen (strain control). Strain control mimics the oedometer test and, an d, sin since ce ra radi dial al st strress and po pore re pr pres essur suree ar aree mea measu sure red, d, all allow owss mea measu sure remen mentt of the theK K 0 value in normally consolidated soils and Poisson’s ratio. Tests with anisotropic reconsolidation are denoted CAU or CK0U for undrained and CAD or CK0D for drained shear. Volumetric strains should be minimised during reconsolidation so that further disturbance to the specimen is minimised. This is achieved by avoiding excessive stress excursions and plastic strains during reconsolidation. It is normal practice to reconsolidate a specimen initially isotropically and then anisotropically to simulate the most recent stress path experienced by the soil. The reconsolidation stress path must be specified by the engineer and requires a good grounding in geology in order to estimate the ‘greenfield’ stress history of a soil and, in built-up areas, a knowledge of the stress history resulting from construction (e.g. loading, excavation or dewatering). Figure dewatering). Figure 3.5 shows 3.5 shows a typical greenfield stress path from initial deposition of a heavily over-consolidated clay: 66 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.


Figure 3.5 Typical stress path and reconsolidation of a ‘greenfield’ heavily over-consolidated clay q K 0 consolidation due to increasing

overburden from geological deposition

K 0 swelling due to decreasing

overburden from geological erosion B A

p′

E D

C

K 0 reconsolidation due to recent

geological deposition to current in situ stress state at E

Field stress path Sampling stress path (approx.) Reconsolidation stress path

by K 0 unloading to D to form K 0 consolidation in a normally consolidated state followed by K a heavily over-consolidated clay, with some K some K 0 reconsolidation to point E resulting from more recent deposition of superficial deposits. On sampling, the clay is likely to suffer some effective stress reduction to point A, for instance. The reconsolidation task is to bring a specimen of the clay back to point E, but if reconsolidated directly back to E (along path AE), this would recreate the correct stress state but not the stress path and recent stress history. The recent stress history of the specimen should be followed, but without large stress excursions excursions that would cause exces excessive sive disturbance. disturbance. For example, the specimen could be reconsolidated initially isotropically to B, then anisotropically by reducing q to point C, followed by K 0 unloading to D and K 0 reloading to the in situ state at E, thereby recreating the geologically recent stress reversal at point D. Similarly, Figure Figure 3.6 3.6 shows shows an app approp ropria riate te rec recons onsolid olidat ation ion st stre ress ss pa path th for a typ typical ical greenfield gree nfield lightly over-consolid over-consolidated ated clay. The specime specimen n could be recon reconsolida solidated ted initial initially ly isotropically to B, then under axial compression to point C, followed by K 0 consolidation to D and K and K 0 unloading to the in situ stress state at E. Note that, due to the inevitable sampling disturbance of a lightly over-consolidated clay, while the stress state may be Figure 3.6 Typical stress path and reconsolidation of a ‘greenfield’ lightly over-consolidated clay

D

q K 0 consolidation due to increasing C

overburden from geological deposition

Field stress path Sampling stress path (approx.) Reconsolidation stress path

K 0 swelling due to

decreasing overburden from geological erosion E

A B

p′



restored, the void ratio will have reduced, resulting in a higher apparent OCR than for the in situ soil. On completion of reconsolidation, creep strains (or secondary compression), which can affect small strain measurements, may continue to develop for some time, particularly in high-quality samples. These should be monitored and the shear stage started only when creep strains have fallen to an acceptable rate, e.g. axial strain less than 0.05% per day as suggested by Jardine by Jardine et (1991), ideally at the end of each stress path stage but certainly et al . (1991), at the end of the whole reconsolidation stage. Clayton stage. Clayton and Heymann (2001) adopted (2001) adopted a tighter criterion of 0.01% per day and found that, provided reconsolidation had not led to yielding, the measured stiffness of Bothkennar Clay and London Clay were unaffected by the recent stress history. Consequently, more attention should be focused on allowing specimens to creep (perhaps for a period of weeks) than on following complex drained loading paths during reconsolidati reconsolidation. on. The added benefit of allow allowing ing creep to complete is that it can mitigate, to some extent, the effects of earlier sample disturbance.

Shear stage The st The stra rain in ra rate te sho shoul uld d be mu much ch lo lowe werr tha than n si simpl mply y to en ensur suree equ equal alisa isati tion on of por poree pressures, particularly in the early part of the shear stage when the small strain behaviour of the specimen needs to be captured. An initial axial strain rate as low as 0.05% per hour is typical, increasing to 0.2% per hour once a vertical strain of 0.2% has been passed. Such low strain rates allow the effects of creep to become significant – hence, the importance of minimising creep at the consolidation stage. An alternative is to employ stress rather than strain control in the early part of the test.

Stiffness measurement In the externally instrumented conventional triaxial cell, the axial strain will only become reasonably accurate at strains larger than about 0.5% (less for soft soils) due to compliance effects. With local strain measurement, stiffness measurement is more accurate in the 0.002 to 0.1% strain range (see Figure 3.4), where stiffness is particularly sensitive to st stra rain in lev level el,, the there refor foree mo most st of the st stif iffn fness ess dec decay ay cu curv rvee is obt obtai ained ned in thi thiss wa way. y. However, to locate the upper plateau of the curve at strains below 10 ( 0.002% in clays, 0.0001% in sands), bender element testing needs to be included. Obtaining the complete stiffness decay curve is not only a requirement for using strain-dependent stiffness constit cons tituti utive ve mode models, ls, but also all allows ows app approp ropria riate te sti stiffn ffness ess valu values es for par particu ticular lar str strain ain levels to be selected for non-strain-dependent stiffness models. It is very useful to include an unload–reload cycle in the shear stage in order to obtain the elastic unload/reload stiffness, as well as Poisson’s ratio in drained tests.

Anisotropic stiffness Provided the bedding planes in a specimen are horizontal, three different shear moduli G moduli G can be determined with three pairs of bender elements (refer to Pennington to Pennington et et al ., ., 1997, and Lings et Lings et al ., ., 2000): g

Gvh : vertical shear waves oscillating in the horizontal direction between the bender elements at the specimen ends



g

g

Ghv : horizontal shear waves oscillating in the vertical direction between a bender element pair placed on radially opposite sides of the specimen Ghh : horizontal shear waves oscillating in the horizontal direction between a bender element pair placed on radially opposite sides of the specimen (offset from the other radially opposite pair).

Isotropic consolidation test Consolidation properties can be measured during the consolidation stage of triaxial tests or by dedicated isotropic consolidation testing on samples with a lower aspect ratio of one (to shorten consolidation times). In conventional triaxial cells the consolidation is isotropic (in terms of stress, but not necessarily in terms of strain) as opposed to onedimensi dim ensional onal (K 0 ). So, the so soil il pr prope opert rtie iess der deriv ived ed ar aree for is isotr otrop opic ic ra rath ther er th than an K 0 consolidation. Bulk modulus K ′ can be obtained with precise pressure control and volume change measurement (although its decay with strain is very difficult to measure accurately), as can the Modified Cam Clay parameters k a and nd l , provided the specimen is consolidated along both an unload/reload line and the normal compression line, but all more accurately with local strain measurement.

Anisotropic consolidation test Stress path triaxial cells can perform anisotropic consolidation tests by stress or strain control, including K 0 consolidation which is equivalent to an oedometer test. There aree sev ar sever eral al adv advant antag ages es of con consol solid idat ation ion te test sting ing in a tr triax iaxia iall ce cell ll ov over er a st stan andar dard d oedometer: g

g g g

g

less specimen specimen disturbance disturbance than that associated associated with with inserting an oedometer oedometer confining ring into clay no boundary effects effects associated associated with specimen specimen bedding into a confining confining ring degreee of saturation degre saturation of the specimen in the apparatus apparatus can be measured measured measurement measur ement of radial stress stress and pore pressure pressure for complete determinati determination on of stress state larger specimens specimens include include more more of the soil’s soil’s fabric.

Permeability test Direct measurement of permeability is also possible in low to intermediate permeability soils in a triaxial cell, with the advantages over the permeameter that total stresses and pore pressures equivalent to those in the field can be applied to the specimen and saturation of the specimen can be verified. The effective stress can be changed in stages to assess the stress- or void ratio-dependency of permeability. The test can even be perform fo rmed ed as an add addit itio ional nal st stag agee in a st stand andar ard d tri triax axia iall co compr mpres essio sion n te test st,, bet betwe ween en th thee consolidation and shear stages, although it will take longer for steady-state conditions to be established established in a 2 : 1 aspect ratio specimen specimen rather rather than a short specimen specifically specifically prepared for permeability testing. The derivation of permeability values is covered at the end of Section 3.3.7. 69 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.


k

d e t a m i t s E

u

E

s t s e t y r o t a r o b a l n o m m o c m o r f e l b a n i a t b o s r e t e m a r a p e h t f o y r a m m u S 2 . 3 e l b a T

d e n i a t b o s r e t e m a r a P

: n n e l i o o i p t t a m a r d i a u l s t o f a s o s n o t g c n n i e r c i u m d 0 p e r n K o r t u o : s i o e s a t g i e c a n M u s t s A

k e n l o : i p t n o m a r i t a u s t a d f a i s l o o t g s n n i n e r o u m d 0 c e c r n K i p u : o s i o r a t e t e c g a o s M u s t s I ′

p

y l e t a m i x o r p p A

s : ′ n E : s o : n s i o e n i r s s i t t o s e a r s s d n p e i l i o r m p o t s o a c m n d i o c l d o e c c o s n d i n i p e o a r n o r c d i t n a r o e r s p U D I

y l e t a r u c c a y l e v i t a l e R

, ) i c y a s l C h ) t t ′ g m c n n a e , e ′ C r c t w m c ( s , , ; e ′ r ′ k u c c e u , c s , v ′ , i : a ′ l s w s n e w : o m : n : e i o n i n r s s e i t o o p k i e a m s r r : s s d s o ′ t i p u s l e e s c r o K e r l m s e p s t p s a o r u y m i c p m n l x t o o c u i o i a l d e i c d b c n e r c o i n o a d u i p d e p m e e ′ , a o r h i n r k m i n d t t l r a c a n i r o u e r ′ , w D I s B U ( P D w

n e m i c e p S

y a l C

s u t a r a p p a y r o t a r o b a L

n i a r t s g l n i a t s n r e t e t h x ) e t t ( n g t s e n r e m e t t r s l e r a u i a x s a e a e h i r T m s r . o F 1

′

E

: n o i s s e r p m o c d e n i a r D E

∗

d k n c a o S R

: n o i t : a n d i o i l s s ′ o s e r n n p i o o c m t a c o r i c s p ′ n o d ’ r , t ′ p e n o o n s i s s s i , a r i n 0 o D P A K

′

n

: n o i s s e r p m o c d e n i a r D n

u

E

′

+ E

e + v o e b v d e a o ′ o E b s a a : : s n o n a i o : t i a n d s s i i o l e r s o s p s e n m r o p o c m c c i d o c p e n d o r i e t a r i n o s d a i n r n U D A

y a l C

′

E ,

c ′

,

′

,

c w

: n o i s s e r p E m o + c e d v o e b n a i a r s D A

∗

d k n c a o S R

r h o k t i t c g t w n h i o r n i e t t w d s s e m t r e t e a s t l r e h s t u a i s l n s x a a e o a i n e i f x s r f t a e i m i r g s t s h t n t i u n e a a t r d r a p t g e a a s m r s n i u s l d i a n a c c e e r c c a r p t o t S l s t s s a r . o F 2



k ′

,

′

E E

u

c ′

p

s

,

d d e e o ′ o ′ ′

E E E

,

c ′

,

′

,

c w

′

r

′

,r

r a e h s y t i u n i t n o c s i c d h t , ′ , g ′ c c n e r ′ , ′ , t w w s

c w d

h h t t g g n n e e r r t t s s l e a c a u f d r i e s t e n R I d d

0

G +

e v o b a s A

∗

l l A

t n e r d e n d e n p e e b d h n t i i a r w t s t s e s e t t t e l n l a e p i x m m s a e o s e i r l n T e c f r i f . o F t s 3

∗

k y d c a n l a o C S R

y a l C

s s e n f f i t s t ) s 0 e t K r ( e i c t e m m o o n d o g e c n i O e r t . o s t 4 F e

x o b r a e h s ( t s ) e r t a r e a h e s h s g n t i c r e r d i n D a . 5

d k n c a o S R

r a e h s e c a f r e t n i d g n i a n l t s a e u t d h i s t g e r n r e o r F t s

l e d e b fi t o e n h t y o a t m e n t u e t i t m i s c n e o p s c e e r o h t t t r a o h t ) . r 2 e 2 . b 3 m n e o m i t e r c e t S u e b e , s d ( e y t i s t l e a t u n q e h m i g i c h e a p s t e o h e t l r p o m f y a c s a o r t u t c l c u a c f fi f o i e d y e r r g e e v s d i d h e i c t a h c i d w , n d i n e a h t s r e o v f e e i h s c a a c y e a h m t y s l r t s l a e u t c i y r t o r a t p a r s o i b i s a h l T e s . e d l h t e fi m e o r h t f n d i e l i n o i a s t b e o h t s r f e o t e e v m i t a a . r t a n n o p e i t i e s e h r d T p n e o ∗ r c



3.3.2 3.3 .2

How can How can the the other other stan standar dard d labor laborat atory ory tes tests ts be be used used to obt obtain ain soil and rock parameters?

Triaxial cells are the most versatile and accurate of the common laboratory test methods. Other methods include the oedometer and direct shear test which are generally generally less accurate but often more economical and can provide useful parameters in some instances, as described in this section. A summary of the parameters that can be obtained accurately from all these common laboratory test methods is provided in Table in Table 3.2. 3.2.

Oedometer test The standard oedometer with incremental loading provides limited information on the stress–strain behaviour of soil specimens due to the small number of equilibrium data points. Alternatively, constant rate of strain (CRS) oedometers (described in in ASTM, 2012) apply constant rates of vertical strain to the specimen, slow enough to be comparable with strain rates in the field. The pre-consolidation stress s ′p of a specimen can be identified more accurately because a continuous plot of void ratio/ln p ratio/ln p′ is obtained rather than an incremental plot. There are several advantages of the triaxial cell over the oedometer for measuring K 0consoldiation stiffness, as described in Section 3.3.1, but these advantages are likely to be less significant in firm to stiff clays. Permeability values derived from oedometer tests are generally too low and as much as three orders of magnitude in error. Many of the drawbacks of the Casagrande oedometer can be overcome by using the hydraulic oedometer (described in in BSI, 1990) for 1990) for either consolidation or permeability testing. It is generally more expensive than using a triaxial cell but its advantages over the triaxial cell are control of the drainage direction and the possibility of using larger specimens (refer to Head to Head and Epps, 2014). 2014). The oedometer can also be used to measure the compressibility of intact rock specimens.

Direct shear test Since neither pore pressures nor principal stresses can be determined in standard equipment, the direct shear test should not be used for the general determination of shear strength parameters for FE analysis. However, measurement of the following parameters is particularly suited to the direct shear test: g g g g

interface shear strength d between between soil or rock and other structural materials residual resid ual shear streng strength th of rock rock or soil shear streng strength th of rock disconti discontinuities nuities shear strength strength of coarse-grain coarse-grained ed soils, provided provided that that the in situ density (for ( for peak strengths) and particle size distribution can be recreated.

3.3.3 3.3 .3

How Ho w ar are e para paramet meters ers obt obtain ained ed fr from om in sit situ u tes tests? ts?

In laboratory testing, elements of soil or rock are tested where stresses are uniform and can be represented by a single stress point or a single stress–strain relationship, whereas with in situ testing different locations around the test experience different stress paths and stress levels. 72 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.


Consequ Cons equent ently ly,, th thee no non-u n-uni nifor form m st stre ress ss st stat ate, e, co combi mbine ned d wi with th ma mate teri rial al st stif iffn fness ess an and d strength, influence test results and prevent the direct measurement of parameters. Individual parameters must be derived by analytical or empirical means, making assumptions about the other parameters and stress state. Table 3.3 summarises 3.3 summarises the parameters that can be determined from common in situ test methods. The parameter interpretation methods can be classified as analytical (stresses and an d st stra rain inss in the soi soill ca can n be cal calcu cula late ted d wit with h som somee as assum sumpt ption ions) s),, sem semii-ana analy lytic tical al (approximate analysis with broad assumptions) and empirical (direct comparison with struct st ructure ure perf performa ormance nce or lab labor orato atory ry tes testt res result ults). s). The anal analytic ytical al int interpr erpreta etatio tions ns ar aree generally more accurate but some have not been applied widely, so should not be relied on alone for parameter determination. Even the analytical interpretations make use of assumptions about geometry, boundary conditions and soil behaviour. It is important to know these assumptions when determining parameters. In particular, most interpretations atio ns assume undr undrained ained soil behaviour in tests in clay and drai drained ned behaviour in tests on sands. In intermediate soils (e.g. silts, laminated soils, mixed soils), such interpretations should only be applied with great care and engineering judgement. Empirical interpretations are soil and rock-type dependent, so are reasonably accurate usually only for the types or even specific locations and loadings from which they were derived. Site-specific empirical correlations still represent the most reliable method of interpreting in situ test data and they should be derived wherever possible. Analytical interpr int erpreta etation tionss pr provi ovide de an alte alterna rnativ tivee and the they y ha have ve the adva advantag ntagee of allo allowing wing the engineer to assess the impact of the different assumptions on the derived parameter.

3.3.4 3.3 .4

How Ho w are are par paramet ameters ers ob obtai tained ned fr from om the the pre pressu ssurem remeter eter tes test? t?

A pressuremeter is a cylindrical probe usually installed vertically into the ground such that it applies a uniform horizontal pressure to the ground via a flexible rubber membrane (Figure (Figure 3.7). 3.7). The radial pressure and deformation of the expanding cavity in the ground are recorded which allows interpretations of the stiffness and, in weaker soils, the strength of the ground. Calibrati Calibr ation on pr proced ocedur ures es are ess essenti ential al and ar aree des descri cribed bed in tes testt sta standar ndards ds and and Clarke (1995). For instance, the measurement of unload/reload shear modulus requires very precise measurements, so small inaccuracies in the measuring system will result in large errors. There are three main types of pressuremeter, depending on their method of installation, which are described later in this section. Further guidance on each type can be found in Clarkee (1995). Clark Pressuremeter tests (PMTs) can be performed using strain and/or stress control, depending on the type. Stress control is better in the initial stiff elastic phase of the test and then strain control (typically 1% per minute) during the plastic phase in order to record a good number of data points on the stress–str stress–strain ain curve. The press pressure ure at soil failur failuree (when cavity strain increases at constant pressure) is called the limit pressure pL – if it is not 73 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.


Table 3.3 Summary of parameters obtainable relatively accurately from common in situ tests Soil type

Parameters obtainable relatively accurately1

1. Pressuremeters (SBP, HPD and CPMT) For more accurate testing of strength, stiffnesss and s h . stiffnes SBP is particularly recommended

Clay Sand

s h , c u , G

Rock

G

2. Seismic test For accurate measurement of very small strain stra in stiffnes stiffnesss

All

G0

3. Piezocone penetration test (CPTu) For profiling and more approximate strength stre ngth testing of soil using site-specific correlations

Clay Sand

4. Flat plate dilatometer (DMT) For profiling and more approximate testing of strength, OCR and K 0 in soils

Clay Sand

5. Standa Standard rd penetr penetratio ation n test (SPT) For more approximate strength testing using site-specific correlations in soils and weak rock

Clay Sand Rock

6. Plate load test (PLT) For approximate strength and stiffness testing

Clay Sand Rock

7. Permeability tests (all types) Good permeability testing is notoriously difficult but, with care, SBP, packer and pumping tests are the most accur accurate. ate.

Clay Sand Rock

In situ test method

Analytically

Semianalytically

Sitespecific2

s h (CPMT only), w ′ , c , G

c u , OCR ′

w

OCR, K 0 , c u w ′

c u ′

w

s c

c u , E u ′ ′ w , E E ′ k (SBP) (SBP) k (pumping (pumping test) k (packer (packer test)

1

The parameters listed in this table are those considered to be relatively accurately obtainable for FE analysis in some cases. It is difficult to grade the relative accuracy of each test method and parameter as this depends on many other facto factors. rs. Many other appro approximat ximate e rela relationsh tionships ips exis existt betw between een these test results and other param parameters, eters, but these should not generally be considered as providing accurate input parameters for FE analysis, but more as ‘otherr sourc ‘othe sources es of param parameters’ eters’ for para parametric metric study and parameter validation. validation. 2 Site-specific empirical correlations are the most reliable and should also be used where possible to validate the analytical relationships. For parameters not listed above, site-specific correlations may be less reliable because test results are significantly influenced by secondary soil and rock properties.



Figure 3.7 Pressuremeter apparatus and typical test data Pressuremeter curve: Unload–reload loops on loading portion of curve Borehole

Probe

or, on unloading portion

p e r u s s e r P

Loading Unloading

Cavity strain εc Test pocket

p e r u s s e r P

‘Lift-off pressure’

Cavity strain εc

reached, it can be estimated by extrapolating the stress–strain curve ( p ( pL in the Me ´ ´ nard pressuremeter is different and is defined as the pressure required to double the volume of the cavity). Failure cannot normally be achieved in PMTs in rock. Two or thr Two three ee unl unload/ oad/re reload load loops are performed performed to meas measure ure the shea shearr modu modulus lus G. Unload–reload loops on the unloading portion of the test have become more common because there is less creep and so less interference on the main pressuremeter curve. However, similar results should be obtained whether loops are performed during loading or unlo unloadin ading. g. G is det deter ermi mined ned fr from om th thee sl slope ope of the un unloa load– d–re reloa load d lo loop. op. St Stra rain in-dependent stiffness can also be derived from the curvature of good quality loops (see example in Section 8.2.3). Creep characteristics can also be measured using the pressuremeter. If the pressure is held constant at any stage during the test, the probe continues to expand. The rate of creep decreases with time, and a plot of creep against log(time) becomes a straight line after about a minute. Therefore, the pressure needs to be held constant for only 2 to 3 minutes in order to determine the creep rate. Results Resu lts ar aree pr prese esent nted ed as a pr pres essur surem emet eter er cu curv rvee (i. (i.e. e. vo volu lume metri tricc ex expan pansi sion on ag again ainst st pressure). With direct cavity displacement measurement (as in self-boring pressuremeter (SBP), high-pressure dilatometer (HPD) and cone pressuremeter (CPMT)), results are presented as pressure against cavity strain 1c (also called radial strain 1r ), derived from (r − r0 )/ )/rr0 , where r where r is the current radius of the cavity and r and r 0 is the original radius of the 75 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.


cavity at in situ state. Note that this is not necessarily the same as the radius of the pressuremeter probe or the cavity radius at the start of the test. r test. r 0 can be approximated as the probe radius and this is often done initially, but once the in situ lateral stress s h0 h0 has been interpreted from the data, the new r new r 0 value at the point where the in situ lateral stress was restored is used to recalculate strains. The strain at r0 is called the reference strain. Of all the in situ test methods, the press pressureme uremeter ter test has the most well-defined boundary conditions, which allows more analytical interpretations of soil parameters to be derived and permits its straightforward simulation by FE analysis (see Section 3.4.2).

Pre-bored pressuremeters (PBP): high-pressur high-pressure e dilatometer (HPD) and Me´ nard pressuremeter (MPM) In a conventional borehole a test pocket, usually at least 2 metres in length, is formed with smooth, vertical sides at the location of each test. Therefore, the PBP is only suited to soils where where the walls of the the test pocket pocket will not collapse collapse prior prior to instal installatio lation n of the probe. probe. In the MPM, radial displacement is determined from the change in volume of a waterfilled cell and a nd only stress control is possible. The HPD (also called flexible dilatometer) dilatometer) was developed to test weak rocks but can be used in dense sands (with the advantage that the test can be performed to beyond peak failure) and firm to hard clays that are too stiff for other pressuremeters. It measures displacement in a similar way to the SBP, using six strain-gauged feeler arms, allowing both stress and strain control to be employed. The MPM is more commonly used for direct design methods using empirical correlations lati ons rather than for para parameter meter determination, determination, and can also be used as a profiling tool. It is, however, possible to obtain more fundamental soil parameters from analysis of its field curve, particularly if the test procedure is modified. The main drawbacks of the PBP are the ground disturbance resulting from forming the borehole and the likelihood of forming a geometrically imperfect test pocket. The horizontal total stress s h at the start of the test is virtually zero, so relatively large cavity strains are needed to bring the pressure back to its in situ value, which makes the measurement of in situ s h rather uncertain.

Self-boring pressuremeter (SBP) To minimise the problem of borehole disturbance, the SBP was developed, which is the superior of the pressuremeter types. Versions with a cutter can be used in all soils up to very stiff consistency and soft rock, but not in gravels and stony soils, while versions with a drilling bit can be used in rock. There There are British and Frenc French h versions of the device and they differ in several respects. The most significant difference is that radial displacements are measured with three or six strain-gauged feeler arms equi-spaced at mid-height in the British device which allow strain control testing, while the French SBP uses volume change measurement as in the Me ´ ´ nard device. The horizontal total str stress ess s h at the start of a test in clay should be approximately equal to the in situ value if the SBP has been installed with only minimal disturbance, which 76 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.


makes the SBP the only pressuremeter type that can make reasonably accurate measurements of in situ horizontal total stress s h in clay. In sands, significant disturbance is usually caused by installation of the SBP so the initial s h may not be the in situ value. The SBP is the most accurate of all the pressuremeters because the ground disturbance should be small and recoverable. But the amount of disturbance is heavily dependent on the skill and experience of the operator, operator, so the test should be performed by special specialist ist subcontractors.

Cone pressuremeter (CPMT) The cone pressuremeter (CPMT), or full displacement pressuremeter, comprises a pressuremeter module mounted behind a standard CPT 15 cm2 cone and friction sleeve. Radial displacement is measured by three feeler arms. The CPMT is pushed into soil as part of the CPT operation and, at the required depth, the cone is halted and the pressuremeter test undertaken, thus providing a means of interpreting stiffness and strength parameters from the pressuremeter test as well as strength parameters and profiling from the CPT results. The horizon horizontal tal total stress s h at the start of the test can be higher than the in situ value due to the effects of cone penetration. However, a method of interpreting s h0 h0 in sand has been derived (as described later in this section) which puts the CPMT at an advantage over the other pressuremeters in this respect because s h0 h0 cannot be determined reliably in sands using the other pressuremeter types. It is the most practical and economical of the pressuremeters, but the initial penetration of the cone causes soil disturbance which must be accounted for in CPMT-specific interpretations of the data.

Soil parameters in clay The in situ earth pressure coefficient K 0 is determined from the in situ total horizontal stress s h0 method, Marsland and h0 which is most commonly interpreted using the lift-off method, Marsland Randolph (1977) method, or by reconstruction of the pressuremeter curve. The lift-off is a break in the initial slope of the cavity strain–pressure curve (Figure 3.7), so it can only be interpreted from pressur pressuremeters emeters with direct cavity displacement measurement (i.e. SBP SBP,, HPD, CPMT). The interpretation is more accurate in the SBP because the initial pressure is close to the in situ s h , while the interpretation is rather uncertain in the HPD and CPMT due to soil disturbance. Even with the SBP, as the stiffness of the soil increases the method becomes more uncertain because the deformations of the instrument become similar simi lar to the initi initial al defo deforma rmation tionss of the soil. Whittle soil. Whittle et et al . (1995) describe (1995) describe a method for the six-arm SBP which also allows ground disturbance and anisotropy to be estimated. Houlsby and Withers (1988) proposed (1988) proposed a method of determining s h0 h0 from CPMTs which, with the application of a correction to account for the finite length of the pressuremeter, has been found to produce results similar to those measured with an SBP (Yu, (Yu, 2004). 2004). Undrained shear strength cu can be interpreted from plane strain analytical solutions of both the loading and unloading stages of pressuremeter tests. Generally, cu values 77 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.


obtained from the unloading stage are more reliable because the effects of soil disturbance are less significa significant. nt. However, However, due to high str strain ain rates and the assumption of infinite pressuremeter length, cu values interpreted from pressuremeter tests tend to be higher than th an tho those se mea measu sure red d in ot other her in si situ tu and la labor borat atory ory te test stss (Yu and Co Coll llin ins, s, 199 1998 8; Prapaharan et Prapaharan ., 1989). 1989). In contrast, the total stress approach of the analytical methods et al ., tends to under-estimate cu for heavily over-consolidated clays (Yu and Collins, 1998). Therefore, overall, cu values interpreted from pressuremeter tests should not be used in isola isolation. tion. A differ different ent interpretation interpretation method for for c cu is needed for the CPMT as derived by Houlsby and Withers (1988). It is a method that has not been applied extensively in practice, so should be used with caution. Yu (2004) found that cu was over-estimated by as much as 10%, at least in part due to the assumption of infinite pressuremeter length. The shear modulus modulus G is obtained directly as half the slope of the unload– unload–reloa reload d loops on G is the cavity strain–pressure plots. The average slope of the unload–reload loop is determined from a line drawn between the two apexes of the loop, in order to obtain average G over the range of shear strain. In accurate pressuremeter tests, the strain-dependency of G can be ass asses essed sed fr from om th thee cur curva vatu ture re of the un unloa load– d–re relo load ad lo loops ops,, as sho shown wn by example in Section 8.2.3, between shear strains of about 0.01% and 1%. By combining PMT-derived G values with seismic test measurements of G G 0 , the complete decay curve of stiffness with strain can be estimated. The corresponding mean effective stress p′ should be estimated (in order to normalise G for stress). Note that p′ is assumed to remain remai n constant and equal to the in situ value in undra undrained ined tests, but will incr increase ease during drained pressuremeter tests. The derived stiffness is the shear modulus in the horizontal plane Gh , which is directly applicable to, for example, the analysis of retaining walls. For situations involving an element of vertical deformation, e.g. under a spread foundation or around an axially loaded pile, Gv is applicable and the value of this parameter would need to be determined taking into account any anisotropy in the soil. As with all in situ tests, the pressuremeter does not measure an ‘element stiffness’ as in a laboratory test on a specimen of soil, because the extent of soil that influences the stiffness value is unknown. Strains undergone by the soil also vary strongly with radial dist di stanc ancee fr from om the pr prob obe, e, so a re refer feren ence ce she shear ar st stra rain in mus mustt be ar arbit bitra raril rily y sel selec ecte ted d as representa repr esentative tive of G for for th thee so soil il.. Th This is ref efer eren ence ce va valu luee is of ofte ten n ta tak ken as th thee str trai ain n meas me asur ured ed by th thee pr pres essu surrem emet eter er (i (i.e .e.. at th thee pr pres essu surrem emet eter er su surf rfa ace ce)) an and d Houlsby (2001) justifies this choice by demonstrating that G is strongly influenced by the soil stiffness close to the pressuremeter. However, the measured stiffness will be a little high beca be caus usee of th thee hi high gher er (s (sma mall ll str trai ain) n) sti tiff ffne ness ss of th thee so soil il fu furt rthe herr awa way y fr from om th thee pressuremeter. Note that, in clay, the tangent modulus G modulus G from from a pressuremeter curve corresponds theoretically with the secant modulus (Figure (Figure 3.8) 3.8) in a laboratory test (Muir (Muir Wood, 1990), 1990), although Clarke (1995) found some variation in secant stiffness between triaxial and SBP derived values in London Clay. 78 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.


Figure 3.8 Secant and tangent stiffness

q

Tangent dq stiffness dεa

Secant q stiffness εa

εa

If a constitutive model requires equivalent laboratory tangent stiffness values, then the pressuremeter values would need to be estimated using the equation: dGs Gt = Gs + g dg

(3.2)

(Muir Wood, 1990)

where Gt and where G and G Gs are the tangent and secant shear moduli, respective respectively, ly, and g is the shear strain.

Soil parameters in sand Shear strength w ′ can be interpreted from plane strain analytical solutions of both the loading loa ding and unl unloadi oading ng st stage agess of pr press essure uremete meterr tes tests ts (Hughes et al ., . , 19 1977 77;; Houlsby ., 1986), as demonstrated by example in Section 8.2.3. Both methods require estiet al ., ′ mation or, preferably, separate measurement of the critical state shear strength w cv of the sand. The method based on the loading stage also interprets a value of dilation angle c but is sensitive to installation disturbance. Both methods use the assumption of zero elastic strain and a linear relationship between volumetric and shear strains which leads w ′ in medium-dense and loose sands. The assumption of infinite to under-es under-estimat timation ion of w pressuremeter length over-estimates w ′ but this has a lesser effect when deriving w ′ from the unloading stage due to the small cavity contraction (Yu, 2004). A semi-analytical interpretation of the CPMT based on large laboratory calibration chamber tests was derived by Schnaid by Schnaid and Houlsby (1992) to (1992) to obtain peak shear strength ′ ′ ′ w p and in situ horizontal effective stress s h0 . Determining s h0 in sands from the other pressuremeter types is difficult and rarely can the lift-off method be employed, so test simulation probably remains the most reliable method to estimate s ′h0 in these cases. The shear modulus G modulus G is is obtained in the same way as described above for clays, but note that p that p′ cannot be assumed to be constant in drained conditions and its variation must be 79 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.


estimated estima ted in orde orderr to tak takee ac accoun countt of the str stress ess-dep -depende endency ncy of sti stiffn ffness. ess. s v remains constant const ant while while Bellotti et al . (1989) propose proposed d the foll follow owing ing equa equatio tion n to es estima timate te s h′ during a pressuremeter test in sand: ′ ′ average s h′ = s h0 0.2( p ) + 0.2( pu′ − s h0

( 3. 3)

where pu′ is the maximum pressuremeter pressure at the start of unloading. Note however where p that s h around a CPMT is likely to be much higher than around other pressuremeter types.

Soil parameters in rock The pressuremeter can measure the shear modulus G in the same fashion as pressuremeters in soil. It is also particularly suited to the measurement of the stress–strain behaviour io ur of fr frac actu ture red d ro rock ck mas masses ses for va vali lida dati tion on of co cons nsti titut tutiv ivee mo model del par parame amete ters rs by simulation of the test. This is provided that the pressuremeter probe diameter is at least six times the discontinuity spacing.

3.3.5 3.3 .5

How Ho w can can stif stiffne fness ss be be measu measured red usi using ng in in situ situ seis seismic mic tes testin ting? g?

As described in Section 3.3.1 for using bender elements in laboratory triaxial testing, shear wave velocity vs provides a direct determination of the very small strain shear modulus G0 . Measured G0 values can be combined with pressuremeter measurements to derive the full stiffness decay curve. Seismic tests can be undertaken by several different means as shown in Figure in Figure 3.9. 3.9. Anisotropy of stiffness can be assessed by measuring vertically propagating, horizontally oscillating shear wave velocity vvh using the downhole or uphole technique, together with horizontal shear wave velocity oscillating in each direction vhv and vhh using the crosshole technique (Fioravante (Fioravante et et al ., ., 1998). 1998). Figure Figur e 3.9 Common types of in situ seismic testing for G 0 determination

Twin receivers for true interval velocity Downhole

Uphole

Crosshole

Source Receiver Surface (Rayleigh) wave

Seismic CPT or DMT (Downhole)



A useful overview of the seismic method is given by Clayton (2011) and guidance on the use of the seismic CPT is available in Butcher in Butcher et (2005). et al . (2005). Boreholes must have plastic linings grouted into them for good mechanical coupling with the ground, and, for crosshole testing, be surveyed for verticality with an inclinometer to verify the distances between them. The crosshole method requires typically threee in-line boreholes thre boreholes and possibly a fourth orthogonal borehole for anisotr anisotropy opy assessment, with spacings of about 5–7 metres. Pressure wave velocity vp can also be measured at the same time. This is of little use in soils and fractured rock because vp is governed by the low compressibility of pore water (vp ≈ 1500 m/s), but in relatively unfractured and unweathered rock, the compressibility may be lower than water and vp can be used in combination with vs to obtain two isotropic linear elastic parameters. For instance: G0 = r v2s

(3.1 ibid.)

    v2p

Poisson’s ratio

n=

2v2s v2p v2s

−1

(3.4)

−1

An alternative to using holes is the surface wave technique where the velocity vr of Rayleigh waves travelling at the ground surface is measured (shear wave velocity vs ≈ 1.09 vr assuming Poisson’s ratio n = 0.25). vr depends on frequency as well as ground stiffness, so a range of frequencies must be measured. The values of G of G0 obtained from the various Rayleigh wavelengths are combined into a stiffness–depth profile. Due to the interpretation needed, these surface methods have a greater uncertainty than the subsurface methods and the uncertainty increases with depth.

3.3.6 3.3 .6

What para What paramete meters rs can can be be obtain obtained ed fro from m other other in situ situ tes testt methods? Piezocone penetration test (CPTu) Common additions to the CPT are seismic wave Common wave rece receivers ivers to form a seismic cone (SCPT) for shear wave velocity measurement and a pressuremeter to form a cone pressuremeter (CPMT). These tests have the significant advantages of effectively combining two test methods in one while providing perhaps the most cost-effective means of performing a seismic or a pressuremeter test on site. A variant of the CPT for soft soils is the full-flow penetrometer where a T-bar or ball replaces the cone. This has the advantage that penetration resistance is not affected by soil stiffness or stress anisotropy, so more precise correlations with soil shear strength may be obtained. Additionally, if full-flow penetration resistances are compared with cone resistances, soil stiffness or anisotropy may be estimated more accurately than with the CPT alone. Full-flow penetrometers are often used in offshore site investigations. 81 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.


For mo For more re in infor forma mati tion on and det detai ails ls of a st study udy of a wor world ldwi wide de da data tabas basee of fu fullll-flo flow w penetration measurements, refer to Low Low et (2011). et al . (2011). The undrained shear strength c strength cu of clays can be estimated using an empirical cone factor dependent on soil type, soil stiffness, cone type, OCR and anisotropy that is best determined from a site-specific correlation. In sands, both analytical and empirical methods exist to estimate shear strength w ′ values that should be reasonably accurate in silica sands, but slightly under-estimated in compressible calcareous sands.

Flat dilatometer test (DMT) Use of the DMT – also called Marchetti dilatometer – is increasing in geotechnical practice. It comprises a stainless steel blade with a flat 60 mm diameter thin steel membrane mounted on one face. The blade is driven vertically into soils and halted every 20 cm for the membrane to be inflated and the pressure to be measured. Reasonably accurate interpretations of OCR, K OCR, K 0 a and nd cu can be made from the DMT in most clays, except heavily over-consolidated clays, although it is not as accurate as the SBP.. Num SBP Numeric erical al ana analys lyses es by Yu (20 (2004) 04) sho showed wed tha thatt inte interpr rpreta etatio tions ns of w ′ fr from om the ′ DMT in sand are heavily dependent on the rigidity index (G (G0/ p0 ) which leads to much ′ w . The recent development of a seismic dilatuncertainty in the direct interpretation of w ometer (SDMT) will help to overcome this difficulty by allowing G allowing G0 to be measured. Soil stiffness cannot be determined reliably from the DMT because the expansion of the membrane by only 1.1 mm does not reach beyond the disturbed soil around the blade resulting resul ting from initial penetr penetratio ation. n.

Standard penetration test (SPT) The SPT remains the most widely used in situ testing technique and many design correlations and charts have been derived. These are all based on purely empirical correlations due to the difficulties of interpreting the test by analytical means and taking account of borehole disturbance and variations in apparatus and procedures. Corrections must be applied to the N value for overburden stress, energy delivered to the rods and rod length, as described in ISO in ISO (2005) and and Clayton Clayton (1995). (1995). Quite consistent results can be obtained and reliable shear strength parameters derived using site-specific empirical correlations with the same test and drilling methods and equipment. One of the advantages of the SPT is its large database of results in different soils and fractured rocks to facilitate the derivation of correlations. However, non-sitespecific correlations for the SPT are only approximate and are not appropriate for FE analysis.

Plate load test (PLT) Use of the PLT in soil is less common because it is expensive compared with alternatives, but in fractured rock it is one of the only methods of determining stiffness in a vertical orientation. Rather like the pressuremeter, it is better suited to simulation of the test to validate model parameters for fractured rock masses rather than deriving parameters directly. This is provided that the plate width or diameter is at least six times the discontinuity spacing. 82 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.


3.3. 3. 3.7 7

How Ho w is is perm permeab eabil ility ity me meas asur ured ed in si situ tu? ?

Laboratory determination of permeability, particularly in granular soils, can be highly inaccurate. More reliable measurements can be made in situ since a greater extent of the ground is tested meaning that the effects of large-scale heterogeneities are included in the determinati determi nation on of permeabi permeability lity.. There are several methods available, using single boreholes, multiple boreholes or the dedicated self-boring permeameter, as shown in Figure in Figure 3.10. 3.10. The most accurate technique is the pumping test with multiple boreholes, although in clays this test may take weeks and a self-boring permeameter would be more appropriate. In rocks, the packer testt is comm tes commonly only use used. d. The vari various ous tec techniq hniques ues are described described brie briefly fly bel below ow.. Fu Furthe rtherr guidance is given by Clayton et Clayton et al . (1995) and Cashman and Cashman and Preene (2012). (2012).

Variable head tests (rising or falling) These tests are used in relatively permeable soils in a cased borehole or open standpipe piezometer. For a falling head test, clean water is added to the borehole or standpipe to raise the water level and then the water level is recorded regularly to determine the rate of fall until the natural level is restored. The rising head method is very similar but inv in vol olv ves rem emo ovi ving ng wat ater er to lo low wer th thee wate terr le lev vel an and d me meas asur urin ing g it itss rat atee of ri rise se.. Permeability is determined using Hvorslev’s method, based on the time taken for water levels to return to equilibrium, as described in Clayton et Clayton et al . (1995) and Cashman and Preene (2012). Figure 3.10 Com Common mon methods of in situ permeability permeability measurement measurement

Ground level

Water removed

Water added

Constant head supply

Pressured supply

t = 0

Falling head

Groundwater level Piezometer or standpipe tube

Packer Rising head = 0 t =

Packer

Piezometer tip or sand filter

Rising head test

Observation wells

Test well

Falling head test

Constant, measured pumping rate Drawdown

Constant head test

Packer test

Water supplied as described for modified SBP

Water supplied to cavity at controlled rates with constant or variable head Borehole, if required

Borehole, if required

Probe After pressuremeter test, probe is retracted a short distance to form a test cavity in the base of the test pocket

Pumping test

Perforated metal tube

Test cavity

Modified self-boring pressuremeter (SBP)

Self-boring permeameter



Constant head tests The effective stress changes during an in situ permeability test will cause large volume changes chan ges in a plas plastic tic cla clay. y. Con Conseq sequent uently ly,, cons constan tantt ra rather ther tha than n vari variabl ablee head tes testin ting g should be undertaken in such soils. The Gibson method is used to extrapolate a plotted curve of changing flow rate, caused by the clay consolidating or swelling, to a constant steady-state flow rate. The permeability can then be determined from the steady-state flow rate using Hvorslev’s equations as described by Clayton et Clayton et al . (1995). It is better to perform the test in a stand standpipe pipe piezometer rather rather than a cased bore borehole hole because small leakages through any gaps between the casing and the soil will render the measurements useless.

Pumping tests Pumpin Pump ing g te tessts in inv vol olv ve a la larg rger er vol olum umee of gr grou ound nd th than an th thee si sing ngle le bo borreh ehol olee te test stss describ des cribed ed abo above ve so the they y ar aree mor moree rep repre resent sentati ative ve of gr ground ound con conditi ditions ons and are less affected by borehole disturbance. A borehole (or test well) is sunk into the stratum to be measured and water is pumped from it at a constant, measured rate. By measuring the resulting drawdown (i.e. fall in piezometric level) around the test well in a series of observation boreholes, the permeability of the stratum can be estimated. Permeability should be determi determined ned using transient flow (or non-s non-steady-s teady-state) tate) techniques techniques because they allow data to be analysed in real time and reduce the period of pumping required. There is a range of complex manual techniques using different assumptions as summarised in Cashm Ca shman an and Pr Pree eene ne (2 (2012 012). ). Alt Alter erna nati tive vely ly,, si simul mulat atio ion n of pum pumpi ping ng te test stss usi using ng FE analysis capable of transient flow analysis can be employed to determine permeability. This is particularly useful where anisotropic permeability exists. Axisymmetric analyses are sufficient where flow patterns have such symmetry, otherwise 3D analyses may be necessary.

Packer test This is also called the Lugeon test and is intended for permeability testing in rock. Packers are inflated to seal the top and bottom of a test section in a drillhole. Water is supplied to the test section at different pressures and the permeability calculated from the measured flow rate.

Self-boring devices Under continuing development are self-boring devices based on the SBP. There are two devices:

1

2

Self-boring permeameter Self-boring permeameter consisting consisting of a perforated perforated metal tube with an internal internal membrane that is inflated to seal the tube during installation and is then deflated to allow the permeability test to start using a constant flow system to supply water down to the tube (Chandler (Chandler et ., 1990). 1990). et al ., Modified self-boring self-boring pressure pressuremeter meter consisting consisting of a conventional conventional SBP combined with the constant flow system for permeability measurement. To perform a permeability test the SBP is retracted a short distance, leaving a well-defined cavity in the ground. The length of the cavity can be varied to assess the anisotropy of permeability (Ratnam (Ratnam et ., 2005). 2005). et al .,



Constant or variable head tests can be performed in both types by supplying water to the tube at controlled rates and measuring the injection pressure required to achieve a particular flow rate. Both methods can minimise soil disturbance by careful installation but, as with single borehole tests, tests, the disadvantage is that only a small volume of soil is tested. tested.

Deriving permeability values For consolidation and groundwater flow analyses (see Chapter 4) it is necessary to specify ground permeability in the input parameters in order to calculate dissipation times and flow rates, respectively. Permeability in most soils varies with void ratio and therefore with wit h ef effect fectiv ivee st stre ress. ss. A var varia iatio tion n of per permea meabil bility ity wit with h dep depth th can be ent enter ered ed to tak takee account of in situ variations in void ratio, but this will not take account of void ratio changes during an analysis so is only suitable where insignificant changes in void ratio are expected. Alternatively, non-linear relationships between permeability and void ratio or effectiv effe ctivee str stress ess can be adopte adopted d tha thatt vary the permea permeability bility in res response ponse to chang changes es in these variables. This is useful in situations with large stress changes in soft soils where void ratio, and hence permeability, changes occur, such as in the construction of embankments on soft clay as demonstrated in the example in Section 8.4. Where permeability relationships are not available, carefully selected average permeability values should be adopted. Permeability is often anisotropic due to, for example, soil fabric or laminations, with higher horizontal than vertical permeability. Therefore, most groundwater flow analyses allow different permeabilities to be specified for each global axis direction. Large perm Large permeabi eabilit lity y diff differe erences nces can caus causee ill ill-con -condit ditioni ioning ng of the gr ground oundwa water ter flo flow w matrix. Aim to keep the difference to an order of magnitude less than 105. The permeability of intact rock is normally so small compared with the permeability of discontinuities that it can be ignored, except in high porosity rocks such as sandstone. With implicit modelling of discontinuities, the permeability of the discontinuities needs to be smeared across the domain according to the formula k = kdisc w/s where w is the width of each discontinuity, discontinuity, s the discont discontinuity inuity spacing spacing and and k s the k disc the permeability of the discontinuity. With explicit modelling of discontinuities, the permeability of each discontinuity can be specified and the permeability of the intact rock normally ignored. Permeability is notoriously variable and difficult to measure accurately. The variation or measurement error could easily be an order of magnitude. Therefore, it is important to perform parametric studies of permeability in order to estimate a range of plausible outputs rather than rely on a single deterministic value.

3.4. 3.4. 3.4.1 3.4 .1

Paramet Para meter er der deriv ivat ation ion and val valida idatio tion n How Ho w ar are e par paramet ameters ers der deriv ived ed fr from om tes testt resu results lts? ?

As described in Eurocode 7 (CEN, (CEN, 2004) 2004) and its guidebooks (Bond (Bond and Harris, 2008; 2008; Frank et Frank ., 2004; 2004; Orr and Farrell, 2011), 2011), parameter derivation is divided into two steps: et al ., Step Ste p 1: establi establish sh a deri derive ved d valu valuee fr from om each test measure measured d valu value. e. The val value ue should be appropriate for the analysis situation (e.g. axisymmetric/plane strain, stress level, strain level, etc.). 85 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.


Step 2: from all the derived values, values, select a characteris characteristic tic value that is appropriate appropriate to model the soil or rock layer in the analysis. These two steps are described in this section.

Derived values There are many factors to consider in order to determine appropriate derived values, including aspects of ground behaviour, the test type and conditions, the expected stresses and strains during construction and the assumptions of the FE model and constitutive models. Some of the important factors are described here: g

g

g

g

g

g

g

g

g g

Stress level: this affects Stress affects most model model parameters, parameters, so derived values values must be normalised for stress or else be appropriate for the stress levels in the structure to be analysed. Mode of deformation: deformation: rarely rarely does the mode of deformation deformation or stress stress path in laboratory or in situ tests match that around real structures in the field, so judgement is required during the derivation of parameters and assessment of the reliability of input parameters becomes important. On major projects, the results of full-scale field trials or centrifuge tests are useful in this respect. Strain Strai n level: stiffness stiffness at small small strains strains is much larger than the stiffness stiffness at large strains. What was the strain level in the parameter tests and what is the expected strain level in the ground during construction? Specimen volumes: volumes: only small small volumes volumes of soil and rock are tested tested in situ and in the laboratory which may not take proper account of features such as discontinuities, laminations or large particles. Corrections may need to be applied to measured values to take account of these features by comparing, for example, laboratory and in situ shear wave velocity, studying detailed sample descriptions (e.g. fissure spacing) and back-analysis of case studies in similar ground. On major projects the results of full-scale field trials are useful in such cases. Brittleness Brittl eness or ductility: ductility: plastic clays clays and rocks tend to be brittl brittlee (i.e. shear strength strength falls rapidly post-peak) at low confining stresses, while all soils and rocks tend to be ductile (i.e. a small post-peak drop in shear strength) at high confining stresses. Strain Strai n rate effects: effects: strain strain rates rates in soil tests are typically typically far higher than in the field, so they exclude creep effects. Soil deformations in the field may continue for some time following stress change due to the effects of creep and this is an area of ongoing research. Soil ageing: mechanical, mechanical, biological biological and chemical chemical processes processes that are are not, as yet, fully understood lead to improved strength and stiffness with time which might be quite rapid in relatively recent deposits, e.g. man-made earthwork structures. Drainage Drain age conditions: conditions: most soil tests and subsequent parameter parameter derivations derivations create create or assume wholly drained or undrained conditions but this may not actually be the case in the test or in the field. Sample disturbance disturbance:: this should be assessed assessed as described described in Section Section 3.2.4. Accuracy Accura cy of parameter parameter derivation derivation method: method: this is summarised in Tables Tables 3.2 and 3.3 for certain methods. Many other methods exist, some of which are more approximate.



Here are sev Here seven en iss issues ues tha thatt may nee need d par particu ticular lar at attent tention ion when det determi erminin ning g deri derive ved d values of geotechnical parameters for FE analysis: Intermediate principal stress s 2 (including plane strain parameters) In the laboratory, most strength and stiffness properties are measured using a conventional tio nal tri triaxi axial al com compre pressio ssion n tes testt wher wheree the inte intermed rmediat iatee pri princip ncipal al st stre ress ss s 2 equals the minor principal stress s 3 . In plane strain, axisymmetric and 3D analyses, however, s 2 may vary between the values of s 3 and the major principal stress s 1 . The variation of s 2 between the values of s 3 and s 1 is defined by the ratio b (as was shown in Figure 2.2). In plan planee st strai rain n comp compre ressio ssion, n, 0.1 0.15 5 ≤ b ≤ 0.3 0.35 5 appr approxim oximat ately ely (Pot Potts ts and Zdr Zdrav avko kovic vic ´ ´ , 1999), while in other stress states and in axisymmetric and 3D geometries, b will depend on the particular stress and strain conditions across the model. But what corrections are needed to soil parameters determined from triaxial tests where b = 0 (compression) or b = 1 (extension) in order to apply them in a plane strain or other analysis types with different b values? ′ ′ = 1.1 for conve A factor w ps /w tc converting rting triaxial compression compression test-derived test-derived friction angle to a plane strain value is often quoted, which has been found to give reasonable results in bearing capacity calculations (Bolton (Bolton and Lau, 1993; 1993; Oh and Vanapalli, 2008). 2008). In the absence of any other information, this would appear to provide a reasonable, conservative estimate. Note that pressuremeter tests are analysed with the assumption of plane strain deformation, thereby deriving plane strain soil properties directly.

Advanced laboratory testing techniques (e.g. hollow cylinder apparatus, true triaxial testing) allow the independent influences of b and anisotropy on soil parameters to be investigated fully. Nishimura fully. Nishimura et al . (2007) found (2007) found that failure states in London Clay with b = 1 or 0 provided lower bound Mohr–Coulomb failure lines with somewhat higher strengths observed at intermediate values of b. Vaid and Campanella (1974) measured (1974) measured higher cu values in normally consolidated Haney clay in plane strain compression compared with triaxial compression (b = 0) or extension (b = 1). In Cumbria sand, Ochiai sand, Ochiai and Lade (1983) noted (1983) noted a large increase in w ′ of up to 9 from other her si simil milar ar st stud udies ies on san sands ds,, inc inclu ludi ding ng Symes (198 (1983) 3) on Ham b = 0 to b = 1. In ot River sand and Sayao and Vaid (1996) on Ottawa sand, strength and stiffness were noted to increase with b up to about 0.6, and then remain constant or decrease slightly to b = 1. 8

From this small number of tests, it appears that adopting triaxial compression or extension strength values for clay is adequately conservative for other b values between 0 and 1 occurring in FE analysis. Triaxial testing of sands is less common commercially, but it appears that both triaxial extension and compression tests should be performed before determining an appropriate w ′ value for FE analysis, since relying on compression tests alone may be overly conservative. Some failure surfaces, such as as Matsuoka and Nakai (1974) and and Lade Lade (1977) offer (1977) offer the possibility of varying strength with b. 87



Anisotropy As described in Chapter 2, some of the commonly used advanced constitutive constitutive models for soils assume isotropy to avoid the additional complexity of anisotropic stiffness and strength. In which case, it is common practice to measure the anisotropic properties of a soil in order to enter some average values of strength and stiffness into an isotropic model that includes the more critical elements of soil behaviour. Anisotropy of small stra st rain in st stiffn iffness ess can be mea measur sured ed usin using g lab labor orato atory ry or in sit situ u seis seismic mic tes testing ting,, and the assumption is often made that the same degree of anisotropy applies to the larger strain stiffness. Alternatively, anisotropy of both stiffness and strength can be assessed by comparing the results of triaxial compression and extension tests or direct shear tests, but results may also be affected by the variation of s 2 .

Anisotropy is expressed in terms of the angle a , as was shown in Figure 2.3. As for the study of s 2 described above, advanced laboratory testing techniques (e.g. hollow cylinder apparatus) are required for independent control of the a and b values. From these, it appears that cu for clays decreases by as much as 50% from a = 0 to a = 90 for normally to lightly over-consolidated reconstituted Boston Blue clay (O’Neill, (O’Neill, 1985; 1985; Seah, 1990), 1990 ), whi while le max maxima ima occu occurr at a = 0 and a = 90 with values about 40% lower at a = 45 in heavily over-consolidated London Clay (Nishimura et al ., ., 2007). Clays also possess anisotropic stiffness (the higher value can be in the vertical or horizontal direction) and depends on strain level, clay type, structure and stress history, while w only changes marginally up or down with a. 8

8

8

8

8

′

In similar studies on sands (Symes (1983) on medium-loose Ham River sand, Wong sand, Wong and Arthur (1985) on dense Leighton Buzzard sand, and Oda and Oda et al . (1978) on (1978) on dense Toyura sand), w peak was observed to decrease by 5 to 12 from a = 0 to a = 90 and stiffness was also observed to decrease. ′

8

8

8

8

It is unusual for anisotropy in soils to govern FE model accuracy over other aspects of soil behaviour, such as stress- and str strain-depe ain-dependency ndency of stif stiffness, fness, which is why const constituitutive models considering these other aspects of behaviour are more common. However, situations where anisotropy may be particularly important include surface loads (e.g. from shallow foundations or embankments) on soft low-plasticity clays where the use of isotropic shear strength could lead to over-prediction of bearing resistance (Potts (Potts and Zdravkovic ´ ´ , 2001; 2001; Zdravkovic ´ ´ et al ., ., 2002), 2002), and the pull-out resistance of bucket foundations in soft cohesive soils may be over-predicted with an isotropic shear strength (Potts and Zdravkovic ´ ´ , 2001). Dilation angle c The dilation angle c of of soil becomes significant in dense granular soils at relatively low confining stresses and can influence FE analysis outputs. It can be measured (refer to Lees, 2012) 2012) or estimated from the equation: ′

c = w

−

30

8

(3.5)

for quartz sands, c is is generally taken as zero for clays and calcareous sands. In undrained effective stress analyses and confined problems (e.g. pile analysis), specification of a 88



non-zero dilation angle can allow unrealistic increased effective stresses to be predicted that delay failure and which would be non-conservative. Therefore, in such analyses, c should also be set to zero. Initial state parameters, e.g. stress ratio K0 and pre-consolidation stress Initial state state parameters parameters are not fundamental soil parameters parameters but are used to set up initia initiall conditions in an FE analysis, such as stress state and stress history (e.g. location of a yield surface). Consequently, they have a direct influence on FE analysis outputs and their importance should not be under-estimated. It should not be assumed that they are consta con stant nt in ea each ch st stra ratum. tum. Ind Indeed, eed, they ofte often n var vary y wit with h dept depth, h, part particul icularl arly y near the ground surface. The in situ stress ratio K 0 (= s h′ /s ′v ) is an important initial state parameter used in the setting up of initial stresses. It has a particularly significant influence on FE analyses of retaining walls, cut slope stability in clay and shallow foundations in drained conditions. Careful measurement of K K 0 is required, or it can be estimated by correlation with other parameters. One common equation (and often the default equation used in programs) is Jaky’s equation: K 0 = 1 − sin w ′

(3.6)

Note, however, that this equation is appropriate only for normally consolidated soils while over-consolidated soils are likely to have significantly higher values of K 0 (see Appendix 3.1 for approximate equations). Compacted soils may also possess high K 0 values due to compaction pressures, whose effect may be significant on retaining walls. Unfortunately, there is no straightforward way of estimating compaction pressures or of simulating them in FE analysis. Clayton analysis. Clayton and Symons (1992) and and Clayton Clayton et et al . (2013) describee appro describ approaches aches for esti estimatin mating g compac compaction tion press pressures. ures. The pre-consolidation stress defines the stress history and is the maximum effective stress that a soil has previously experienced. It defines the starting location of yield surfaces in some constitutive models (e.g. Modified Cam Clay, cap hardening models). It can be determined from consolidation tests (e.g. triaxial consolidation test or CRS oedometer) where stresses pass from the reloading line to the primary loading line. In heavily overconsolidated clays, very high applied pressures may be required to reach primary loading. When the pre-consolidation stress cannot be measured, it needs to be estimated using knowledge of the geological history of the site. Note th Note that at th thee re reloa loadi ding ng be beha havi viou ourr an and d tr tran ansi siti tion on to pri prima mary ry lo load ading ing pr pred edict icted ed by advance adv anced d cons constit titutiv utivee mode models ls is influ influenc enced ed by the pre pre-con -consol solida idation tion st stre ress, ss, K 0 and Poisson’s ratio. Therefore, it is important to validate that all three parameters provide a sufficiently accurate simulation of such behaviour. Drained parameters from undrained tests Tests on clays are generally undertaken in undrained conditions but many constitutive models require (drained) effective stress parameters, e.g. drained stiffness. For elastic 89 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.


(unload/reload) Young’s modulus, the conversion from undrained to drained value can be made using the equation: ′

E =

2 1 + n′

  3

E u

(3.7)

To obtain other parameters and to validate selections, simulation of the undrained tests should shou ld be per perform formed ed with the eff effecti ective ve st stre ress ss par paramet ameters ers and adj adjust ustment mentss made as necessary to improve the agreement between the real test results and the simulation outputs. Poisson’s ratio, n Drained Poisson’s ratio n′ is rather difficult to measure accurately (Lees, 2012) but most FE analyses are not particularly sensitive to n′ values, in which case it is acceptable to estimate values. In advanced constitutive models, n′ is usually a true elastic parameter and a value in the range 0.1 to 0.25 is usually appropriate. In basic models, such as LEPP models, stiffness behaviour is more sensitive to n′ values and n′ differs between primary loading and elastic unloading/reloading. In the former case, n′ should be in the range 0.25 to 0.4 to include behaviour that is otherwise covered by hardening plasticity in the more advanced models, whereas in the latter case n′ is a true elastic parameter as in the advanced models. Loose sands tend to be at the lower end of these ranges, clays mid to upper and dense sands at the upper end. When using the gravity switch-on method to establish initial stresses and simulate fill placement in subsequent construction stages (see Section 1.4.1), the n′ values are set to establish appropriate initial K 0 values in LEPP models. In undrained conditions (zero volume change), n equals 0.5 but, since such a value would create a singularity in the stiffness matrix, n is set close to 0.5 (usually 0.495) in FE analyses of undrained soil in terms of total stress (see Chapter 4). Cohesion c ′ A low c low c ′ value is sometimes specified in order for a failure criterion to fit observed data, but remember that this gives soil a tensile strength at zero confining stress which is unrealistic. Therefore, a tension cut-off should be invoked in the constitutive model to keep tensile stress at very low values or zero. It is good practice to input a nominal value (e.g. 0.1 kPa) for c for c′ to ensure that initial zero stress states at ground surfaces remain within the yield surface and hence help to avoid calculation problems.

Characteristic values Statistical Statistic al determin determination ation of the characteristi characteristicc value is possibl possible, e, but rarely are there sufficient data available and constitutive model parameters can be rather complex for this approach. Alternatively, characteristic values can be determined by eye from plots of derived values and by using engineering judgement. Eurocode 7 (CEN, 2004) provides an alternative definition of the characteristic value for this purpose, namely a ‘cautious 90 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.


Table 3.4 Factors affecting the degree of caution exercised when establishing characteristic values Narrow caution margin

Wide caution margin

Data Da ta ha hass sm smal alll sc scat atte terr (u (uni nifo form rm pr prop oper erti ties es))

Data ha Data hass la larg rge e sc scat atte terr (v (var aria iabl ble e gr grou ound nd and/or sampling/test effects)

Large number of data points, including from different test types and other sources (e.g. other local tests, published studies, databases)

Small number of data points from a single test type and no other data sources

Experience of soil type

No experience of soil type

Large volume volume of ground ground invo involved lved in in the limit limit state state

Limit stat state e could occur in in small volu volume me of ground

Structure is strong and stiff and able to redistribute loads

Structure is weak and/or flexible and unable to redistribute loads

No ris isk k of pr pree-e exi xissti tin ng fa fail ilur ure e su surf rfa ace cess

Risk Ri sk of pr pree-e exis isti ting ng fa fail ilu ure su surf rfa aces

estimate of the value affecting the occurrence of the limit state’. Caution means how far below (or, in rare cases, above) the mean of the derived values is taken as the characteristic value. The limit state refers to the need to select a characteristic value that is appropriate for the limit state to be analysed. For example, in a serviceability limit state (SLS) analys ana lysis, is, peak soi soill str strengt ength h is ofte often n app approp ropria riate te whi while le in ult ultima imate te lim limit it sta state te (ULS (ULS)) analyses, a post-peak critical state strength is safer. The following procedure is recommended for establishing characteristic values by eye:

1 2 3 4

Plot derived derived values in an appropriate appropriate way way (often against against depth). depth). Draw Dra w by eye or using regress regression ion methods a best fit line through through the data. Draw Dra w the characteristi characteristicc line with an appropriate appropriate caution margin depending depending on the factors listed in Table in Table 3.4. 3.4. Assesss the range of permis Asses permissible sible values values for par parametri ametricc studies studies (see Secti Section on 7.3.3).

Figure 3.11 illust Figure illustra rates tes the dete determin rminati ation on of cha chara racte cteris ristic tic val values ues by ex exampl amplee for an arbitrary set of data. It can be seen that between 0 and 4 metres depth, there are only a few data points from a single test method and they have a wide scatter. Therefore, the characteristic line was drawn with a wide caution margin from the best fit line due to the uncertainty in the value of this parameter. However, below 4 metres depth, many more data points were obtained from three different test methods and the scatter was small. Therefore, the characteristic line was drawn with a narrow caution margin from the best fit line because there was more confidence in the data.

3.4.2 3.4 .2

How Ho w ar are e par paramet ameters ers ass assess essed ed for ac accur curac acy? y?

There is considerable There considerable uncertainty in the characterisa characterisation tion of soil, so it could be dangerous to adopt any derived parameter as representative of the real soil to be simulated without 91 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.


Figure 3.11 Determining characteristic values for an arbitrary data set

0

0

10

Parameter 20 30

40

50

1 2

WIDE CAUTION MARGIN

3 4 m : h t p e D

5 6 7 8 9

Test method 1 Test method 2 Test method 3

NARROW CAUTION MARGIN

10

Best fit line drawn by eye or regression analysis

Characteristic value line drawn with suitable margin from best fit line

appropriatee valida appropriat validation. tion. There are several methods of param parameter eter validation, as describ described ed here, and all should be employed to some extent at the end of the parameter derivation process.

Soil test simulation Some soil tests can be simulated straightforwardly by FE analysis and this will check that th at th thee der deriv ived ed par parame ameter terss an and d con const stit ituti utive ve mo model del re repr prese esent nt th thee soi soill be beha havio viour ur recorded in soil tests. This is particularly important in tests where soil behaviour transitions sit ions fr from om re reload loading ing to prim primary ary load loading ing beha behaviou viourr to che check ck tha thatt the cons constitu titutiv tivee model and input parameters predict this transition realistically. Soil test simulation also provides an opportunity to adjust the model parameters to achieve a better fit with the test data. Among the laboratory tests, triaxial and oedometer tests can be simulated straightforwardly, either with simple FE models of unit length and only one or two elements, or even with single-point algorithms, since uniform stresses can be assumed. Idealised single-element singleelement models are shown in Figure in Figure 3.12. 3.12. Single-point algorithms cannot be used for time-dependent consolidation properties because a drainage path length is required. 92 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.


Figure 3.12 Idealised laboratory test simulations Applied stress or displacement Centre-line Applied stress or displacement Centre-line

Arbitrary specimen dimensions unless simulating time-dependent consolidation

Cell pressure

Zero soil density

Axisymmetric simulation of oedometer test

Axisymmetric simulation of triaxial test

For more detailed studies, multi-element models that simulate end-effects and other test conditions more accurately may be preferable. Of the in situ tests, since penetration tests are rather complex to simulate, only PMTs and PLTs can be simulated in a routine manner. FE models of these tests can be set up in an idealised way as shown in Figure in Figure 3.13. 3.13. With advanced constitutive models or for greater accuracy, full simulation of the pressuremeter is necessary, as described in Figure in Figure 3.14. 3.14. In situ permeability tests are also well suited to simulation by FE analysis. For tests in boreholes, 2D axisymmetric analyses can be performed provided that ground strata are horizontal and that permeability is transversely isotropic (i.e. the same in all horizontal directions but can be different from the vertical direction). In test simulations, the precision of the analysis needs to be significantly higher than used typically in other analyses. Specify a maximum equilibrium error of 0.01% or less, rather than the more typical 1%. With automatic step-sizing this will also increase the number of data points, thereby giving a well-defined curve on graphical plots for comparison with test data.

Plausibility check While test simulations may verify that the constitutive model and its parameters are representative of the soil test conditions, what if the test results themselves were in error? A check should be performed to see whether the derived parameters are plausible, i.e. that they lie within permissible ranges. Typical values that can be referred to, based on soi soill de desc scrip ripti tions ons and cha chara racte cteri risa satio tion n te test sts, s, ha have ve bee been n pu publi blish shed ed wi widel dely y (e (e.g .g.. Brinkgreve et Brinkgreve ., 2010; 2010; Day, 2000; 2000; Look, 2007; 2007; Schnaid, 2009 (Chapter 2009 (Chapter 7)). et al ., 93 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.


Figure 3.13 Idealised in situ test simulations Centre-line

p

Unit height

r

Distance to remote boundary ~30r

Set initial stresses

Probe radius r Axisymmetric simulation of a pressuremeter test (length/diameter ratio higher than 6) Applied displacement Centre-line r

Plate radius r

Distance to remote boundaries >6r

Axisymmetric simulation of a plate load test

3.4.3 3.4 .3

Are Ar e the there re oth other er so sourc urces es of par parame ameter ters? s?

Much of thi Much thiss ch chapt apter er has de desc scrib ribed ed the mos mostt ac accu cura rate te bu butt co comme mmerc rcia iall lly y av avai aila lable ble methods for obtaining geotechnical parameters parameters for FE analysis. Nevertheless, Nevertheless, FE analysis often needs to be performed without an adequate site investigation. This section summarises summari ses some reaso reasons ns for a lack of site inves investigat tigation ion information and possible means of overcoming them, before describing other potential sources of parameters that may provide sufficiently accurate model parameters for FE analysis. These techniques are also useful in providing further validation of model parameters.

Possible reasons for insufficient site investigation information Highly variable or gravelly ground Highly variable soil is difficult to divide into identifiable layers for parameter testing while the presence of gravels or cobbles prevents high-quality sampling and the use of many in situ and laboratory testing techniques. 94 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.


Figure 3.14 Full pressuremeter test simulation Centre-line

Test pocket will need to be supported by applied equivalent in situ horizontal stress until test is simulated

Interface elements may be required at the corners of the test pocket to allow deformation away from the fixed vertical boundary

Probe radius

Detail p

p

Pressuremeter test Axisymmetric simulation

Remote model boundaries or, for calibration chamber simulation, at chamber boundaries

Axisymmetric simulation (detail)

To overcome this, large volumes of the ground can be tested in situ, for example by seismic testing, large diameter PLT or by trial excavations. Monitored structures in similar ground conditions can be back-analysed by FE analysis to obtain macro properties for variablee soils. variabl Insufficient funds available or site investigation completed earlier Advanced site investigation techniques can be expensive and may be unaffordable on some projects. Also, the site investigation may have been completed earlier without having ha ving FE anal analys ysis is in mind and only basic tes testt re resul sults ts and low low-qua -qualit lity y sam samples ples are available. Of course, such a paucity of information would require very conservative characteristic values of parameters, bringing into question the value of performing FE analysis at all. Alternatively, by back-analysing similar case studies or simply estimating advanced parameters, preliminary FE analyses could be used to demonstrate the potential benefit of performing further advanced parameter testing. If the potential savings in construction costs outweigh the initial costs of further investigation, then it may be possible to secure funds for additional testing.

Other sources of parameters Having taken all possible steps to obtain geotechnical parameters by high-quality site investigation even in difficult circumstances, the alternative is to use other means to obtain parameters, as described here. These sources should be used in combination, not in isolation, to derive parameters parameters by differ different ent means, thereb thereby y impr improving oving their reliability. reliability. Site-specific empirical relationships After Aft er per perform forming ing a smal smalll numb number er of adva advanced nced tests to obta obtain in ac accur curat atee par parame ameters ters,, deriving site-specific empirical relationships with more basic tests allows a high number of reasonably accurate parameters to be obtained economically. 95 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.


For example, a small number of SBP tests and stress path triaxial tests may be undertaken to derive accurate strength and stiffness parameters. By comparing these parameter et erss wi with th mo morre ba basi sicc te tesst res esul ults ts,, su such ch as SP SPT T N va valu lues es,, ad adja jace cent nt to SB SBP P te tesst or triaxial test sampling locations, existing empirical relationships between N and derived parameters can be refined for the specific site. Provided the same equipment, procedures and personnel are employed for all the other SPTs in that soil layer, reasonably accurate derivations of parameters could be made from a large number of inexpensive SPTs across the whole site. Case study parameters It is likely that high-quality site investigation data or derived parameters for ground strata beneath major towns and cities have been reported in case studies somewhere. These provide a valuable alternative source of parameters, but remember that parameters can vary significantly even within a geological stratum, and different stress states, stre st ress ss pa paths ths and str stress ess his histor tories ies also affect affect tes testt re result sults. s. Ther Therefor efore, e, car caree is nee needed ded to identify subdivisions within strata. Parameters should not simply be adopted from a casee stu cas study dy with without out con conside sidering ring lik likely ely vari variat ation ions, s, per perhaps haps base based d on cha chara racte cteris risati ation on tests, and their effects on FE analysis results. For example, example, Pantelidou and Simpson (2007 (2 007)) no noted ted a co cons nsis isten tentt ve vert rtic ical al va vari riat atio ion n of soi soill par parame amete ters rs of th thee Lon London don Cl Clay ay acros ac rosss cent central ral Lond London, on, clos closely ely fol follow lowing ing its geol geologi ogical cal subd subdivis ivision ions. s. Ho Howev wever, er, vari vari-ation at ionss we were re al also so no note ted d wi withi thin n su subdi bdivi visi sions ons wh which ich co could uld be ca cate tego goris rised ed to a cer certa tain in extent on the basis of Atterberg limits. This shows that even when using a case study from the same geological subdivision, it cannot be assumed that the parameters will be the same. Case study monitoring data Case studies providing monitoring data for structures built in similar ground conditions also provide a valuable source of parameters. Back-analysis of a case study can be used to obtain and validate model parameters, but the case study needs to contain sufficient detail on the design and construction sequence in order to simulate the case study conditions sufficiently accurately. Databases of soil parameters Several parameter databases have been developed and they are growing in number and size all the time. Most contain particular parameters for a specific geological stratum, city/r cit y/regio egion n or samp sampling ling/te /test st meth method, od, such as the Nor Norweg wegian ian Geot Geotech echnica nicall Ins Institu titute te (NGI) block sample database (Karlsrud (Karlsrud et al ., ., 2005), 2005), and many are integrated into a geographical geogr aphical information information sys system tem (GIS) to identif identify y site investigation investigation locations. Some are geared toward specific constitutive models, e.g. Duncan e.g. Duncan et et al . (1980) for (1980) for the Duncan and Chang model. Databases can give an indication of the permissible range of soil parameters but not a single, accurate value. Ideally, detailed information on the source and method of obtaining each parameter, as well as visual descriptions and characterisation test results results should be avai available lable in a data database, base, so that the user can assess its reliability reliability and suitability for a specific project.



Appendix 3.1 Some equations equations that may be useful in the valida validation tion of model or initia initiall sta state te parameters.

Equations for estimating K 0 For normally consolidated soils, Jaky’s equation: K 0 = 1 − sin w ′

(3.6 ibid.)

provides quite a reliable stress ratio. For over-consolidated soils, other, more approximate equations have been proposed to estimate K 0 , such as (from Wroth, (from Wroth, 1975): 1975): ′

K 0oc

n

′

        

m

= OCR 1 − sin w −

3 1 − K nc 1 + 2K nc

−

1 − n′

3 1 − K 0

(OCR − 1)

= ln

1 + 2K 0

(3.8)

OCR 1 + 2K nc 1 + 2K 0



(3.9)

′ where m = 0.0022875 0.0022875PI /s v′ and K nc PI + 1.22, OCR = s v,max nc is obtained from Equation 3.6.

Equation 3.8 has pro Equation provided vided a reasonable predictio prediction n for a number of soils up to an OCR of ′ acceptable table range of 0.1 to 0.25, while Equat Equation ion 3.9 about 5 and provided n lies within an accep was proposed for OCR values above 5 and requires an iterative solution. There are also the following empirical relationships: ′

sin w ′ K oc 0 = (1 − sin w )OCR

K 0oc K 0nc

 

(3.10) (Mayne and Kulhawy, 1982) 1982)

2

= OCR

(3.11)

(Schmidt, 1966) 1966)

Correlations Corr elations between c u and drained strength or pre-consolidation stress of clays



cu = sin w ′ c′ cot w ′ +

  1 + K 0 ′ s v 2

(3.12)

for normally consolidated clays cu 0.8 = 0.23OCR ′ s v0

cu = 0.22s p′

(3.13)

(Jamiolkowski Jamiolkowski et et al ., ., 1985) 1985)

(3.14) (Mesri, 1975) 1975)

where s ′p is the maximum pre-consolidation stress. 97 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.


Elastic relationships E ′ G= 2(1 + n′ ) E ′

′

K =

(3.15) (3.16)

3(1 − 2n′ )

G′ = Gu since pore water has no shear stiffness, therefore: 1.5E ′ E u = (1 + n′ ) 1 − n′ E ′

′

E oed =

(3.17)

 

(3.18)

(1 − 2n′ )(1 + n′ )

Correlations between G0 and soil states G0 = 70 G0 = 33

(2.17 − e)

2

(1 + e) (2.97 − e) (1 + e)

p′ 0.5 (MPa) for rounded sands

(3.19)

p′ 0.5 (MPa) for angular sands

(3.20)

2

(both from Richart from Richart et et al ., ., 1970) 1970) OCRk G0 = 625 pa p′ 2 0.3 + 0.7e

    

(3.21)

for clays where k is obtained from the graph in Figure in Figure 3.15 (Hardin, 1978) 1978) – 3

Gv 0 = C p (1 + e)

p′ (MPa) pa

(3.22)

where C p is a constant generally between 300 and 600 MPa. From a survey of resonant where C column tests on clays and sands (Clayton, 2011). Figure 3.15 k value value for Equation 3.21 0.5 0.4 0.3 k

0.2 0.1 0 0

20

40

60

80 80

100

Plasticity index I p



REFERENCES

ASTM (2012 (2012)) ASTM D418 D4186/D4 6/D4186M186M-12e1, 12e1, Stan Standard dard tes testt metho method d for oneone-dime dimensio nsional nal conso con solid lidat ation ion pr prope operti rties es of sa satur turat ated ed coh cohesi esive ve soi soils ls usi using ng con contr troll olleded-st stra rain in loa loadin ding. g. ASTM International, West Conshohocken, PA. Bell Be llot otti ti R, Gh Ghio ionn nna a V, Ja Jami miol olko kows wski ki M, Rob ober erts tson on PK an and d Pet eter erso son n RW (1 (198 989) 9) Interp Int erpre reta tatio tion n of mod moduli uli fr from om sel self-b f-bor oring ing pr press essur ureme emeter ter tes tests ts in san sand. d. Ge Ge´ ´ otechnique otechnique 39(2): 269–292. Binns A (1998) Rotary (1998) Rotary coring in soils and soft rocks for geotechnical engineering. Proceedings of the Institution of Civil Engineers – Geotechnical Engineering 131(2): 63–74. Bolton MD and Lau CK (1993) Vertical Vertical bear bearing ing capa capacity city fac factors tors for cir circular cular and str strip ip 30(6): 1024–1033. footings on Mohr–Coulomb soil. Canadian Geotechnical Journal 30(6) Bond A and Harris A (2008) Decoding Eurocode 7 . CRC Press, Abingdon. Brinkgreve RBJ, Engin E and Engin HK (2010) Validation (2010) Validation of empirical formulas to derive Numerical Metho Methods ds in Geote Geotechnic chnical al Engin Engineerin eering g (Benz modell par mode paramet ameters ers for sands. In Numerical and Nordal (eds.)), Taylor & Francis, London, pp. 137–142. BSI (1990) BS (1990) BS 1377-6:1990. Methods of test for soils for civil engineering purposes, Part 6: Consolidation and permeability tests in hydraulic cells with pore pressure measurement. BSI, London. Butcher AP, Campanella RG, Kaynia AM and Massarsch KR (2005) Seismic cone downhole procedure to measure shear wave velocity – a guideline, ISSMGE TC10: Geophysical Testing in Geotechnical Engineering. Groundwater ter Low Lowering ering in Cons Construct truction ion, A Pra Practica ctical l Cashma Cas hman n PM and Pr Preen eenee M (20 (2012) 12) Groundwa Guide to Dewatering, 2nd edn. CRC Press, Boca Raton, FL. CEN (2004) EN 1997-1 Eurocode 7: Geotechnical design – Part 1: General rules. CEN, Brussels. CEN (2007) EN 1997-2 Eurocode 7: Geotechnical design – Part 2: Ground investigation and testing. CEN, Brussels. Chandler RJ, Leroueil S and Trenter NA (1990) Measurements of permeability of London Clay using a self-boring permeameter. Ge Ge´ ´ otechnique otechnique 40(1): 113–124. Pressuremeters eters in Geote Geotechnic chnical al Desig Design n. Bla Clarke Clar ke BG (1995 (1995)) Pressurem Blacki ckiee Aca Academ demic ic and Pr Profe ofesssional, Glasgow. Stand andard ard Pe Penet netra ratio tion n Te Test st (SP (SPT): T): Met Method hodss and Use, CI Clayto Cla yton n CRI (19 (1995) 95) The St CIRI RIA A Report 143. CIRIA, London. Ge´ ´ otechnique otechnique 61(1): Clayto Cla yton n CRI (20 (2011) 11) Stiffn Stiffness ess at sma small ll st strrain: re resea searc rch h and pr pract actice. ice. Ge 5–37. Clay Cl ayto ton n CR CRII an and d He Heym yman ann n G (2 (200 001) 1) Stiffn Stiffness ess of geo geoma mater terial ialss at ve very ry sm small all st stra rains ins.. Ge´ ´ otechnique Ge otechnique 51(3): 245–255. Clayton CRI and Siddique A (1999) Tube (1999) Tube sampling disturbance – forgotten truths and new perspectives. Proceedi Proceedings ngs of the Ins Institu titution tion of Civi Civill Engi Engineers neers – Geot Geotechn echnica icall Engi Engineeri neering ng 137(3): 127–135. Clayto Cla yton n CRI and Sym Symons ons IF (19 (1992) 92) The pr press essur uree of com compa pacte cted d fill on re retai tainin ning g wa walls lls.. Ge´ ´ otechnique Ge otechnique 42(1): 127–130. Clayton CRI, Matthews MC and Simons NE (1995) Site Investigation, 2nd edn. Blackwell Science, Oxford. Earth Pr Press essur uree and Ear EarththClay Cl ayto ton n CR CRI, I, Woo oods ds RI RI,, Bo Bond nd AJ an and d Mi Mili liti tits tsky ky J (2 (201 013) 3) Earth Retaining Structures, 3rd edn. CRC Press, Abingdon. 99 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.


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103



Chapter 4

How are groundwater effects taken into account? 4.1. 4.1.1 4.1 .1

Introduction Int How Ho w are are satur saturat ated, ed, part partial ially ly sat satur urate ated d and dry dry soils soils mode modelle lled? d?

In pr prac actic tical al geo geotech technic nical al FE ana analy lysis sis the po pore re wa water ter and sol solid id par partic ticles les ar aree nev never er modelled separately, each with their own constitutive relationship as in fluid–structure interaction analyses. Rather, the effect of the groundwater is included in the analysis using some assumptions. The soil mass of solid particles, water and air is modelled as a single continuum with a constitutive model that represents the engineering properties of this combined mass. In effective stress analyses, the stresses within the soil mass are divided into pore pressure and effective stress. The modelling of saturated soils is the topic of this whole chapter. Consideration is required of: g g g

g

groundwater pressure groundwater pressure and its direct influence on effective effective stress the influence of any groundwa groundwater ter flow on groundwa groundwater ter pressure any change change in the vo volume lume of a soil, perhaps due to loading loading or unlo unloadin ading, g, because because this requires water to flow in or out of voids so that they can change volume temporal effects in low-permeability low-permeability soils soils where incompressible incompressible groundwater groundwater supports load changes in the short term until they dissipate and load is transferred to the soil skeleton.

How the FE analysis takes into account these effects is covered in the following sections of this chapter. Dry, granular soils are relatively straightforward to model in this respect because there are no groundwater pressures or groundwater flow to consider. The soil is modelled in terms of effective stress and the pore pressure is zero. Note that natural dry-looking clays aree ne ar nev ver tr trul uly y dr dry y an and d ca cann nnot ot be mo mode dell lled ed as dr dry y so soil ils. s. Th They ey al alw ways re reta tain in so some me moisture, so are partially saturated. The behaviour of partially saturated soils is significantly more complicated than that of saturated or dry soils. The soil is composed of three phases (solid, liquid and gas) with the interaction between the liquid water and water vapour/air phases being particularly complex. 105 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.


Due to the dif Due difficu ficulty lty of sim simula ulatin ting g par partia tially lly sa satur turat ated ed soi soill beh behav aviou iour, r, it is com common mon practice, and conservative, to assume the soil above groundwater level is dry with pore pressu pr essure re zer zero o an and d to ado adopt pt the sam samee soi soill pa para ramet meters ers as for the sa satur turat ated ed soi soil. l. Thi Thiss neglects the elevated effective stress arising from pore suctions that otherwise appear to give the soil higher strength and stiffness. Alternatively, partially saturated clay can be modelled in terms of total stress (undrained Method C – see Section 4.2.4) and the stiffness and undrained shear strength set to higher values as appropriate for the partially saturated clay. However, always remember that any change in moisture content would cause significant changes in the stiffness and undrained shear strength and these would not be taken into account when modelling the clay in this simplified way. Also, neither of these simplified approaches would predict volume changes arising from changes in the moisture content of partially saturated fine-grained soils. Consequently, simula sim ulatio tion n of fou founda ndatio tion n hea heave ve on an ex expan pansiv sivee cla clay, y, for ex exam ample ple,, by FE ana analy lysis sis requires specialised approaches not covered by this book. Readers could refer to, for example, Fredlund et al . (2012) and and Gens Gens et al . (2006). (2006).

4.1.2 4.1 .2

Whatt do Wha do the dif differ ferent ent por pore e pre pressu ssure re ter terms ms mean mean? ?

In FE analysis it is useful to divide pore pressure into different terms because there are different options for the way each is calculated.

Steady-state or ‘at rest’ pore pressure This is the por This poree pr pressu essure re ari arisin sing g fr from om con const stant ant hy hydr draul aulic ic bo bound undary ary con condit dition ionss (e. (e.g. g. stationary groundwater level or constant extraction rate from a well). This part of the pore po re pr pres essu sure re do does es no nott ch chan ange ge wi with th ti time me du duri ring ng a di disp spla lace ceme ment nt or co cons nsol olid idat atio ion n analysis. However, it is possible to change the hydraulic boundary conditions in an analysis stage in order to establish a new steady-state pore pressure distribution (see Figure 4.1). 4.1). This pore pressure can be specified directly for horizontal phreatic surfaces when setting up the initial stresses and in subsequent analysis stages, or it can be calculated by a steady-state groundwater flow (or seepage) analysis (see Section 4.3). Horizontal phreatic surfaces occur in hydrostatic conditions, where there is negligible groundwater flow, and can be used to enter the steady-state pore pressure in other situations where hydraulic gradients are not very high. A typical example may be the steady-state pore pressure around a retaining wall to a basement with a dewatering scheme, where it may be sufficiently accurate, provided hydraulic gradients are not too high and changes in ground density are taken into account, to specify phreatic surfaces on each side of the retaining wall and assume a linear variation of steady-state pore pressure toward the wall toe where the pressures should be equal. The use of groundwater flow analysis to calculate rather than specify the steady-state pore pressure would be necessary in more complex cases, e.g. strata with different permeabilities or void ratio-dependent permeability, or where hydraulic gradients are high. 106 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.

How are groundwater effects taken into account?

Figure 4.1 Generi Genericc cont contours ours of por pore e pres pressure sure in typic typical al pro problems blems in low low-perm -permeabili eability ty soil Steady state pore pressure

Excess pore pressure, ue

In situ conditions

Total pore pressure

ue = 0

Impermeable wall Rapid change in hydraulic boundary conditions

Positive ue

End of consolidation

ue = 0

Excavation Excavation and load application

Positive ue Negative (suction) ue

End of consolidation ue = 0

Excess or non-equilibrium pore pressure This is the cha This change nge in po pore re pr pressu essure re fr from om the ste stead ady-s y-sta tate te va value lue cau caused sed by loa loadin ding, g, unloading or a rapid change in the hydraulic boundary conditions in undrained or consolida sol idatio tion n con condit dition ions. s. It can als also o be gen gener erat ated ed by dev devia iator toric ic st stres resss cha change ngess whe where re increased excess pore pressure is generated in normally and lightly over-consolidated soils and decreased (suction) exc excess ess por poree pre pressur ssuree in hea heavily vily over-conso over-consolida lidated ted soils durin du ring g un undr drain ained ed she shear, ar, if the con const stitu itutiv tivee mod model el inc incorp orpor orat ates es suc such h beh behav aviou iour. r. In general terms, hardening models can predict positive excess pore pressure in normally and lightly ov over-con er-consolid solidated ated clays, while enteri entering ng a nonnon-zero zero dilation dilation ang angle le cause causess 107 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.


negative (suction) excess pore pressures to be generated for heavily over-consolidated soils, although the latter is not recommended because it can lead to an over-prediction of undrained strength. Excess pore pressure can be calculated in a displacement analysis in undrained conditions (Method A or B – see Section 4.2.4), or in a coupled consolidation analysis (Section 4.4) when prediction of the variation of pore pressure with time is needed. It is generated in the analysis model by small volumetric strains in the virtually incompressible pore water but the accuracy of excess pore pressure predictions depends to a large extent on the accuracy of the constitutive model (see Section 4.2.5). At typical loading rates in the field, excess pore pressures occur only in low-permeability soils (e.g. clays) and would always be zero in free-draining soils (e.g. sand and gravel). Only at very high loading rates, e.g. seismic loading, might excess pore pressure occur in high-permeability soils such as sand. Excess pore pressures dissipate to zero during consolidation until steady-state conditions are restored (see Figure 4.1).

Transient pore pressure Sometime Somet imess the there re ar aree not st stead eady-s y-sta tate te con condit dition ions, s, for ex examp ample le wit with h a tid tidal al va varia riatio tion n of gr groun oundw dwat ater er lev level, el, or dur during ing a pum pumpin ping g tes testt or ra rapid pid dr draw awdo down wn of a re reserv servoir oir.. Consequently, Consequ ently, the groundw groundwater ater pressure is changing with time due to changing hydraulic hydraulic boundary conditions and the steady-state pore pressure becomes transient pore pressure. This is distinct from the temporal dissipation of excess pore pressure occurring during consolidation as described in the preceding paragraph. To handle transient pore pressures in an FE analysis, either extreme steady states can be considered (e.g. in tidal conditions adopt hydrostatic pore pressure with the phreatic level at the highest and lowest astronomical tides) or time-dependent changes in groundwate terr flo flow w an and d po pore re pr pres essu sure re in re resp spon onse se to ti time me-d -dep epen ende dent nt hy hydr drau auli licc bo boun unda dary ry conditions can be predicted using transient groundwater flow analysis. The predicted values are therefore transient rather than steady-state pore pressures. It is preferable to perform such analyses in a separate stage to displacement analyses to avoid unnecessary complexity, but in exceptional cases a transient groundwater flow analysis can be coupled with a displacement and consolidation analysis in order to predict the timedependent depen dent gro ground und displ displace acement ment caus caused ed by tempo temporal ral chan changes ges in hydr hydraulic aulic boun boundary dary conditions.

Total or active pore pressure This is the actual value of pore pressure at a point at a particular time and is the sum of the steady-state and excess pore pressure (see Figure 4.1). In the case of a transient groundwater flow analysis, the total pore pressure would be the sum of the transient pore pressure and excess pore pressure. Where possible, only one of these pore pressure terms should be changing in each analysis stage, to avoid unnecessary complexity. In exceptional cases, they may vary simultaneously. 108 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.


4.2. 4.2. 4.2.1 4.2 .1

Draine Drai ned d an and d un undr drai aine ned d an anal aly yse ses s Whatt do the term Wha termss dr drain ained ed and und undra raine ined d mean mean? ?

Undrained means no water flow, so the volume of voids in a saturated soil remains unchanged and the soil density remains constant. Since water is essentially incompressible in comparison with the soil skeleton, any imposed loading is transferred directly to the pore water (as excess pore pressure) and the effective stress remains constant, while any attempt to impose volume change is resisted by excess pore pressure changes. In an undrained analysis the excess pore pressure is not allowed to dissipate. Although, in reality, consolidation and dissipation of excess pore pressure starts immediately on loading, it is sufficiently slow in some cases for undrained conditions to be a reasonable assumption for field problems. These cases are where the rate of loading is high relative to the soil’s permeability, which occurs in most construction activities in stiff clays, or during particularly high rates of loading (e.g. earthquake accelerations) in any soil. Drained means that water is free to flow through the voids of a soil such that void volume changes can occur and no excess pore pressures develop. Any loadings result in total stress changes that equal effective stress changes. These conditions occur when the rate of loading is low relative to the soil’s permeability, which occurs in most construction activities in sands and gravels. Modelling undrained conditions in FE analysis is not without its pitfalls, as described in Section 4.2.4. Modelling drained conditions is more straightforward.

4.2.2 4.2 .2

When Whe n are are drain drained ed or or undr undrain ained ed assu assumpti mptions ons app appro ropri priate ate? ?

In this section, ‘rate of loading’ is taken to also include rates of unloading, hydraulic boundary condition change and shear strain. If the rate of loading is sufficiently slow relative to soil permeability that no significant excess pore pressures are generated, then a drained analysis is appropriate, as summarised in Figure 4.2. 4.2. For most practical cases, this encompasses construction activities in granular soils such as sand and gravel. It does not matter whether short-term or long-term conditions are needed because the output would be the same. Vermeer and Meier (1998) proposed that U . 70% or T . 0.40 in order to adopt the drained assumption, where U = = degree of consolidation, T = = time factor: T =

kE oed 2 g w d

t

(4.1)

where k = permeability, E oed oed = one-dimensional stiffness, g w = weight density of water, d = = drainage path length, t = construction time being simulated. If the rate of loading is sufficiently high relative to soil permeability that no significant dissipation sipa tion of excess pore pore pressure pressure occurs during the loading loading itself, then an undr undrained ained analysi analysiss is appropriate. For most practical cases, this includes construction activities in stiff clays. 109 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.


Figure Figur e 4.2 Selection of drained, undrained or consolidation analyses Drained construction

Undrained construction Short-term outputs

Drained analysis

Undrained analysis

Time-dependent outputs

Partial consolidation during construction Long-term outputs

Undrained analysis (method A)*

Consolidation analysis

For all time periods Consolidation analysis * Method A should only be used with an appropriate, advanced constitutive model. Refer to Section 4.2.5.

An un undr drai aine ned d anal analy ysis prov provid ides es ou outp tput utss fo forr the the sh shor ortt-ter term m ca case se im imme medi dia ate tely ly on comple com pletio tion n of loa loadin ding, g, unl unloa oadin ding, g, hy hydr draul aulic ic bou bounda ndary ry con condit dition ion cha chang ngee or she shear ar straining. Vermeer and Meier (1998) proposed that an undrained assumption may be appropriate when U , 10% or T , 0.10 during construction. It is wrong to assume that all clay soils behave in an undrained manner. In particular, normally or lightly over-consolidated clays may behave in an almost drained manner since often they are quite thin deposits and have sandy or silty laminations that shorten drainage path lengths. If long-term outputs were required from a soil that behaves in an undrained way during construction, it would be wrong to perform only a drained analysis in an attempt to obtain the long-term outputs directly. The reason for this is illustrated by the example of triaxial compression test simulation outputs shown in Figure 4.3. 4.3. An undrained triaxial compression test on a lightly over-consolidated clay under stress control was simulated, which was followed by consolidation to allow excess pore pressures to dissipate. Then a drained triaxial compression test was simulated on the same clay directly to the same stress state. On the graph of deviatoric stress against axial strain, the undrained and drain dr ained ed lin lines es ar aree ini initia tially lly qui quite te clo close, se, wit with h the und undra raine ined d cas casee bei being ng slig slightl htly y sti stiffe ffer. r. However, once yield starts in the undrained specimen as it approaches the failure line, the lines diverge considerably considerably.. This simple exa example mple demonstra demonstrates tes the impo importanc rtancee of simulating the correct stress path rather than establishing the final stress state by a direct path, and why long-term conditions following undrained or partially drained construction conditions cannot be simulated simply by performing a drained analysis. Doing so introduces errors to the outputs and, worse still, could miss potential failure states in undrained conditions. 110 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.


Figure 4.3 Stres Stresss path pathss in the simulation simulation of longlong-term term conditions conditions in low low-perm -permeabili eability ty soils

q


Consolidation Small difference with stress control

n e i l r e u l i a F

Undrained compression

Drained compression

Mean effective stress, p′

q


Undrained compression

Drained compression

Consolidation

Large difference due to yielding in undrained case

Axial strain, εa

Therefore, to obtain long-term Therefore, long-term outputs follo following wing undrained undrained construction construction conditions, conditions, construction must be simulated in undrained conditions using Method A (see Section 4.2.4) such that excess pore pressures are generated, followed by a consolidation analysis (see Sectio Sec tion n 4.4 4.4)) to dis dissip sipat atee ex exces cesss por poree pr press essur ures es for the lon long-t g-term erm cas case, e, as sho shown wn in Figure 4.3. Note that the accuracy of Method A predictions is heavily dependent on the constitutive model adopted for the soil. In cases where only adequate safety against geotechnical failure of a structure constructed in undrained or partially drained conditions is being assessed, it might be acceptable to perform an undrained (Method C) and/or drained analysis, depending on which is critical (see Figure Figure 4.4 4.4), ), with bas basic ic but app approp ropria riate te con constit stitutiv utivee mod models els to chec check k for fail failur ure, e, remembering that displacement and structural force predictions may be inaccurate. In cases where the rate of loading falls between the limits of drained and undrained behaviour, i.e. excess pore pressures develop and then partially dissipate during construction, a consolidation analysis is necessary (see Section 4.4). This tends to occur when simulating construction activities in normally or lightly over-consolidated clays, 111 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.


silts and any mixed soils with a significant clay content. The rate of loading will need to be entered into the FE analysis as will the rate of excess pore pressure dissipation (using soil permeability and hydrau hydraulic lic boundary conditions). Note that a consolidation analysis would normally include a parametric study of soil permeability due to the high degree of uncertainty in this parameter. Where intermediate cases are ‘nearly drained’ or ‘nearly undrained’ during construction, it may be acceptable to undertake a simpler drained or undrained analysis, respectively, where it is conservative, as described in Figure 4.4 and the following paragraphs. Unloading (e.g. excavation, cut slope) causes an immediate negative (suction) excess pore pressure. With time, the soil swells as the excess pore pressure dissipates, effective stress reduces and failure is approached. Therefore, the long-term drained assumption is safety critical. For short-term stability during construction, stiff, heavily over-consolidated da ted cla clays ys can be ass assum umed ed to be un undr drain ained, ed, but no norma rmally lly con consol solida idated ted or lig lightl htly y over-consolidated clays should usually be assumed as drained even in short-term cases. Loading (e.g. emba Loading embankme nkment nt cons constructi truction on on a clay foundation) foundation) cause causess an immed immediat iatee increa inc rease se in ex excess cess po pore re pr pressu essure re.. Wi With th tim time, e, the soi soill con consol solida idates tes as the ex exces cesss po pore re pressure dissipates, effective stress increases and the soil moves away from failure. Therefore, stability increases with time and undrained short-term stability is critical. However, clay fill for embankments may have negative (suction) excess pore pressure on placement, particularly in plastic clays. The effective stress would decrease with time and the soil would approach failure. Therefore, the long-term drained case may be critical for the stability of clay fill in embankments. These unloading and loading cases so far refer to mean total stress changes. Deviatoric stress changes also cause excess pore pressure, and beware of cases where the deviatoric stre st ress ss go gove verns rns the sen sense se of ex excess cess po pore re pr press essur ure. e. Nor Normal mally ly con consol solida idated ted and lig lightl htly y Figure Figur e 4.4 Safety-critical drainage conditions for constructions in low-permeability soils Unloading problems Long-term (drained) case critical except perhaps in very soft normally consolidated clays e.g. basement excavation

e.g. cut slope Loading problems Short-term (undrained) case critical except perhaps in heavily over-consolidated clays

e.g. foundation

e.g. embankment construction



over-consolidated clays generate positive excess pore pressure during undrained shear while heavily over-consolidated clays generate negative (suction) excess pore pressure and there is a sliding scale between these extremes of over-consolidation. Therefore, very soft norm normally ally conso consolida lidated ted cla clay y may generate generate suffic sufficient ient posit positive ive exce excess ss por poree pre pressur ssuree from undrained shear due to unloading to overcome the negative excess pore pressure generated from the reduced mean total stress. In such a case, the factor of safety on stability may actually be lower in the short term. Similarly, a heavily over-consolidated clay may generate sufficient negative excess pore pressure from undrained shear during loading to overcome the positive excess pore pressure arising from the increase in mean total stress. In such a case, stability would be more critical in the long term. How well an FE model can predict excess pore pressure in the face of these conflicting influences depends to a large extent on the constitutive model adopted for the soil. In most cases there will be a high degree of uncertainty regarding excess pore pressures, so careful parametric studies and other validation exercises are required to assess the reliability of outputs. Where a ground model contains some layers requiring drained analysis and some layers undrained analysis, then clearly the FE analysis should be run with the appropriate assumption for each layer concurrently. Where some layers require consolidation analysis and some require drained or undrained analysis, there are two options:

1

2

The layers layers for drained or undrained undrained analysis are are composed of non-consolidating non-consolidating elements (displacement degrees of freedom only) while the layers for consolidation analysis are composed of consolidating elements (displacement and pore pressure degrees of freedom) with appropriate hydraulic boundary conditions at interfaces with other non-consolidating layers (see Section 4.3.3). Perform Per form a consolidatio consolidation n analysis analysis for all soil layers layers with high permeability permeability for the drained layers and low permeability for the undrained layers.

4.2.3 4.2 .3

How Ho w is dr drain ained ed ana analy lysis sis perf perform ormed? ed?

In drained analysis, the pore water does not contribute to soil stiffness because it is assumed free flowing. The engineering behaviour of the soil is governed only by the mechanical properties of the soil skeleton (Figure ( Figure 4.5). 4.5). Consequently, no excess pore pressure is generated and total stress changes equal effective stress changes. Since the pore water contributes no stiffness, a drained analysis is also appropriate for dry soils. Alternat Altern ativ ively ely,, dr drain ained ed beh behav aviou iourr can be sim simula ulated ted in a con consol solida idatio tion n an analy alysis sis (see Section 4.4) by allowing excess pore pressures to dissipate to insignificant levels. Constitu Cons titutiv tivee mod models els sho should uld be for formul mulat ated ed in ter terms ms of effe effecti ctive ve st stres resss usi using ng dr drain ained ed parameters.

4.2.4 4.2 .4

How Ho w is und undrai rained ned ana analy lysis sis per perfor formed med? ?

There is a higher potential for error with undrained analyses, so increased caution and checking of results is required. 113 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.


Figure Figur e 4.5 Methods of simulating drained and undrained soil behaviour Dry or drained soil

Undrained (A & B) V

V

=

V

V

δV

δV (small)

δV (small)

δσ

δσ

δσ

= ∆σ′ ∆u = 0, ∆σ = δV

Undrained (C)

∆σ′ and ∆u calculated

δσ

δV

K ′

V

K ′ = bulk modulus

≈

= ∆u ∆σ′ = 0, ∆σ =

δσ

δV

K w

V

≈

δσ K u

of soil skeleton

K w = bulk modulus of water ≈ 100 to 1000K ′

K u = undrained

Constitutive model in terms of effective stress

Constitutive model in terms of effective stress

Constitutive model in terms of total stress

Common input: E ′, v ′, ϕ′

Common input: E ′, v ′, K w, ϕ′ or c u

Common input: E u, v u (= 0.495 to 0.499), c u

bulk modulus

There are three methods of simulating undrained soil behaviour (not including consolidation analysis with a short time interval):

Method A (effective stress analysis) A high value of bulk modulus K w for the pore water is added into the stiffness of the soil so that volumetric strains are small and excess pore pressures are generated (Figure 4.5). K w is either entered manually or calculated automatically depending on the software, while all other model parameters, including shear strength, are entered as drained effective stress values. This method has the advantage of providing outputs of excess pore pressure, but these are only likely to be reasonably accurate when using appropriate, advanced constitutive models. It should also take account of changes in soil behaviour, in particular undrained strength, due to preceding construction stages because undrained strength is continuously formulated in terms of effective stress. However, the accuracy of the for formul mulat ated ed und undra raine ined d st stre rengt ngth h is dep depend endent ent on the com comput putat ation ion of ex exces cesss por poree pressure and is often detrimentally affected by a lack of effective stress testing data (see Section 4.2.5).

Method B (effective stress analysis, specified undrained strength) This works in the same way as Method A, except that the shear stress is limited by the specified undrained shear strength (and drained shear strength is no longer an input parameter). This removes the potential for a dangerous over-prediction of undrained shear strength when using Method A, which is more likely in basic models such as the LEPP Mohr–Coulomb model, for which Method B can be more appropriate. However, excess pore pressure predictions may become highly inaccurate, so Method B should not 114 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.


be followed with a consolidation analysis, and advanced constitutive models may lose some of their features when using Method B. Furthermore, changes in undrained shear strength due, for example, to consolidation would not be taken into account unless the undrained shear strength were re-specified in a new model.

Method C (total stress analysis) Undrained soil parameters are entered directly into the model with undrained Young’s modulus E u and undrained Poisson’s ratio nu being the common stiffness parameters. Theoretically, nu should be 0.5 for the bulk modulus K u to become infinitesimal, but to avoid numerical problems a value slightly less than 0.5 is adopted, typically nu = 0.495 to 0.499. There is no separate term for the bulk modulus of pore water, so no excess pore pressure is calculated. The undrained shear strength is also entered directly. Method C is app Method appro ropri priat atee for bas basic ic con const stitu itutiv tivee mod models els whe where re un unre reali alist stic ic con condit dition ionss might otherwise be predicted with Methods A or B. It is not suited to advanced soil models, except those formulated in terms of total stress. One disadvantage is that any undrained strength changes occurring due to consolidation would not be accounted for because consolidation is not modelled, so these would need to be specified by the user.

Consolidation analysis with short time interval An alternative method of simulating undrained behaviour is a coupled consolidation analysis (see Section 4.4) with a time interval short enough for the dissipation of excess pore pressure to be insignificant. This is equivalent to Method A or B, depending on whether a drained or undrained shear strength is specified in the consolidation analysis. Remember Rem ember,, how howeve ever, r, tha thatt specif specifying ying an undr undrained ained shear str strength ength in a cons consolida olidation tion analysis has the drawback that the strength is not updated automatically due to the effe ef fects cts of co cons nsol olid ida ati tion on an and d th thee pr pred edic icti tion on of ex exces cesss po pore re pr pres essu sure re is li lik kely to be inaccurate.

Bulk modulus

K

The bulk modulus K of of soil grains is about 30 times greater than K of water, so the change in volume of soil grains is assumed to be zero. This is distinct from K ′ of the soil skeleton which arises from rearrangement of the soil grains. In drained soil, pore water is free to flow so it possesses zero bulk modulus, so K ′ governs volumetric strain and stresses are carried by the soil skeleton. The bulk modulus of undrained saturated soil is governed by K w of the water phase only, because the pore water cannot flow and because K w is significantly higher than K ′ of the soil skeleton. Therefore, in Methods A and B and in coupled consolidation and groundwater flow analyses, K w is set to a value significantly higher than K ′ for the accurate simulation of undrained behaviour but less than the true K w of water to avoid numerical problems. Some programs set K w automatically automatically,, otherwise K w needs to be set by the user. A value of 100 to 1000 times K ′ is appropriate and outputs are not particularly sensitive to values within this range. In Method C, K w is set to zero and the high bulk stiffness of the undrained soil mass is set indirectly by specifying a Poisson’s ratio of 0.495 to 0.499. 115 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.


Undrained shear strength

c u

cu ( (or or s u ) is not a fundamental parameter but depends on stress state and stress history and typically changes during construction activities. In Method A it is calculated by the constitut cons titutive ive model and is not an inpu inputt par paramete ameter, r, so can tak takee acc account ount of chan changes ges in the stress state. In Methods B and C it is an input parameter that needs to be appropriate for the stress history and stress state at any point during the analysis.

The dilation angle should be set to zero when using Method A otherwise failure can be prevented due to the continuous generation of negative (suction) excess pore pressure, and hence increased effective stress, during shear.

4.2. 4. 2.5 5

Why Wh y is th the e pr pred edic icti tion on of c in Method A often inaccurate? u

The prediction of excess pore pressure is critical to predicting accurate effective stress paths and failure at realistic values of cu . The problem is that only advanced models appr ap propr opria iate te for the soi soill and mod model el con condit dition ionss ar aree cap capab able le of pr prod oduci ucing ng rea reason sonab ably ly accurate predictions of excess pore pressure in response to changes in mean total stress and, in particular, deviatoric stress. Basic models can generate highly inaccurate stress paths pa ths,, lea leadin ding g to the cal calcul culat ation ion of err errone oneou ouss cu value valuess ba based sed on eff effecti ective ve st stre ress ss parameters. Figure 4.6 shows the prediction of undrained shear strength in a lightly over-consolidated da ted cl clay ay by Me Meth thod od A fo forr a pu pure rely ly de devi via ato tori ricc lo load adin ing. g. Th Thee LE LEPP PP mo mode dell wi with th Mohr–C Moh r–Coul oulom omb b fai failur luree crit criteri erion on is a ba basic sic mod model el wit with h no har harden dening ing pr prop operti erties, es, so predicts a vertical stress path in undrained deviatoric loading. No excess pore pressure is generated, so effective stress is over-estimated leading to dangerous over-prediction of the undrained shear strength when the stress path reaches the effective stress failure line. Figure 4.6 The prediction of c in a lightly over-consolidated clay using Method A ( Mansik Mansikkama kama¨ ki, 2015) u

Deviatoric stress, q Failure line (effective stress)

Range of predicted undrained strengths by different hardening models and parameters

Actual undrained strength

LEPP Mohr–Coulomb model stress path

Undrained domain of different hardening models and parameters Mean effective stress, p′



Hardening models predict the excess pore pressure that lowers the estimate of undrained shear strength, but this is not an exact science. Different models and parameters will produce a range of possible stress paths around the true stress path, as indicated by the grey shaded area. Method A also predicts cu changes due to consolidation, but these might also be wrong if the wrong stress path is predicted. A further potential source of error is the unavailability of effective stress parameters for low-permeability soils due to the expense of obtaining them. Simulation of the undrained tests (see Section 3.4.2) can be used to back-calculate effective stress parameters. Further test simulations should be performed to check the generation of cu for similar stress state, stress history and stress path to the conditions simulated in the main analysis model, e.g. plane strain, axisymmetric, loading/unloading, primary loading, etc. The dilation angle should always be set to zero in undrained Method A analyses to avoid the negative (suction) excess pore pressures being generated that can cause undrained shear strength to be over-estimated. To help identify any errors, always check that the output of deviatoric stress is less than the estimated value of prevailing undrained shear strength: s 1 − s 3 ≤ 2 cu

(4.2)

as demonstrated for generic foundation and retaining wall examples in Figure 4.7 and 4.7 and in the example in Section 8.3.4. In both examples, the shaded areas indicate where a higher shear strength has been mobilised in the FE analysis than should be available according to the strength profile shown. In which case, the constitutive model parameters, or the constitutive model itself, should be revised until the mobilised strength everywhere is less than or equal to the strength profile. The strength profile should be based on site investigation data, but should also take account of any consolidation or hydraulic boundary condition changes since the site investigation was undertaken. It may be a best estimate of undrained shear strength, or a characteristic or design line, as appropriate for the aims of the FE analysis. Figure 4.7 Checking mobilised undrained shear strength in undrained (Method A) analysis 0

10

20

30

>2c u 40

40

50 c u: kPa

40

>2c u

60 30

Contours of σ 1 – – σ σ 3

Best estimate, characteristic or design line

40

Contours of σ 1 – – σ σ 3

Depth



4.3. 4.3. 4.3.1 4.3 .1

Ground Grou ndw wat ater er flo flow w an anal aly yse ses s Whatt types Wha types of gro groun undw dwat ater er flow flow ana analy lysis sis are per perfor formed? med?

In groundwater flow (or seepage) analyses, the soil skeleton is assumed rigid so only pore pressure degrees of freedom are considered at the nodes and seepage equations are used in the FE analysis. Hydraulic boundary conditions are used to define the problem (see Section 4.3.3). The continuity equation must be satisfied at all locations, which means that volumetric flow rates into an element must be the same as volumetric flow rates out of an element (plus any sources or sinks). The relationships between flow rates in saturated soil and permea per meabil bility ity in the thr three ee axi axiss dir direct ection ionss (i.e (i.e.. Dar Darcy’ cy’ss la law) w) ar aree sub subst stitu ituted ted int into o the continuity equation. In cases of the same permeability in all directions, this equation becomes Laplace’s equation as used in other fields of engineering.

Steady-state analysis The hydraulic head and soil permeability remain constant everywhere with time.

Transient analysis The hydraulic head changes with respect to time in order to model, for example, seasonal or tidal variations, establishing steady-state conditions (e.g. initiating dewatering or a pumping test, rapid drawdown). Boundary conditions are defined as a function of time.

Coupled flow and displacement To avoid unnecessary complexity, it is usually preferable to perform a groundwater flow analysis to establish the steady-state pore pressure distribution separate to a subsequent analysis stage of displacement or consolidation analysis. In some cases, more usually involving transient groundwater flow analysis, it is required to predict both displacement and pore pressure changes due to time-dependent changes in hydraulic boundary conditions, e.g. to predict displacement and stability of a reservoir embankment during rapid drawdown. In such cases, a groundwater analysis and displacement or consolidation analysis are coupled in the same analysis stage. In such cases, the total pore pressure is calculated directly, then excess pore pressures can be calculated at the end from the steady-state pore pressure. In some programs it may also be possible to include a void ratio or effective stress-dependent permeability in order to simulate the effect of volume change on permeability and, in turn, its effect on groundwater flow.

4.3.2 4.3 .2

Why is it dif difficu ficult lt to sim simula ulate te unc unconfi onfined ned flo flow? w?

Unconfined flow occurs where the phreatic surface forms a flowline. A common example is seepage through an embankment dam. These cases are more difficult because the FE analysis needs to predict the position of the phreatic surface. Above the phreatic surface, permeability reduces because the soil becomes partially saturated and the water has a smaller volume through which to flow. So, one method employed in some programs involves reducing the permeability of the soil when specified threshold pore pressures (typically zero or small suction values) are reached. On the compressive side of the threshold, saturated permeability is adopted in the analysis, while on the tensile side a signifi sig nifican cantly tly re reduc duced ed per permea meabil bility ity is ado adopte pted. d. Som Somee pr prog ogra rams ms inc includ ludee un unsa satur turat ated ed 118 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.


groundwater groundw ater flo flow w anal analysi ysis, s, in which case the soil-w soil-wate aterr char charact acterist eristic ic curv curvee (SWC (SWCC) C) describing the relationship between saturation ratio and suction needs to be obtained, as does the relationship between permeability and saturation ratio. Such relationships are not straightforward to obtain accurately for soils, so some parametric study is necessary to assess the likely variation of outputs within permissible ranges of seepage analysis input parameters. Refer to, for example, Fredlund et al . (2012).

4.3.3 4.3 .3

Whatt do the dif Wha differ ferent ent hyd hydrau raulic lic bou bounda ndary ry condi conditio tions ns mean mean? ?

Hydraulic boundary conditions affect excess pore pressure in consolidation analyses (see Section 4.4) and steady-state or transient pore pressure in groundwater flow analyses. Either prescribed flows or changes in pore pressure can be specified and they can be constant cons tant (for stea steady-s dy-stat tatee gro groundw undwate aterr flo flow w anal analysi ysiss and conso consolida lidation tion anal analysi ysis) s) or time-d tim e-depe epende ndent nt (for tr tran ansien sientt gr grou oundw ndwat ater er flo flow w an analy alysis sis). ). The They y ar aree not use used d in the drained and undrained analyses described in Section 4.2. Most programs adopt default boundary conditions at mesh boundaries, so the user must be aware of these in order to decide where user-defined conditions would be more appropriate.

Closed, no-flow or impermeable boundary Closed boundaries allow zero flow across the boundary in either direction and no dissipation of excess pore pressure. Interface elements may also form this boundary at impermeable structures within a mesh when structural elements are otherwise assumed permeable by default. A closed closed bo bound undary ary is usuall usually y ado adopte pted d at the bottom bottom bound boundary ary of a mesh, unl unless ess perh perhaps aps to allow dissipation of excess pore pressure to a hard but permeable layer represented by the bottom boundary. Also, a vertical boundary to a mesh that forms an axis of symmetry should be closed since no horizontal flow should occur across the axis of symmetry.

Open, permeable, free-draining or seepage boundary An open boundary allows unrestricted flow across a boundary in either direction. This is the usual flow boundary condition at the vertical and top mesh boundaries (except axes of symmetry). If the top surface of a mesh is submerged, then the water pressure at an open boundary is determined by the depth of the overlying external water. If a top surface is above the phreatic level, then the open boundary becomes a seepage surface allowing water to flow out through this boundary if the water level coincides with this open downstream boundary during the analysis. In which case, water pressure would be zero but the elevation head would vary in non-horizontal surfaces, so a seepage boundary is not necessarily an equipotential or a streamline. In consolidation analyses, excess pore pressure is zero at an open boundary and the total pore pressure equals the steady-state value.

Infiltration/extraction Infiltration/e xtraction boundary An inflowing or outflowing unit volume per unit time (per metre width in the out-ofplane direction in plane strain analysis) is specified on an infiltration or extraction line boundary specified in the geometry, e.g. a vertical line for an extraction well or the 119 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.


ground surface to simulate rainfall infiltration (where soil has sufficient permeability to absorb the water continuously, otherwise a conditional rainfall/evaporation boundary would be required). Rather like distributed loads, these flow rates are converted by software into equivalent nodal flows.

Drain A drain is like a seepage boundary except that it is placed inside the mesh. In consolidation analyses, the excess pore pressure is zero and the total pore pressure equals a spe specifi cified ed hea head d val value ue or the st stead eady-s y-sta tate te por poree pr press essur ure. e. Su Such ch bo bound undary ary con condit dition ionss are commonly used to simulate vertical wick drains installed in a soft clay foundation to hasten consolidation during embankment construction, as illustrated in Section 8.4.

Sources/sinks A source (inflow) or sink (outflow) is a flow rate applied at a node. In 3D FE analysis, this represents a point source or sink while in 2D FE analysis it simulates flow in or out of an infinite line perpendicular to the plane of the analysis.

Precipitation/evaporation This is a conditional boundary condition. For precipitation, specified inflow occurs up to a specified threshold total pore pressure at which this prescribed pressure takes over and no further flow occurs. If the pore pressure is more tensile than the specified threshold, then the specified inflow occurs. Similarly, for evaporation, the specified outflow occurs until un til the por poree pr press essur uree bec becom omes es mo more re ten tensil silee tha than n a spe specifi cified ed thr thresh eshold old tot total al po pore re pressure, at which point the pore pressure is fixed at the threshold value.

Prescribed head or pressure Total pore pressure, head or excess pore pressure can be the nodal degree of freedom in FE analysis software and the boundary conditions need to match. Programs allow users to prescribe head or pressure on lines, or pressure gradients on areas and volumes, and the software interpolates the values at nodes. Normally, the user can choose between specifying an incremental change in pressure or the accumulated value. Some examples of hydraulic boundary conditions applied to common groundwater flow and consolidation analyses are shown in Figure 4.8. 4.8. Note that where a phreatic surface is used to define the steady-state pore pressure, hydrostatic conditions are assumed below this surface and no groundwater flow analysis is performed. However, some programs also use the phreatic surface as a tool for defining the prescribed head on model boundaries for groundwater flow analyses. In such cases, check that the prescribed head or pore pressure on the model boundaries has been defined correctly.

4.4. 4.4. 4.4.1 4.4 .1

Con onso soli lida dati tion on an anal aly ysi sis s When Whe n is a co conso nsolid lidat ation ion ana analys lysis is nec necess essary ary? ?

In low-permeability soils, excess pore pressure dissipates slowly. Therefore, soil volume change and effective stress change occur over time, even when loadings and hydraulic boundary conditions are constant. This means that the strength and stiffness of the soil are also changing with time. 120 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.


Figure 4.8 Typical hydraulic boundary conditions in common applications Closed (impermeable wall) Open with prescribed head or pressure

Open Open Open with prescribed head or pressure

Open with prescribed head or pressure, or closed if plane/line of symmetry

Open with prescribed head or pressure

Open with prescribed head or pressure

Closed*

Closed*

Supported excavation

Embankment dam

Closed (impermeable wall) Open with prescribed head or pressure

Extraction line

Closed (plane/line of symmetry)

Open Open

Open Vertical drains

Closed*

Closed*

Cofferdam

Embankment construction (consolidation analysis)

Line/plane of symmetry Open

Closed

Open

*Can be set to open when significant flow or dissipation of excess pore pressure is expected across the bottom boundary, e.g. to a porous rock forming the bottom boundary.

Closed*

Shallow foundation (consolidation analysis)

Where construction activities occur in undrained conditions, they can be simulated with an und undra raine ined d ana analy lysis sis,, bu butt a con consol solida idatio tion n ana analy lysis sis is req requir uired ed to sim simula ulate te the sub sub-sequent dissipation of excess pore pressure in order to predict longer term behaviour. Wheree cons Wher construct truction ion acti activitie vitiess occur in part partially ially dra drained ined cond condition itions, s, a cons consolida olidation tion analysis is required at all stages, as described in Figure 4.2. Typical problems where a consolidation analysis may be necessary (all in low-permeability soils) include: g g

g g g

g

deformation deforma tion of soft soft soil soil to compare compare outputs with site monitoring monitoring data influenced influenced by consolida consolidation tion (e.g. the redistribution of forces in retaining walls for excavations in clay) progressive progr essive failure embankment stability stability during construction construction (see example example in Section 8.4) long-term predictions predictions following construction in undrained undrained conditions conditions (to dissipate dissipate excess pore pressure following an undrained Method A analysis) (see example in Section 8.3) construction activities activities in partially drained drained conditions, i.e. excess pore pressures pressures are are both generated and dissipate significantly during construction time, typically in normally and lightly over-consolidated clays or mixed soils with a significant clay content. 121



The dissipation of excess pore pressure depends on soil permeability, rate of loading (when loading and consolidation occur concurrently – see Section 4.4.3) and hydraulic boundary conditions. Soil permeability is always an uncertain parameter, so where a prediction of consolidation time is required, a parametric study of permeability is necessary in order to estimate a permissible range of consolidation times. The types of hydraulic boundary conditions for consolidation analysis are described in Section 4.3.3. Their location relative to areas of excess pore pressure also affects dissipation rates since this determines drainage path lengths. Pore pressure fixities are specified in terms of steady-state, excess or total pore pressure, depending on the software.

4.4.2 4.4 .2

Whatt is co Wha coupl upled ed con consol solida idatio tion n ana analys lysis? is?

Many of the assumptions of undrained analysis (Methods A and B – see Figure 4.5) apply in a consolidation analysis too. The pore water is assumed virtually incompressible by adding a high K w into the soil stiffness, the soil particles are assumed incompressible while the K ′ defines the bulk modulus of the soil skeleton. Then the set of equations describing groundwater flow (Section 4.3.1) and the set of equations describing equilibrium as used in displacement analysis are integrated in a coupled fashion to simulate time-dependent flow of water and volume change. Nodes have both displacement and pore pressure degrees of freedom with both hydraulic and displacement boundary conditions required to define a problem. Construction stages can be defined in terms of a period of time over which the dissipation of excess pore pressure is predicted by the FE analysis. This is particularly important when defining loading rates in a construction stage. Alternatively, a target minimum excess pore pressure or degree of consolidation can be specified in order to calculate the time taken to reach the target. To integrate the equations over time, consolidation is broken down into individual time steps. These time steps need to be small enough for an accurate solution but if too small, large fluctuations in the calculated excess pore pressures can result. Suitable time steps vary from a few seconds for laboratory test simulations to several days for field observations. Many programs have automatic time step control. Where only manual setting is available, an initial choice is made and outputs of excess pore pressure examined. If large fluctuations occur, time steps can be changed by an order of magnitude up and down until satisfactory outputs are obtained. As time passes, dissipation rates slow so time steps are progressively increased.

4.4.3 4.4 .3

Can loa loadin ding g and and cons consoli olida datio tion n be perf perform ormed ed in in the the same same stage stage? ?

Generally, it is preferred to simulate the development of excess pore pressure during loading, unloading or a change in hydraulic boundary conditions in one analysis stage and then to simulate the dissipation of excess pore pressure in a subsequent analysis stage with the appropriate pore pressure boundary conditions to avoid the complexity of simultaneous application of both types of boundary condition. In such a case, the first stage must be an undrained analysis (Method A) and not Method B because excess pore 122 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.


pressure predictions can be highly inaccurate, and not Method C either because no excess pore pressures are determined. However, both loading and consolidation can be performed in the same stage (where steady-state pore pressure remains unchanged but excess pore pressure varies). This is needed to simulate construction activities where excess pore pressures are generated but also partially dissipate within the construction time. In this case, the time period needs to match the construction time so that the loading rate is approximately the same as that occurring in the field. In some cases, it may be conservative to assume undrained conditions during construction (see Figure 4.4) if this is preferred. It is als also o po possib ssible le for gr groun oundw dwat ater er flo flow, w, con consol solida idatio tion n an and d dis displa placem cement ent all to be analysed together (in which case volume changes are based on the change in total pore pressure) but it may not be possible to distinguish accurately between excess and steadystate pore pressure changes. Such coupling is useful, for instance, in the analysis of rapid draw dr awdo down wn of a res reserv ervoir oir to pr predi edict ct def deform ormat ation ion due to con consol solida idatio tion n an and d st stabi abilit lity. y. However, near failure, convergence of the calculation may be difficult to achieve. REFERENCES

Fredlu Fred lund nd DG, Ra Raha hard rdjo jo H an and d Fr Fred edlu lund nd MD (2 (201 012) 2) Unsatur Unsaturat ated ed So Soil il Mec Mechan hanics ics in Engineering Practice. Practice. Wiley, Hoboken, NJ. Gens A, Sanchez M and Sheng D (2006) On constitutive modelling of unsaturated soils. Acta Geotechnica 1(3): 137–147. Mansikkama ¨ ¨ ki J (2015) Effective stress finite element stability analysis of an old railway embankment on soft clay, PhD thesis, Tampere University of Technology, Finland. Vermeer PA and Meier CP (1998) Stability and deformations in deep excavations in cohesivee soils siv soils.. Proceed Proceedings ings of the Inte Interna rnationa tionall Con Confer ference ence on Soil Soil-Str -Structu ucture re Inte Intera raction ction in Urban Civil Engineering, Engineering, Darmstadt, October.



Chapter 5

How are geotechnical structures modelled? 5.1. 5.1. 5.1.1 5.1 .1

Struct Stru ctur ural al ge geom omet etry ry Whatt types Wha types of ele elemen ments ts are use used d for for st struc ructur tures? es?

There are several element types available to model structures, each of which works in a different way (see Table (see Table 5.1). 5.1). It is important to understand how each element type works in order to select the most appropriate element. In this section, a brief description of the common element types is provided together with typical applications. The pros and cons of using continuum elements for structures are covered in Section 5.1.2.

Spring element The most basic type is the spring element, which connects one node to a fixed point there the reby by pr prov ovidi iding ng the no node de wit with h som somee st stiffn iffness ess aga agains instt tra transl nslat ation ion in a par partic ticula ularr directi dir ection on,, or con connec nects ts two nod nodes es in the mo model del tog togeth ether er pr prov ovidi iding ng an axi axial al st stiffn iffness ess between them. It can extend or compress according to the displacement of the nodes connec con nected ted to it and generat generates es a com compr press essiv ivee or ten tensil silee axi axial al for force ce ac accor cordin ding g to the spring stiffness specified for it. For a linear spring, the force simply equals the spring stiffness k times spring extension or compression ( f ( f = ku ku). ). The k value for a strut or cable, which has units force/displacement (typically kN/mm), can be calculated from Equation 5.1. k=

EA L

(5.1)

where E is where E is the Young’s modulus of the material, A material, A is is the cross-sectional area of the strut, cable, etc. and L is the length of the strut, cable, etc. The spring element is used for structures that act in axial tension or compression but require no interaction with the model except at each end. Typical examples would be a strut supporting a retaining wall or the free length of a ground anchor. If the strut were supported by other means not included in the analysis model, by a shear wall for example, then the spring could be connected to a node fixed in space and an appropriat pri atee spr spring ing st stiffn iffness ess ass assign igned, ed, as des describ cribed ed und under er Suppor Supports ts to re retai tainin ning g wa walls lls in Section 5.1.4.



Table 5.1 Summary of element types and structural applications Characteristics

Applications

2D/3D sp sprring element

Provides an axial stiffness between two nodes.

For linear structures without interaction with the ground along their length, e.g. ground anchor free length, strut support.

2D ba bar/ r/me memb mbra rane ne el elem emen entt

Tran ansl slat atio iona nall de degr gree eess of freedom at nodes. Axial stiffness only. Zero bending stiffness.

For membrane-like structures interacting with the ground in 2D model models, s, e.g. geote geotextile xtiles. s. Approximate modelling of closely spaced linear structures, e.g. ground anchor fixed length, soil nails and rock bolts.

3D bar element

Translational degrees of freedom at nodes. Axial stiffness only. Zero bending stiffness.

For linear structures in 3D models where bending stiffness is neglected (but interfaces with the ground are poorly modelled without modifications to the model), e.g. ground anchor fixed length, soil nails, rock bolts.

3D beam element

Translational and rotational degrees of freedom at nodes. Axial and bending stiffness.

For linear structures in 3D models with bending stiffness, e.g. capping beam.

2D plate element

Translational and rotational degrees of freedom at nodes. Axial and bending stiffness.

For planar structures orientated in the out-of-plane direction in 2D models, e.g. embedded retaining wall, tunnel lining, raft foundation.

3D membrane element

Tran ansl sla ational degrees of freedom at nodes. Axial stiffness only. Zero bending stiffness.

For membrane-like structures in 3D models, e.g. geotextiles.


How are geotechnical structures modelled?

Table 5.1 Continued Characteristics

Applications

3D pla late te/s /sh hel elll el elem emen entt

Tran ansl sla ati tio ona nall an and d rot ota ati tio ona nall degrees of freedom at nodes. Axial and bending stiffness.

For planar structures in 3D models, model s, e.g. embedd embedded ed retaining wall, tunnel lining, raft foundation, shear wall.

2D continuum element

Transl sla ational degrees of freedom at nodes.

For all structures interacting with the ground orientated in the out-of-plane direction in 2D models, e.g. embedded retaining wall, tunnel lining, raft foundation, also closelyspaced structures in the outof-plane direction.

3D continuum element

Transl sla ational degrees of freedom at nodes.

For all structures interacting with the ground (see Section 5.1.2).

Bar element The bar element (also called the truss element when there are no mid-side nodes), rather like the spring element, can only sustain axial load. However, it can be curved and connected to the mesh at other nodes along its length so that it changes shape and deforms in response to deformations around it. The axial stiffness is defined in terms of the material Young’s modulus and section area EA EA.. Since it has no rotational degrees of freedom and stress and strain are assumed constant across the section, it can have no bending stiffness and forms pinned connections with other elements. It can only resist bending to the extent that any bending mobilises tension in the bar element (with updated coordinates – see Sectio Sec tion n 1.4 1.4.3 .3). ). Bar elements elements cou could ld be used to rep repres resent ent lin linear ear gr groun ound d str structu ucture res, s, e.g. ground anchors (fixed length), soil nails and rock bolts, in 3D analyses, if the bending stiffness of these structures need not be considered. The drawback is that the element has no surface area so it models ground–structure interfaces poorly, although some programs have the facility to establish an elastic zone of equivalent structure volume in the ground to obtain more realistic interface behaviour (see Section 5.1.2). Linear ground structures can also be modelled using bar elements in 2D analyses but only if the structures are closely spaced and, even then, only to an approximate degree (see Section 5.1.5). 127 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.


Bar elements can be used to represent struts and cables, as can spring elements, but bar elements have the advantage that their engineering behaviour is specified using a constitutive model that can incorporate elastic-plastic behaviour and, in 2D axisymmetric analyses, the significant resistance provided by hoop forces is taken into account. The use of bar elements to represent planar structures, e.g. geotextiles, is described under Membrane element. element.

Beam elemen elementt The beam element is similar to the bar element except that it has rotational as well as displacement place ment degrees of freedom and stress and stra strain in can vary across across the section. Therefore, Therefore, it can model the rotational, or bending, stiffness of a structure as well as axial stiffness, defined as EI as EI and and EA EA,, respectively, where I where I is is the second moment of area of the section. The line of the beam element itself represents the central axis, or locus of centroids of transverse cross-sections. The neutral axis (where bending stresses are zero) coincides with the central axis in straight beams but not in curved beams. There are two common beam bending assumptions that may be adopted for beam elements: g

g

Euler–Bernoulli Euler–Bern oulli theory theory (equivalent (equivalent to Kir Kirchho chhoff ff theory for plates and shells): transverse planes remain normal and flat after deformation, so no transverse shear deformation occurs. Suitable for cross-sectional areas very small relative to beam length. Timoshenk Timo shenko o theory (equivalent (equivalent to Mindlin theory for plates plates and shells shells): ): transverse planes remain flat but can rotate away from the normal during deformation, so transverse shear deformation can occur. This is suitable for larger cross-sectional areas relative to beam length.

For more complex behaviour, continuum elements should be used. In 3D an anal aly yse ses, s, be beam am el elem emen ents ts ma may y be us used ed to re repr pres esen entt li line near ar st stru ructu cture ress su such ch as capping and waling beams to a retaining wall, or other linear structural features. The use of beam elements in 2D analyses is described under plate and shell elements.

Membrane element Membrane elements are the 3D equivalents of bar elements and, as such, have three coordinates per node, are planar and have no volume. They have translational but no rotational degrees of freedom, so can deform, extend and sustain tension and compression (if allowed) in directions within their plane but have no bending stiffness. Their most mo st com commo mon n use is in sim simula ulatin ting g pla planar nar soil re reinf infor orcem cement ent suc such h as ste steel el st strip ripss in reinforced earth walls and geotextiles, in which case they are set to resist only tensile forces. In 2D plane strain and axisymmetric analyses, the membrane is equivalent to the bar element, resisting forces in its plane, including in the out-of-plane direction.

Plate and shell elements Plate and shell elements are planar like membrane elements, but they have rotational as well as displacement degrees of freedom at their nodes so that they possess bending 128 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.


stiffness defined in terms of EI EI . The difference between plates and shells is rather subtle and there are different definitions in the literature. A plate element is sometimes defined as having bending stiffness but no axial stiffness (i.e. no membrane action) while the shell element has in-plane membrane behaviour coupled with bending behaviour – so called because they are intended for shell structures where membrane behaviour dominates over bending. Nonetheless, some programs do contain ‘plate elements’ with both axial and bending stiffness. A plate element is sometimes considered as being flat while a shell element is curved but, again, this definition does not hold everywhere. For the modelling of geotechnical structures it is essential to have both axial and bending stiffness. Plate and shell elements are used to represent structural walls, such as embedded retaining walls and shear walls, tunnel linings and spread foundations where the bending stiffness needs to be simulated. There are some assumptions associated with plate and shell elements which should be valid whenever they are used. They include: g g g g

g

thickness small thickness small compared compared with with length and and width bending deflection small compared with thickness the mid-surfa mid-surface ce is the neutral neutral surface surface during bendin bending g stresses str esses normal normal to the mid-surfac mid-surfacee are small small comp compare ared d with the bend bending ing stresses stresses and are assumed constant or zero through the thickness (plane stress condition) linear strain strain distribution with depth. depth.

Two common plate bending theories exist which differ in the way they calculate out-ofplane displacement: g

g

Kirchhoff theory: theory: out-of-plane normals normals remain straight straight and normal to the surface, surface, so any point on the mid-surface during bending only deflects in the normal direction to the undeformed surface and no shear deformation occurs. This is suitable for very thin plates. Mindlin theory: out-of-plane out-of-plane normals remain straight straight but can rotate rotate relative relative to the surface, so shear deformation can occur. This is suitable for thicker plates, which includes most geotechnical structures. It can also be used for thin plates but accuracy deteriorates when plates become very thin due to numerical illconditioning.

For more complex behaviour, continuum elements should be used, particularly for thick structures and for detailed analysis around connections. When changing the material properties of a plate or shell element already installed in an earlier construction stage, the same ratio of bending to axial stiffness EI /EA must be EA must maintained, otherwise the resulting change in effective section depth would cause an unrealistic change in bending moment. Plate and shell elements in 2D analyses are equivalent to beam elements except that they are formulated in terms of plane strain and axisymmetric versions of 3D plate theory. 129 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.


Stresses are generated in the out-of-plane direction (due to the prevention of strains in thiss dir thi direct ection ion)) whi which ch infl influen uence ce the beh behav aviou iourr of pla plate te and she shell ll elem element entss wit within hin the analysis plane via the Poisson’s ratio. Hence, for structures that are relatively flexible in the out-of-plane direction, e.g. sheet pile walls, Poisson’s ratio should be set to zero in isotropic material models. Anisotropic (orthotropic) models allow the stiffness in each direction to be specified explicitly (see Section 5.1.6).

Continuum elements Continuum elements are the area or domain elements in 2D or the volume or solid elements in 3D. This is the element type normally used to represent the soil or rock but they can be used to represent structures too and there are advantages to this, as covered in Section 5.1.2. The 3D version is the fundamental continuum element because the three displacement degrees of freedom are equally weighted in each direction, while the 2D version results from a simplification to the FE formulation in the out-of-plane direction. The 2D continuum element can be used in plane strain or axisymmetric analyses to represent structures with thickness, e.g. an embedded retaining wall, tunnel lining, raft foundation or, in an axisymmetric analysis, a single pile. The 3D continuum element can be used to simulate the same structure types in 3D meshes.

5.1.2 5.1 .2

Should con Should contin tinuum uum or non non-co -conti ntinuu nuum m elemen elements ts be be used used for for structures?

Structures that have significant volume in reality, such as piles, diaphragm walls, piled walls, raft raft foundations and and tunnel linings, can be modelled either with continuum elements (volume elements in 3D, area elements in 2D) or with non-continuum (line or surface) elements, e.g. membrane, shell. The pros and cons of each are summarised in Table 5.2. 5.2. Due to their ease of use and tendency to be more conservative in some soil–structure interaction problems, non-continuum elements are used more widely than continuum elements in routine analysis work. However, non-continuum elements should only be used when the structure can be represented in this way with sufficient accuracy. They have certain assumptions (see Section 5.1.1) and are intended for thin-walled structures, so as structures become thicker, the non-continuum elements become less suitable. Step changes in thickness, voids and connections may also become difficult to model. Structures with only a small volume, such as sheet pile walls, are more suited to simulation with non-continuum elements. Note that moment-reducing effects (Figure ( Figure 5.2) 5.2) still occur in steel sheet piles to some extent due to their corrugated section but will not be taken into account with non-continuum elements. Refer also to Section 5.1.6 regarding the anisotropy of sheet pile walls (and other structure types).

5.1.3 5.1 .3

How Ho w ar are e gro groun und–s d–stru tructu cture re int interfa erfaces ces mod modell elled? ed?

At int interfa erfaces ces bet betwe ween en the gr grou ound nd and str struct uctur ures, es, rel relat ativ ivee mo move vemen mentt can occ occur, ur, e.g e.g.. settlement of soil immediately behind a deflected retaining wall. Relative movement can also als o occ occur ur at dis discon contin tinuit uities ies in ro rock. ck. Ho Howe weve ver, r, the req requir uireme ement nt for com compa patib tibili ility ty of displacements in an FE analysis prevents separation, overlap or slippage along planes in a mesh. Therefore, interfaces need to be modelled with dedicated elements, which has 130 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.


Factors tors to consi consider der in the use of cont continuum inuum or non-c non-continu ontinuum um structural structural elemen elements ts Table 5.2 Fac Continuum elements

Non-continuum elements

Setting up

Takes longer bec eca ause full geometry needs to be defined and material parameters are less straightforward, particularly for 2D models.

Easier set-up because only planar geometry of structure needs to be defined define d and secti section on prop propertie ertiess are input directly as material parameters.

Elem El emen entt si size ze

The tr The true ue th thic ickn knes esss of str truc ucttur ures es is often small compared with the overall geometry, which can require a large number of elements in order to avoid high aspect ratios.

The geometry of non-continuum elements is suited to structures that are thin relative to the overall geometry.

Self-weight

Use true density.

The true volume is not represented, so the specified self-weight of structures in the ground should be the net additional weight of the structure over and above the ground weight occupying the true structure volume in the FE model (see Figure (see Figure 5.1). 5.1).

Genera Gen erall ac accur curacy acy

Should be mor Should more e ac accur curat ate, e, particularly in cases of complex geometry or where the assumptions of non-continuum elements do not hold. The moment-reducing effects of interface friction are included (see Figure 5.2).

Should be reasonably accurate for simple geometries and where the inherent inher ent assump assumptions tions of the elements hold. Often more conservative than continuum elements in soil–structure interaction problems because the moment-reducing effects of interface friction are not taken into account (see Figure 5.2).

End En d bea earrin ing g

The en The end d be bear arin ing g of ax axia iall llyy lo load aded ed structures, e.g. piles and retaining walls, is simulated appropriately because the true contact area with the ground is defined (see Figure 5.3). 5.3).

Negligible end bearing is obtained because the element has no contact area with the ground. Plate or shell elements may be added at the end to simulate the width of the structure or some programs have the facility to establish an elastic zone of equivalent structure width in the ground to obtain more realistic end bearing failure loads, but the input parameters should be validated carefully (see Figure 5.3). 131



Table 5.2 Continued Continuum elements


Interface stresses on linear structures

The contact area between structure and ground is accurately represented so interface friction is modelled more accurately, particular parti cularly ly in axiall axiallyy loaded prismatic linear structures such as piles, soil nails, rock bolts, ground anchor fixed lengths.

In 2D analysis, the contact area becomes continuous in the out-ofplane direction, so specifying interface element friction properties accurately is more difficult. In 3D analysis, beam and bar elements have zero contact area, so would not simulate interface friction correctly. Some programs have the facility to establish an elastic zone of equivalent structure volume in the ground to obtain more realistic interface behaviour with the ground, but the input parameters should be validated carefully.

Per erme meab abil ilit ityy

The per The erme meab abil ilit ityy ca can n be de defin fined ed as for soil and rock elements.

Structures are treated as permeable because the ground on each side of a structure has common nodes. Interface elements are required around the structure to separate the ground nodes on each side and render the structure impermeable.

Output

Obtaining outputs of structural forces involves some manipulation of the output data, e.g. integrating the output stresses on sections to obtain axial force, shear force and bending moment (see example in Section 8.2.4).

Direct output of structural forces.

the advantage of allowing the specific properties of the interface to be assigned to those elements rather than to the elements on either side of the interface. The interface can be modelled with thin continuum elements, as shown in Figure 5.4, 5.4, with appropriate stiffness and strength properties but this requires a lot of mesh refinement to avoid unacceptably high aspect ratios. Interface (or slip) elements of zero thickness are the element types most commonly used to simulate ground–structure interfaces. They have the advantage of allowing infinitely thin elements without significant refinement to the mesh. 132 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.


Figure 5.1 Specifying structure self-weight for continuum and non-continuum elements Continuum elements and reality

External structure

Structure self-weight

Use true structure density


Specify full structure weight


Structure element Internal structure

Use true structure density


Specify net structure weight

Ground occupying true structure volume

The nodes on each side of an interface element are coincident because, in reality, the structure and the ground are in contact. Relative elastic shear and normal displacements are governed by the shear stiffness (K ( K s ) and normal stiffness (K ( K n ) of the interface. interface. Determining these stiffness values is not straightforward because they have different units (kN/m3) to standard material stiffness values and no parameter testing methods are available. Figure 5.2 Influence of interface friction on structures and comparison of modelling approaches Continuum elements and reality

Interface friction

Lever arm

M ≠ 0


Interface friction

No lever arm

= M =

0



Figure Figur e 5.3 Accounting for end bearing in structural elements Continuum elements and reality

Non-continuum element options

Base elements End bearing

No end

End bearing

bearing

Structure elements only

Structure ‘volume’

Added base elements

End bearing

Automatic stiffening of continuum elements within structure volume

Figure Figur e 5.4 Options for modelling ground–structure interfaces No interface modelling (rigid interface)

Structure

Ground

Thin continuum elements

Structure

Ground

Interface elements

Structure

Ground Nodes coincident at interface

Structure

Ground

No relative displacement at interface

Structure

Ground

Thin continuum element deforms allowing relative displacement at interface

Structure

Ground

Interface elements allow separation and sliding at interface



If they are too low, excessive deformation at the interface will be calculated, too high and the large difference in stiffness between adjacent elements may cause numerical ill-conditioning leading to stress oscillations in the output. However, provided that the order of magnitude of these values is in the range of the material stiffness either side of the interface, then outputs may not be too sensitive to the values adopted. This should be checked by a parametric study. The elem The element ent thi thickn ckness ess als also o pla plays ys an imp import ortant ant role in the st stiffn iffness ess of the ele elemen ment. t. Geomet Geo metric ricall ally, y, the elem element ent ha hass zer zero o thi thickn ckness, ess, but a vir virtua tuall val value ue is re requi quire red d as a material parameter for the calculation of elastic deformations. High values result in large elastic deformations, so the value should be small, particularly for large normal stresses. However, if the value is too low, then numerical ill-conditioning may occur. Appropriate values typically lie in the range 1 to 10% of the interface element length but a parametric study allows appropriate values to be identified. A Coulomb friction criterion (Equation 5.2) defines the change from elastic to plastic behaviour in terms of effective stress, while a simple limiting shear stress is used for undrained analyses in terms of total stress. If the calculated shear stress is less than the failure stress, then the interface element binds the ground and structure together while allowing some relative elastic deformation. If the shear stress reaches the failure stress, then permanent slippage occurs at the interface. Also, where any calculated tensile stress at the interface exceeds the tensile strength (c (c′ tan w ′ ) or a specified tension cut-off, the interface element allows separation between the ground and the structure and K n and K s reduce essentially to zero. If gap closure occurs in a subsequent analysis stage, the program needs to record the amount of separation so that compressive stresses are not restored at the interface until the same amount of separation is reversed and contact at the interface element is re-established. ′

′

′

t = c + s n tan w

(5.2)

The interface shear strength can be measured by, for example, a laboratory direct shear test. In the absence of test data, interface friction is typically adopted between a half and the full internal shear strength of soil, depending on the characteristics of the structure (e.g. material, installation method). If a smooth interface is required, perhaps when simulating certain laboratory tests, interface elements with very low or zero K s and shear strength can be used. Where relative movements at interfaces are quite small, sufficiently accurate outputs may be obtained without interface elements. This can be checked by running an analysis with and without interface elements and comparing the outputs. Where relative movements are quite large, interface elements will certainly be needed and refinement of the mesh around the interface may be required too due to the high stress and strain gradients in these regions. Interface elements introduce additional complexity to FE analysis models, so it is worthwhile running an analysis model prior to installing interface elements to check that it 135 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.


completes all the analysis stages satisfactorily. In setting up interface element geometry, particular care is required at connections between structures to ensure that the interface elements do not change the intended connection type.

5.1.4 5.1 .4

How Ho w are are the the comm common on geot geotech echnic nical al stru structu cture re type typess model modelled led? ?

Much of the guidance on how to model a specific problem is common to different structure types, such as about using continuum or non-continuum elements, and is provided throughout this chapter and, indeed, throughout this book. This section provides some useful tips that are mostly exclusive to specific structure types. More guidance on particular structure types can be obtained from published case studies by others who have applied FE analysis to similar problems.

Supports to retaining walls (including ground anchors) The common supports to retaining walls include struts, slabs and ground anchors. The stiffness of these supports has a significant and non-linear effect on outputs of retaining wall deflection and bending moment. Support stiffness is often an uncertain parameter due to thermal effects, concrete shrinkage and connection details (see Section 5.1.7), so a para pa ramet metric ric stu study dy of sup suppo port rt sti stiffn ffness ess is oft often en re requi quire red d in FE an analy alyses ses of sup suppo ported rted retaining walls. Linear supports can be modelled with spring or bar elements and slabs with plate or shell elements, as described in Section 5.1.1. Where supports interact directly with the ground, as shown in Figure 5.5, 5.5, the entire support should be modelled. When a slab or strut spans between two identical retaining walls such that an axis of symmetry can be used as a model boundary (see Section 1.2.4), then only half the slab or strut length need be modelled reacting against a fixed point on the axis of symmetry. In many cases, a slab or strut is supported by other structural members, such as shear walls and bracing, before loads are eventually transferred back to the ground through foundations remote from the retaining wall being simulated (Figure 5.5). Either these additional structural members and foundations are included in the model or they can be substituted for spring elements with an equivalent stiffness. The spring element option is more straightforward for the geotechnical modeller when the structural supports are complex but requires some analysis by the structural engineer in order to provide appropriate spring stiffness values va lues.. Fu Furth rther er ite itera ratio tions ns bet betwe ween en the geo geotec techn hnical ical and str structu uctural ral mod models els wou would ld als also o be required until the deflections and loads at the interface between the two models match. The fixed lengths of ground anchors need to be modelled with continuum elements in order to simul simulate ate the anch anchor–g or–grou round nd interf interface ace mor moree acc accura urately tely.. Altern Alternativ atively, ely, some programs adopt non-continuum elements with a function that automatically sets the groun gr ound d wit within hin the spe specifi cified ed volu olume me to the pr prope operti rties es of the gr grout out ma mater terial ial.. Th This is is acceptable provided that the input parameters have been validated carefully. A pre-stress in an anchor or strut is imposed as illustrated in Figure in Figure 5.6. 5.6. Axial forces are applied at the node at each end (or at one end if the strut intersects a model boundary on an ax axis is of sy symm mmet etry ry)) in th thee di dire rect ctio ion n th tha at th thee pr pree-st stre ress ss wi will ll im impo pose se it itsel selff on th thee 136 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.


Figure 5.5 Support types

Supports interacting directly with the ground

Full simulation of support structure

or equivalent spring supports

More complex structural support

connected structure or ground before activating the anchor or strut. In the subsequent analysis stage the anchor or strut is activated and the axial forces removed. The rebound of the connected structure or ground will cause the pre-stress to be transferred to the anchor or strut elements. The outputs of anchor or strut force should approximately match the initial applied loads and the applied loads can be adjusted in subsequent re-runs until the desired pre-stress is achieved. The outputs of anchor or strut force in later stages are likely to be somewhat above or below the pre-stress due to the effects of the other construction activities being simulated.

Geosynthetics and reinforced soil/earth walls Membrane elements are commonly used to represent geosynthetics and the steel strips of reinforced earth walls, with interface elements placed between the membrane elements and soil. This method works quite well for steel strips but not so well for geosynthetics, particularly geogrids. This is because soil sits within the apertures of a geogrid leading to a different type of interaction between soil and geogrid than simulated by membrane and interface elements. It is also difficult to select appropriate stiffness and strength values for polymers because they are heavily dependent on rate effects and creep. 137 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.


Figure Figur e 5.6 Applying pre-stress to ground anchors and struts

1. Install fixed length and apply pre-stress loads

2. Install free length

3. Release loads (pre-stress transferred to anchor)

1. Apply pre-stress loads to strut connections

2. Install strut

3. Release loads (pre-stress transferred to strut)

As with all geotechnical FE analyses, the construction sequence needs to be simulated to obtain realistic stress distributions which, for multi-layered reinforced soil walls, can mean many construction stages in order to place each layer, although some simplification may be possible as described in Section 1.4.2. Simulating the stabilising effect of geogrids is a particular challenge, as described by Lees (2017). In order to simulate membrane action in geosynthetics, it is essential to use updated coordinates in the FE analysis, as described in Section 1.4.3.

Spread foundations Usually a 3D analysis is necessary when simulating spread foundations although 2D plane strain or axisymmetric models are appropriate for certain geometries and loadings. The axisymmetric assumption is suitable for circular foundations or as an acceptable approximation of a square foundation (with the radius set to achieve the same equivalent foundation area as the square foundation) but only with a vertical, concentric applied load. The plane strain assumption is suitable for long strip foundations with closely spaced or continuous uniform loading along their length. Inclined or eccentric loads and moments are permitted but only acting in the plane of the analysis. Analyses involve applying either load or displacement to foundations. The former tends to be used for simulating foundations in service while the latter tends to be used in order to obtain a failure load (or resistance) for the foundation from the output. 138 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.


Figure 5.7 Options for idealised simulation of spread foundations Flexible foundation (load control): Smooth FOUNDATION

Ro ug h FOUNDATION Prescribed vertical displacement Prescribed vertical load Horizontal fixity Vertical displacement tied between nodes

Rigid foundation (displacement control): Smooth FOUNDATION

Ro ug h FOUNDATION

Rigid foundation (load control): Smooth FOUNDATION

Ro ug h FOUNDATION

When certain assumptions are appropriate (see Section 5.3.1), it is not necessary to model the foundation itself. If a foundation is assumed perfectly rigid, a uniform displacement can be imposed on the soil nodes where the foundation base would be, or the nodes tied to displace uniformly in the case of applying a load, as shown in Figure 5.7. 5.7. If assumed perfectly flexible, the foundation load is applied at the soil nodes where the foundation base would be. Similarly, a perfectly rough soil–foundation interface can be sim simula ulated ted with without out inc includ luding ing the fou found ndat ation ion its itself elf by con constr strain aining ing the hor horizo izonta ntall displacement of the soil nodes coincident with the base of the foundation while the nodes at the sides of the foundation can be constrained to displace with the foundation base. For a perfectly smooth interface, the same nodes would be unconstrained. 139 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.


Piles Particular care is needed with axially loaded piles when specifying the properties for the interface elements between piles and the ground. Pile installation disturbs the ground immediately around piles such that its properties differ from those of the undisturbed ground gro und enco encounter untered ed duri during ng site inv investi estigat gations ions.. Ther Therefor efore, e, inpu inputt par paramete ameters rs mus mustt be validated by back-analysing pile load tests and the test pile types, loads and ground conditions must be similar to the piles being simulated in the main model. Interface elements are also required for laterally loaded piles to avoid tension being generated where the pile deflects away from the ground. Piles need to be modelled with continuum elements in order to simulate the pile–ground interface more accurately, although some programs adopt special beam elements that take an arbitrary orientation through the 3D solid elements representing the ground. Interface line elements between the beam nodes and virtual nodes within the surrounding solid elements model interaction with the soil and spring elements define the base resistance, while plasticity in the ground is disabled around the beam elements within the specified pile radius (refer to Tschuchnigg to Tschuchnigg and Schweiger, 2015). 2015 ). This method is acceptable provided that the input parameters have been validated carefully, particularly for pile groups. For single piles, a 2D axisymmetric analysis is possible only in the specific case of a circular section (or a square section approximately if the radius is set to achieve the same surface area in contact with the soil) pile with vertical orientation, a vertical, concentric applied load and horizontal soil layers and groundwater level. A pile with a non-vertical applied load or a moment needs to be simulated with a 3D model. The constrained conditions created by a loaded pile in an FE analysis can result in unrealistic raised effective stresses and over-predicted failure loads if the dilation angle of the soil is set above zero. Therefore, in FE analyses of piles, a dilation angle of zero or a carefully set dilation cut-off are required. Clearly 3D models are required for the analysis of pile groups and piled rafts. A constitutive model for the soil that accounts for non-linear strain-dependent stiffness becomes important for more accurate simulation of pile–pile interaction. The validation of pile group analyses is more difficult because load tests on pile groups are rare, so published case studies of similar conditions and other analysis methods (e.g. boundary element method) are more common sources of validation data.

Ground improvement and grouting Ground Groun d imp impro rove vemen mentt can be div divide ided d int into o dif diffus fusee met method hodss (wh (wher eree a hom homoge ogeneo neous us material is formed, e.g. compaction, consolidation and grouting) and discrete methods (where inclusions are created that remain separated from the ground, e.g. stone and concrete columns, deep mixing and jet grouting). Depending on the size and spacing of inclusions relative to the dimensions of the problem, either each inclusion can be modelled by FE analysis in a similar fashion to piles, or the beneficial effects of the 140 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.


inclusion inclus ionss can be sme smear ared ed ac acro ross ss the gr groun ound d and defined defined in ter terms ms of the imp impro rove ved d properties of the ground, as for diffuse methods. With the smeared approach, rather than attempt to simulate the ground improvement activity itself, it is common practice simply to substitute the in situ material for one with appropriately enhanced density, strength and stiffness. Ground improvement may also have an impact on ground permeability that would need to be considered in consolidation and groundwater flow analyses. Obtaining constitutive model parameters for improved ground prior to improvement can, of course, be difficult. Ideally, field trials will be performed so that parameters can be obt obtain ained ed fr from om tho those. se. Cas Casee stu studie diess of sim simila ilarr gr groun ound d imp impro rove vemen mentt pr proje ojects cts als also o provide a valuable source of parameters. Post-improvement testing provides a means to validate input parameters, but only after completion of the construction activity. Jet grouting, permeation grouting and compaction grouting in soil and consolidation grouting in rock are all forms of ground improvement whose beneficial effects can be simulated in the same way (i.e. material substitution). Jet grouted structures are brittle in ten tensio sion n wh which ich ma may y re requi quire re spe specia ciall con consid sidera eratio tion n in the con const stitu itutiv tivee mod model el (se (seee Section 5.2.1) in detailed studies of such structures. Compensation grouting is a form of hydrofracture or soil-fracture grouting intended to control ground movements. It is commonly used between an advancing tunnel and the ground surface below existing structures to compensate for stress relief and ground loss and to control tunnelling-induced settlements. There are two main approaches to simulating compensation grouting: ‘prescribed strain’ or ‘prescribed pressure’, as summarised by Wisser et al . (2005). (2005). The latter and Addenbrooke et al . (2002) describe in detail prescribed pressure approaches using interface elements that open out to represent the growing grout body. Soga et al . (2000) adopted a similar approach using continuum elements to represent the grout body.

Tunnels A distinction can be made between soft ground tunnelling and rock tunnelling. In soft ground tunnelling significant stresses are taken by the tunnel lining and there are larger deformations, including ground surface settlements in the case of relatively shallow tunnelling. In rock tunnelling a greater proportion of tunnel-induced stresses are supported by the ground and design and simulation is more focused on stress relief, rock quality deterioration and local instability. The remainder of this section on tunnels will focus on soft ground tunnelling. In addition to tunnel lining deformation, the main cause of tunnelling-induced ground deformations is ground loss. Ground loss is the extra volume of soil that is excavated over and above the tunnel volume due to stress relief and partial closure of the excavation. It is ex expr press essed ed as a per percen centag tagee of the tun tunnel nel vo volum lumee and typical typical va value luess ar aree 2– 2–3% 3% for conventional tunnelling methods without adequate support and 0.5% for modern earth pressure balance (EPB) tunnel boring machines (TBM), although this can increase to 141 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.


about 1% in more difficult ground conditions. Therefore, simulation of both the tunnel lining and the tunnelling process are critical elements to obtaining sufficiently accurate predictions of ground deformation. Theree ar Ther aree th thre reee co comm mmon only ly us used ed me meth thod odss of si simu mula lati ting ng tu tunn nnell ellin ing. g. Th Thee fir first st tw two o described in the following paragraphs are idealised methods intended for 2D plane strain models or 3D models where the tunnel is activated in its entirety in a single stage. The third method is a full 3D simulation of the advancing tunnel.

1

2

3

Volume loss control Volume control method: method: soil elements inside the tunnel tunnel are deactivat deactivated ed and the lining elements activated in the same analysis stage. In the next stage, the volume loss is prescribed, usually by specifying a circumferential contraction of the tunnel lining. For circular linings the percentage contraction is approximately half the percentage volume loss owing to the geometry of a circle. The actual volume loss achieved should then be checked in the outputs. Since volume loss is an input parameter, this method is better suited to cases where the volume loss can be determined for a particular tunnelling method and ground conditions. This method is used more commonly in the simulation of TBM tunnels. Load reduct reduction ion method: method: this is also also called the b -method -method or a-method (Panet (Panet and Guenot, 1982). The soil elements to be excavated are deactivated and an artificial support pressure is applied to the faces of the excavation in the same analysis stage prior to installation of the lining. The pressure is a proportion ( b ) of the in situ stress s 0 prior to tunnel excavation. Either the stress from the excavated soil elements is completely deactivated and the pressure bs 0 is applied or (1 − b )s 0 of the excavated soil elements’ stress is deactivated, as is possible in some programs. Further support pressure reductions can be imposed, if required, when simulating time effects (e.g. consolidation, creep). In a subsequent stage the tunnel lining is activated and the remaining support pressure removed. Here the proportion of unloading prior to lining construction is prescribed and the volume loss is calculated. The b value value is not straightforward to determine – it is based on experience of the tunnelling method in particular ground conditions and experience of applying the method in FE analysis. Trial values can be used and the outputs compared with expected outcomes. Higher b values values (typically up to about 0.7 for conservative predictions of structural forces) are appropriate for earlier lining installation, higher lining forces and less ground deformation. Lower b values values (typically down to about 0.2 for conservative predictions of ground movements) are appropriate for later lining installation, lower lining forces and more ground deformation. This method is commonly used in the simulation of sprayed concrete lining (SCL) tunnels, as well as TBM tunnels. Step-by-s Stepby-step tep method: method: this meth method od involves involves a full simulation simulation of the tunn tunnel el excavation and lining installation steps in a 3D FE analysis, without the idealised input parameters of the previous two methods. Many analysis stages are required in order to simulate an advancing tunnel so it is very time-consuming to set up and to run. Therefore, this method is normally used only for detailed studies rather than more routine analyses.



When simulating tunnelling in soft clay, it may be necessary to include grout pressures in TBM tunnelling (from the injection of grout to fill the annular space between the bore and the lining – the lining would need to be removed temporarily to apply this pressure) or other support pressures, e.g. compressed air, to prevent ground movements and heave being over-predicted. These additional pressures can be adjusted to achieve the expected ground loss or lining forces. Tunnel linings may be simulated with continuum or non-continuum elements with the associat asso ciated ed adva advantag ntages es and disad disadvant vantages ages descri described bed in Sectio Section n 5.1.2 5.1.2.. Join Jointed ted circ circular ular tunnel tun nel lin lining ingss can be sim simula ulated ted as con contin tinuou uouss lin lining ings, s, but wit with h an effe effecti ctive ve seco second nd moment of area I e calculated according to Equation 5.3 ( Muir Wood, 1975) 1975 ) to help account for the additional flexibility introduced by the joints. I e

=

I j + I

2

 4

n

(I e  I

,

n .

4)

(5 3) .

where I is is the second moment of area of a segment section, n is the number of segments and I j is the effective second moment of area at a joint (which is difficult to determine because it depends on hoop thrust and joint geometry so is often assumed as zero or close to zero). This approach should obtain a reasonably accurate compressible stiffness of the lining and, hence, outputs of hoop thrust, and reasonably accurate predictions of ground movement. The prediction of lining bending moment may be inaccurate because the effect of the joints is smeared around the lining rather than being considered explicitly. Hinges can be considered to simulate joints and obtain more representative lining bending moment distributions while more accurate outputs can be obtained with springs placed at joints between segments simulated using shell elements to model the specific characteristics of the lining joints (refer to, for example, Li et al ., ., 2015, 2015, for cast iron bolted tunnels and Wang and Wang et al ., ., 2012, 2012, for concrete segmental linings). The main difficulty in simulating SCLs is in the constitutive behaviour. The lining is loaded immediately on application and as the sprayed concrete (shotcrete) is still curing, it gai gains ns st stre rengt ngth h an and d st stiffn iffness ess whi while le per perfor formin ming g its sup suppo portin rting g ro role. le. Sh Shotc otcre rete te als also o exhibits creep behaviour and post-peak softening in both compression and tension. The simplest method to model the lining is to use a linear elastic model with an artificially low stiffn st iffness ess tha thatt is inc increa reased sed in sub subseq sequen uentt st stag ages es to ac accou count nt for sho shotcr tcrete ete cur curing ing (e. (e.g. g. Mo 2005). However, this can over-predict structural forces in the lining ¨ ¨ ller and Vermeer, 2005). due to high tensile stresses. More advanced constitutive models have been developed to account for more aspects of shotcrete behaviour, such as Scha as Scha ¨ ¨ dlich and Schweiger (2014) thatt inclu tha includes des hard hardening ening/softe /softening ning plas plasticity ticity,, time-d time-depend ependent ent str strengt ength h and stiff stiffness, ness, cree cr eep p an and d sh shri rink nkag age. e. Fur urth ther er de desc scri ript ptio ion n of th thee mo mode dell llin ing g of co conc ncre rete te is gi giv ven in Section 5.2.1.

5.1.5 5.1 .5

How are How are stru structu ctures res mo model delled led in 2D 2D with with the the plan plane e stra strain in or or axisymmetric assumption?

Judging whether a 2D plane strain or axisymmetric assumption is appropriate for a particular problem was covered in Section 1.2.1. As far as structures are concerned, certain 143



geometries are suited to these assumptions. Structures with a long, straight horizontal dimension and a uniform prismatic section (e.g. a long strip foundation, floor slab or diaphragm wall) are suited to the plane strain assumption, while similar structures whose long dimension is circular around a vertical axis of symmetry (e.g. a vertical shaft or single pile) are suited to the axisymmetric assumption. Deriving the input parameters and interpreting output for such structures is relatively straightforward. Similar structures can be composed of a series of identical, closely spaced discrete structures (e.g. contiguous piles) arranged in a straight line (quasi-plane strain) or in a circle about a vertical axis of symmetry (quasi-axisymmetric). Deriving the input parameters and interpreting output for such structures is less straightforward and the primary aim of this section is to provide guidance in this area. As the spacing between structures in the out-of-plane direction increases, it becomes more difficult to justify adopting the plane strain or axisymmetric assumption. With a small spacing, 2D models are acceptable for predicting deformations and stability at a global level. With a larger spacing and for detailed studies of such structures at any spacin spa cing, g, a 3D mod model el is nec necessa essary ry.. Fo Forr ins instan tance, ce, in a sim simula ulatio tion n of a lon long, g, st stra raigh ightt reinforced earth wall with closely spaced steel strip reinforcement, a 2D plane strain FE analysis study of overall stability or surface settlement may be acceptable but a verification of adequate safety against reinforcement sliding would require a 3D analysis because a plane strain analysis would not simulate the detailed soil–reinforcement interaction correctly. If closely spaced structures are connected by beams orientated in the out-of-plane direction, tio n, thi thiss hel helps ps to mai mainta ntain in the pla plane ne str strain ain or ax axis isymm ymmetri etricc con condit dition ion.. Co Comm mmon on examples are capping and waling beams used in embedded retaining walls to distribute load from supports and to connect wall elements together.

Input and output in 2D plane strain models Figure 5.8 shows 5.8 shows a true plane strain case with a retaining wall and supporting slab of continuous section. If using continuum elements to model the structures, then the geometry is simply a 2D section and the structural elements have the same thickness and material properties as the actual structure. If using line elements, their geometry should coincide with the mid-plane of the structures they represent and the section properties are specified in the input parameters for the structural materials. Note that outputs of stru st ruct ctur ural al fo forc rces es su such ch as ax axia iall fo forc rce, e, be bend ndin ing g mo mome ment nt an and d sh shea earr fo forc rcee fr from om bo both th methods will be provided per unit length in the out-of-plane direction. Figure 5.9 shows 5.9 shows a quasi-plane strain case with closely spaced piles forming a contiguous piled wall and struts at a larger spacing connected in the out-of-plane direction via a cappin cap ping g bea beam. m. Fir First, st, the st struc ructur tural al sect section ion pr prope operti rties es nee need d to be con conve verted rted to the their ir equivalent plane strain values by dividing them by their spacing. The values will be expressed per unit length in the out-of-plane direction. The conversion for the piles is shown in Table in Table 5.3. 5.3. Note that the pile spacing is the distance between each pile, typically centre-to-centre, and not the size of the gap between them. 144 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.


Figure 5.8 Example true plane strain structure

Slab thickness d s Section properties entered directly

Wall thickness d w Slab thickness d s Wall thickness d w 3D

2D plane strain (continuum elements)

2D plane strain (non-continuum elements)

Figure 5.9 Example quasi-plane strain structure

Strut spacing 12 m

Spring stiffness = 4.2 kN/mm/m k =

Spring stiffness = 4.2 kN/mm/m k =

Strut stiffness k = 50 kN/mm Contiguous piled wall: 0.6 m diameter piles at 0.9 m spacing 3D

Wall thickness d = =

Equivalent wall section properties per unit length entered directly

EI

12

EA



Table 5.3 Exampl Example e wall pro propertie pertiess per pile and their plane strain strain equiv equivalent alent Single pile properties

Equivalent plane strain properties

Diameter B = 0.6 m Area A = 0.283 m2 Second moment of area I = 6.36 × 10 − 3 m4 Young’s modulus E = 20 × 106 kN/m2 6 EA = 5.66 × 10 kN 3 2 EI = 127 × 10 kNm Weight density g = 25 kN/m3 Pile weight = 7.1 kN per m length

Divide single pile properties by spacing (0.9 m) 6 EA = 6.29 × 10 kN/m 3 2 EI = 141 × 10 kNm /m Wall weight 7.9 kN per m length per m run



If using plate or shell elements to model the wall, the plane strain properties on the right of Table 5.3 can be entered directly as input for the material model. If using continuum elements to model model the wall, an equivalent equivalent thickness d thickness d of of the 2D continuum element needs to be derived. In order to model both the axial and bending stiffness correctly, there is only one unique combination of d of d and E and E values. values. From the ratio EI ratio EI /EA EA,, a unique value of can be obtained from Equation 5.4. d can d =

  12

EI

(5.4)

EA

Therefore, in this example, d = 0.52 m and E can can only be 10.9 × 106 kN/m2. Also, the weight density for the wall material should be g = 15.2 kN/m3 in order to obtain the same equivalent wall weight. To convert the structural force outputs per unit length in the out-of-plane direction to values per pile, they are simply multiplied by the spacing. For example, an axial load of 150 kN/m would be equivalent to 135 kN per pile in this example, while a bending moment of 400 kNm/m would be equivalent to 360 kNm per pile. In thi thiss ex exam ample ple the st strut rutss wil willl be re repr presen esented ted by spr spring ing ele elemen ments ts and the st struc ructur tural al engineer provided a spring stiffness value of k = 50 kN/mm per strut to represent the support from the structure. At a spacing of 12 m, the equivalent plane strain value is k = 4.2 kN/mm per m run. Depending on the software being used, the k value may be entered directly or the parameters E , A, L and s may need to be entered, which the program uses to calculate k according to Equation 5.5. unitt len length gth = k per uni

EA Ls

(5.5)

where L is where L is the member member length length and s and s is is the t he spacing. In this exam example, ple, A A,, L L and and s s could could be set 2 artificially to 1 and then an E an E value value of 4200 kN/m entered to obtain the requir required ed k k value. value. In the output, a strut force of 80 kN/m, for example, would be equivalent to a strut force of 960 kN per strut.

Input and output in 2D axisymmetric models Figure 5.10 shows 5.10 shows a true axisymmetric case with a circular shaft supported by a retaining wall of uniform section and a solid slab near the top of uniform thickness. The spokes drawn on the figure divide the structure into sectors of angle one radian each to help illustrate how input and output data are sometimes expressed in axisymmetric analyses, i.e. per radian. If using continuum elements to model the structures, then the geometry is simply a 2D section from the central axis of symmetry and the structural elements have the same thickness and material properties as the actual structure. If using line elements, their geometry should coincide with the mid-plane of the structures they represent and the section properties are specified in the input parameters for the structural materials. 146 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.


Figure 5.10 Example of a true axisymmetric structure Wall thickness d w

Slab thickness d s Section properties entered directly

Slab thickness d s

Wall thickness d w

3D



Figure 5.11 shows Figure shows a qu quas asii-ax axis isym ymme metri tricc ca case se wi with th cl clos osely ely sp spa ace ced d pi pile less fo form rmin ing g a contiguous piled wall supported by ground anchors at a spacing of 36 connected in the out-of-plane direction via a waling beam. First, the structural section properties need to be converted to their equivalent axisymmetric values. For continuous structures in the out-of-plane direction, properties are expressed per unit hoop length as shown for the piles in Table in Table 5.4. 5.4. 8

If using plate or shell elements to model the wall, the properties on the right of Table 5.4 can be entered directly as input for the material model. If using continuum elements to model the wall, an equivalent thickness d of the 2D continuum element needs to be derived using Equation 5.4. Note that it would be essential to adopt an anisotropic model for a contiguous piled wall to avoid unrealistic hoop forces being generated (see Section 5.1.6). In this example, d = 0.65 m was obtained. E can can be calculated from the equivalent EA value, where A is simply d times times the unit arc length of the wall centreline as ill illus ustr trat ated ed in Figure Figure 5.12 5.12,, giv giving ing E = 13.6 × 106 kN/m2. Als Also, o, the weight weight den density sity 3 for the wall material should be g = 5.1 kN/m in order to obtain the same equivalent wall weight. To convert the structural force outputs per metre in the out-of-plane direction to values perr pi pe pile le,, th they ey ar aree si simp mply ly mu mult ltip ipli lied ed by th thee sp spa aci cing ng.. Fo Forr ex exam ampl ple, e, an ax axia iall lo load ad of 125 12 5 kN kN/r /rad ad wo woul uld d be eq equi uiva vale lent nt to 15 150 0 kN pe perr pi pile, le, wh whil ilee a be bend ndin ing g mo mome ment nt of 230 kNm/m would be equivalent to 276 kNm per pile. The equ The equiva ivalen lentt pr prop operti erties es of spo spoke ke-lik -likee st struc ructur tures es suc such h as the gr grou ound nd anc anchor horss ar aree expressed per radian as shown in Table 5.5. 5.5 . An output of anchor force of 390 kN/rad, for example, would be equivalent to an anchor force of 246 kN per anchor. 147 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.


Figure 5.11 Example of a quasi-axisymmetric structure Spring stiffness = 23.8 kN/m/rad k =

Wall thickness d = =

EI

12

EA

2D axisymmetric (continuum elements) Spring stiffness = 23.8 kN/m/rad k = Anchor spacing 36°

Contiguous piled wall: 0.75 m diameter piles at 11.6° or 1.2 m hoop spacing

3D

5.1.6 5.1 .6

Equivalent wall section properties per unit length entered directly 2D axisymmetric (non-continuum elements)

How Ho w is is geom geometr etrica icall anis anisotr otropy opy in st struc ructur tures es hand handled led? ?

Geometrical anisotropy occurs most often in embedded retaining walls, but can occur in any structure type where structural sections are discontinuous, non-uniform or profiled in one direction. Embedded retaining walls are continuous in the vertical direction where the maximum bending resistance is required, but have discontinuities in the horizontal direction in the plane of the wall. Piled walls have gaps or softer infill between them and even diaphragm walls have joints that reduce their axial and bending stiffness in the Table 5.4 Exampl Example e wal walll pro properti perties es per pile and their axisy axisymmetri mmetricc equiv equivalent alent Single pile properties

Equivalent axisymmetric properties

Diameter B = 0.75 m Area A = 0.442 m2 Second moment of area I = 15.5 × 10 − 3 m4 Young’s modulus E = 24 × 106 kN/m2 6 EA = 10.6 × 10 kN 3 2 EI = 37.2 × 10 kNm Weight density g = 25 kN/m3 Pile weight = 4.0 kN per m length

Divide single pile properties by spacing (1.2 m) 6 EA = 8.84 × 10 kN/m 3 2 EI = 310 × 10 kNm /m Wall weight 3.3 kN per m length per m run



Figure 5.12 Calculation of example wall section area for a unit length sector Centreline length 1 m A =

Internal shaft radius 5.68 m

= d =

0.65 × 1.0 = 0.65 m2

0.65 m

horizontal direction. Sheet piles have a profile that gives them significantly lower stiffness in the horizontal direction than the vertical. In 2D plane strain analyses of wall sections, this does not present a problem (provided the plane strain assumption is appropriate, e.g. away from the corners of an excavation) because strains in the out-of-plane horizontal direction along the wall are zero in any case, so isotropic material properties can be assumed adopting the bending and axial stiffness in the vertical direction. However, in 2D axisymmetric analyses, the generation of hoop forces in isotropic wall elements would lead to wall deflection being governed by hoop stress rather than wall bending and a significant under-prediction of wall deflection and bending, as illustrated by example in Section 8.3.4. Therefore, in 2D axisymmetric FE analyses, it is essential that anisotropic (orthotropic) constitutive models are adopted in situations where hoop stresses are generated in anisotropic structures. Similarly Simila rly,, in 3D an analy alyses ses of app appro roxim ximat ately ely pla plane ne st stra rain in def deform ormat ation ionss (e. (e.g. g. nea nearr the centre of a long retaining wall), isotropic structural properties should be acceptable, but in non-plane strain situations, e.g. near corners of excavations, anisotropy should be included in the model. For piled walls, the individual piles can be modelled with gaps between them or, when using shell elements, appropriate anisotropic axial and bending Table 5.5 Example anchor properties per anchor and their axisymmetric equivalent Single anchor properties

Equivalent axisymmetric properties

Area A = 600 × 10 − 6 m2 Young’s modulus E = 200 × 106 kN/m2 3 EA = 120 × 10 kN Free length 8 m 3 k = 15 × 10 kN/m Pre-stress 200 kN

Divide single anchor properties by spacing (0.63 rad) 3 EA = 190 × 10 kN/rad Free length stays the same at 8 m 3 k = 23.81 × 10 kN/m/rad Pre-stress 317 kN/rad



stiffness stiffn ess par paramet ameters ers shou should ld be sele selecte cted. d. Whe Where re onl only y isot isotro ropic pic ma mater terial ial beha behaviou viourr is available in an FE analysis program, the rotational degrees of freedom in the nodes of shell elements at excavation corners can be released to reduce the amount of error. More information on the importance of modelling structural anisotropy is provided by Zdravkovic by Zdravkovic ´ ´ et al . (2005). (2005). When specifying anisotropic structural properties, the local axis directions of the structural elements must be known and particular care is needed to ensure that the local axes are set up in a consistent way to help avoid errors in the correspondence of material parameters and axis directions. The difference in stiffness between two perpendicular directions should be limited to a factor of about 20 to avoid ill-conditioning, even if the true difference in stiffness is greater.

5.1.7 5.1 .7

How Ho w ar are e st struc ructur tural al con connec nectio tions ns mod modell elled? ed?

In modelling terms there are three main types of struc structural tural connection connection avai available, lable, namely roller (simple), pinned and full (fixed) and methods of simulating these connection types are illustrated in Figure in Figure 5.13. 5.13. Actual structural connections are likely to fall somewhere Figure 5.13 Modelling idealised connection types Roller (simple) connection

Pinned connection

Full (fixed) connection

Continuum/ continuum elements Nodes tied in one direction

Continuum/ non-continuum elements

Non-continuum/ non-continuum elements

Beam or shell element extended into continuum element Nodes tied in one direction

One side must be bar or membrane element Nodes tied in one direction

Beam or shell elements only

or hinge specified between beam or shell elements



between these idealised categories, e.g. a full connection is likely to allow a small degree of movement or ‘play’ that is difficult to simulate. The most appropriate of the idealised connection types needs to be selected and, where there is uncertainty, connection types can be changed in order to assess their effect on the key outputs. The choice of connection type is important because it has a large effect on the outputs of structural forces and deflections. Some structural connections may transfer only tensile or compressive loads and this also needs to be reflected in the setting up of these connections in the FE model. Connections Connect ions also influ influence ence the ov overa erall ll stif stiffness fness of stru structur ctures. es. Fo Forr ins instance tance,, the axia axiall stiffness of a strut support to a retaining wall may be calculated from the section and material properties of the strut, but flexibility at the connection between the strut and retaining wall, due to packing, for example, could significantly reduce the true stiffness of the strut that may not be reflected in the FE model due to the assumption of a pinned or full connection. This could result in an over-prediction of strut force and an underprediction of wall deflection at the strut. Connection stiffness is a common uncertainty that needs to be considered when interpreting analysis outputs. Where elements are joined at a single node, only connections between beam and plate/ shell elements transfer moment and model fully fixed connections. When joining a plate/ shell or beam element to a continuum element, no moment is transferred. A full connection can be simulated approximately by extending the non-continuum element two or three elements into the continuum, as shown in Figure 5.13. There are certain assumptions inherent in non-continuum elements (see Section 5.1.1) that often become invalid near connections, so detailed outputs from connections may be unreliable. Furthermore, there are assumptions regarding the geometry of connections tio ns inh inher erent ent in the use of no non-c n-cont ontinu inuum um elem element ents. s. Co Conne nnecti ctions ons ar aree ass assum umed ed to occur at the central axis of structural members, as shown in Figure 5.14, 5.14, which may nott be the cas no casee in re reali ality ty an and d whi which ch af affec fects ts the predicti prediction on of str struct uctur ural al for forces ces an and d deflections.

Figure 5.14 Typical connection between non-continuum elements Central axes of members Spring element

Shell element

FE model

Inherent assumptions

Reality



5.1.8 5.1 .8

How Ho w ar are e dis distri tribut buted ed loa loads ds app applie lied d in in an an FE FE model model? ?

Distributed loads can be specified in an FE analysis program (e.g. line loads in units of force/distance and area loads in units of force/area), and although they may be displayed in the graphical interface as such, remember that loads can only be applied at element nodes. Load arrangements are converted (automatically in most programs) to equivalent point loads at nodes. However, the resulting equivalent loads, particularly in a coarse mesh, may not represent the intended load adequately. So it is important to check how distributed loads have been represented by equivalent point loads in the outputs, and to make refinements to the specified loading or the mesh geometry appropriately. This is another reason to have finer meshes in loaded areas, in addition to where there are steep gradients of stress and strain, as covered in Section 1.3.2.

5.1. 5. 1.9 9

What Wh at ar are e si sing ngul ulari ariti ties es? ?

A singularity is a point where artificially high or low output is calculated due to some assumptions in the model. A typical example would be very high values of shear force at a point load obtained from an elastic shell element representing a spread foundation (as illustrated in Section 8.2.4). In reality, the point load would be distributed over an area rather than concentrated at a single point. Rather than being infinitely thin, the founda foun dation tion wou would ld hav havee thick thickness ness allo allowing wing greater greater redi redistrib stribution ution of str stress. ess. Also, the reinforced concrete of the foundation would not be elastic but would crack or yield at high stresses and the stresses would be redistributed. Other potential singularity sources include corners with small or large angles, pinned supports and connections and folds in plate or shell elements. Some singularities can be removed relatively straightforwardly, for example by replacing a concentrated load with a more realistic distributed load, as shown by example in Section 8.2.4. Others are more difficult to remove. With experience, singularities in outputs can be recognised as such. This is important to avoid designing structures to resist artificially high outputs of stress.

5.2. 5.2. 5.2.1 5.2 .1

Struct Stru ctur ural al ma mate teri rial als s Can lin linear ear ela elast stic ic mode models ls be be used used for for conc concret rete e and and grou grouted ted structures?

Con oncr cret etee an and d gr grou out, t, bo both th re rein info forc rced ed an and d un unre rein info forc rced ed,, ar aree very co comm mmon only ly us used ed mater ma teria ials ls in ge geot otec echn hnic ical al st stru ructu cture res. s. Ra Rath ther er li like ke so soil il an and d ro rock ck,, th they ey ar aree co comp mple lex x materials, but since they are much stronger and stiffer than soils and soft rocks, geotechnical techn ical mat material erial char characte acterist ristics ics are mor moree lik likely ely to go govern vern soil– soil–stru structur cturee inter interacti action on behaviour and it is usually possible to simplify the concrete or grout constitutive model to a simple, homogeneous, isotropic, linear elastic model. However, to recognise when thiss is no thi nott an app appro ropri priat atee ass assum umpti ption on,, it is nec necess essary ary to und unders erstan tand d the asp aspects ects of concrete and grout behaviour not included in the linear elastic model. Therefore, as well as providing guidance on the use of linear elastic models, this section also summarises the simulation of other aspects of concrete and grout behaviour and non-linear models. Moree com Mor compr prehe ehensi nsiv ve gu guida idance nce on FE an analy alysis sis of con concre crete te str struct uctur ures es is pr prov ovide ided d by Rombach (2011) and and Brookes Brookes (2016). (2016) . 152 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.


For the remainder of this section, the term concrete is taken to mean both concrete and grout.

Linear elastic model Concrete is very stiff compared with soil and soft rocks so the strains in concrete are likely to be small and within the approximately linear initial portion of the stress–strain curve (Figure (Figure 5.15). 5.15). Therefore, in most cases, a linear elastic model is sufficient to model concrete structures in soil–structure interaction analyses. In a uniaxial compressive stress state, normal concrete behaves in an approximately linear elastic way up to a stress of about 40% of the compressive strength according to Eurocode 2 ( CEN, 2004). 2004 ). Exceptions to this would be when concrete stresses and strains are likely to lie outside the linear portion of the curve. This can be checked by running an FE analysis with a linear elastic model for the concrete initially and identifying any parts of the model where non-linear behaviour would be expected. Also, there are certain cases, such as sprayed concrete (shotcrete), where significant stresses can be experienced by the concrete before it has cured, or unreinforced jet grouted structures, where significant tensile strain may occur, that require more advanced constitutive models. These cases are covered later in this section. When ado When adopti pting ng an iso isotr tropi opicc lin linear ear ela elast stic ic mo model del,, on only ly two st stiff iffnes nesss par parame ameters ters ar aree required, namely Poisson’s ratio n and Young’s modulus E . n is usually between zero, which is appropriate for concrete in tension, and 0.2, which is appropriate for concrete in compression. For bending, which has both tension and compression, the n value is often not critical but outputs of structural forces can be checked to see if they are influenced significantly.

Figure 5.15 Typical stress–strain curve for concrete in compression

σ c

Initial linear stiffness

εc



There are a number of factors affecting the E value. value. As well as the particular concrete mix, concrete stiffness is influenced by age, shrinkage, creep, temperature, cracking and bedding-in effects. Under sustained stress the concrete will creep and it is common practice to reduce the E E value by about 50% in long-term cases to help take account of creep and cracking. Like soil and rock, concrete properties are also affected to a certain extent by the level of confinement. A further important factor to consider is shrinkage strains in concrete which could lead to significant changes to the dimensions of large concrete structures, or large stress changes if the structures are prevented from shrinking. Dong ing. Dong et et al . (2016) used (2016) used a relatively simple thermal strain approach to obtain satisfactory predictions of shrinkage effects in concrete slabs interacting with a diaphragm wall.

Reinforcement There ar There aree two ap appr proa oache chess to mod modell elling ing re reinf infor orcem cement ent ba bars rs in con concr crete. ete. Usu Usuall ally, y, a smear sme ared ed app appro roac ach h is tak taken en wh where ere the pr prop operti erties es of the st steel eel ar aree sme smear ared ed ac acro ross ss the sectio sec tion n to cre creat atee a sin single gle ma materi terial al wit with h the com combin bined ed st stren rength gth and sti stiffn ffness ess of the concrete and steel parts of the section, and this is clearly more straightforward to model. Alternatively, for very detailed studies of soil–structure interaction or connection details, an explicit approach is taken where the concrete and steel materials are discretised as in the rea reall str struct uctur uree (th (thee rei reinfo nforc rceme ement nt as bar elements elements wit within hin the con contin tinuu uum m or she shell ll elements for the concrete). A rigid bond is normally assumed between the concrete and the reinforcement but this may not actually be the case in anchorage regions or at laps between bars.

Cracking A particular feature of concrete behaviour is cracking as the material goes into tension. This affects the stress distribution in the concrete as well as the stiffness so is important in detail det ailed ed st stud udies ies of con concre crete te beh behav aviou iourr an and d mak makes es the st study udy of pla plast stic ic beh behav aviou iourr in reinfor rein forced ced conc concrete rete ra rather ther complicated. complicated. The stiff stiffness ness of cra cracked cked concrete concrete sectio sections ns is influenced by the reinforcement arrangement and orientation of the cracks and cannot be cal calcul culat ated ed eas easily ily.. On Onee op optio tion n is to at attem tempt pt to mo model del the cra cracki cking ng and there there ar aree essentially two approaches. The discrete crack approach requires contact or gap elements to be set up in the model at the start, and crack opening and propagation are predicted based on pre-defined criteria. Thee dr Th draw awba backs cks of thi thiss app appro roac ach h ar aree tha thatt the loc locat ation ion an and d cr crac ack k dir direct ection ion ar aree pr preedetermined in the analysis and the material parameters are difficult to measure. The sme The smear ared ed cra crack ck app appro roac ach h is mor moree com common mon and is use used d in ad advan vanced ced mo model delss for sprayed concrete and jet grout. Here a tensile failure surface is included in the constitutive model, which causes the material to soften on crack formation.

Non-linear models There are some occasions when non-linear geometrical behaviour may influence structural behaviour, such as in the buckling of slender columns and struts. Such instances 154 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.


are rare in geotechnical structures but may need to be considered for slender piles with significant lengths exposed above ground level. More com More common monly ly,, a non non-li -linea nearr con const stitu itutiv tivee mod model el ma may y be re requi quire red d wh when en con concr crete ete cracking needs to be considered, for instance in SCL or in some jet grouted structures subject sub jected ed to ten tensil silee str stress esses es (e. (e.g. g. a jet gr grou outed ted sla slab b sub subject jected ed to upl uplift ift pr pressu essure) re).. As described descri bed unde underr Tunnels in Sec Sectio tion n 5.1 5.1.4, .4, adv advan anced ced con const stitu itutiv tivee mod models els ha have ve bee been n develo dev eloped ped for spr spray ayed ed con concre crete te (sh (shotc otcret rete) e) an and d gr grout out,, e.g e.g.. Sch Scha a ¨ ¨ dlich and Schw Schweiger eiger (2014), that include tension softening to take account of concrete cracking. Sprayed concrete models should also include time-dependent strength and stiffness to take account of curing following load application. Concrete loaded early in its curing process is also subject to significantly more creep than concrete loaded when curing is substantially complete. To obtain input parameters, compression tests on shotcrete are needed at different ages. Time-dependent properties are not so relevant for jet grouting because these structures tend to be stressed only after curing.

Fluid concrete As described in Section 1.4.2, approximate installation effects can be considered and, for concrete structures, this may require the simulation of static, fluid concrete. Since fluid concrete has no strength, it is modelled simply by applying its self-weight to the model and, if necessary, its fluid pressure to any non-horizontal surfaces in contact with the concrete. As well as bored piles and diaphragm walls, the installation effect of a thick concrete raft may need to be considered, for instance. The ground response to the weight of fluid concrete in a freshly cast raft differs somewhat from that under the same weight of stiff, hardened concrete.

5.2.2 5.2 .2

Can lin linear ear ela elast stic ic mode models ls be be used used for for stee steell stru structu cture res? s?

Steel structures are also quite commonplace in the ground. Examples include tubular or pipe piles for foundations and sheet pile walls. Steel is also used in ground anchors, soil nails, reinforced earth and rock bolts.

Steel material behaviour The typical stress–strain curve for steel in uniaxial tension in Figure 5.16 shows 5.16 shows that the elastic portion of the curve below the yield stress is quite linear, so the modelling of steel material within its elastic range is well suited to a simple linear elastic model. If the plastic behaviour of steel needs to be modelled, the yield criterion that best relates multi-axial stress states to the uniaxial behaviour shown in Figure 5.16 is generally considered to be the von Mises criterion, although the Tresca criterion can also be used. Often, a simple LEPP model is used to model steel because it captures the initial linear elastic portion of behaviour well and then the steel deforms plastically at constant yield stress which is conservative for material behaviour. Alternatively, a hardening model could be adopted in order to simulate the hardening of the steel material up to its ultimate strength. 155 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.


Figure 5.16 Typical stress–strain curve for steel in tension Strain hardening

σ

Yield stress

Elastic

ε

Sheet pile walls One of the advantages of FE analysis over limit equilibrium methods is that flexibility is included in the calculations, which can lead to economies in design. As described in Section 5.3.1, while a flexible wall has higher deflections, the bending moment is actually lower and cost savings can be made in the structural sections required to resist the bending moment, provided that the higher deflection is acceptable. If performing plastic design of sheet piles, which is permitted for some pile sections in Eurocode 3, remember to consider geometric effects. While sheet piles will resist higher bending moments beyond the initial yield moment (and therefore sustain permanent deformations), only certain sections will be able to develop the full plastic moment resistance, and possibly even slightly higher due to strain hardening, and these are called high rotation capacity sections. Low rotation capacity sections suffer local buckling before the full plastic moment resistance is developed, and all sections will eventually soften due to geometric buckling effects. Such softening behaviour can be incorporated into the materi ma terial al mod model el for the pla plate te or she shell ll elem element ents. s. Alt Altern ernat ativ ively ely,, wh when en mod modell elling ing hig high h rotation capacity sections, plasticity can be allowed only up to the first occurrence of plastic moment so that the softening portion need not be modelled. More information on the FE analysis of sheet pile walls with plastic hinges can be found in BourneWebb et Webb et al . (2011).

5.3. 5.3. 5.3.1

Soil–s Soil –str truc uctu turre in inte terrac acti tion on How does rel relativ ative e soil/ soil/stru structur cture e sti stiffness ffness influ influence ence outp outputs? uts?

The engineering behaviour of two materials of very different stiffness (e.g. soil and concrete) interacting with each other is significantly more complicated than that of one material alone. For instance, an elastic beam on fixed supports can be assumed to have a deflection under a certain load that is inversely proportional to the beam’s bending stiffness EI , while the bending moment remains essentially unchanged in spite of variations in EI (Figure 5.17). 5.17). 156 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.


Figure 5.17 Elastic behaviour of a simple structure Displacement Low EI : • High deflection

High EI : • Low deflection • Bending moment unchanged EI

When a structural member transfers load to or from soil, the situation is more complicated, as illustrated in Figure in Figure 5.18. 5.18. While increasing structural stiffness relative to the soil leads generally to less deflection of the structure and soil, the bending moment is not constant but increases due to the greater ability of the member to distribute loads. As relative structural stiffness decreases, the soil redistributes more stress and the bending momen mo mentt in the st struc ructur turee dec decrea reases ses.. Th Thee rel relat ation ionshi ships ps bet betwe ween en re rela lativ tivee st stiffn iffness ess an and d deflection or bending moment are non-linear, as shown for the range of typical graphs in Figure in Figure 5.19. 5.19. Even against the log of EI , bending moment and displacement usually plot as curves, whose shape depends on the particular characteristics of each application. It is possible to make simplifying assumptions in certain cases of soil–structure interaction analysis, as shown in Figure 5.20. 5.20. If a structure is sufficiently stiff relative to the ground, it can be assumed perfectly rigid and represented as a rigid body (linear elastic material with high stiffness) or by prescribed displacements (see Figure 5.7 for spread foundations) in the FE model. If a structure and its foundations have sufficiently low stiffness relative to the soil, the structure can be assumed perfectly flexible. The structural loads are then independent of settlement because the structure has no ability to transfer Figure 5.18 Soil–structure interaction behaviour

High relative structure stiffness: • Lower deflection • Higher bending moment

Low relative structure stiffness: • Higher deflection • Lower bending moment



Figure 5.19 Influence of relative structural stiffness on soil–structure interaction outputs Bending moment

Displacement

Log EI

Log EI

loads, so ground deformation can be predicted merely by applying the structural loads to the ground and ignoring the structure. In most cases, the stiffness of the structure relative to the ground will lie well within these extreme ext remess so that the structure structure will deform with the ground ground and it will have have the ability to redistribute some of the loads. The structural loads on the ground then depend on deformation and both the structure and ground stiffness will need to be modelled correctly in the FE analysis. In complex cases, the ground and geotechnical structures are normally simulated in a separate FE analysis model from the superstructure. Therefore, some iterations between the separate models are needed until deformations and loads are in sufficient agreement. The structural model may use a simpler method to simulate the ground, such as springs, as discussed in Section 5.3.2.

Figure 5.20 Idealised assumptions in soil–structure interaction Perfectly rigid structure

Perfectly flexible structure

tlemen t on a plane Se t tl

Structure loads independent of settlement

Perfectly rigid ground

Stiff structure

Loa oad d transfer

Loa Lo ad transfer

Structure loads dependent on settlement



Finally, there are cases, usually on hard rock, where ground deformations are sufficiently small to be ignored so that the structure can be assumed to have fixed supports. Calculating the relative stiffness of a structure compared with the ground in order to decide dec ide wh wheth ether er any sim simpli plifyi fying ng ass assum umpti ption onss can be ma made de is ra rathe therr dif difficu ficult lt wit witho hout ut setting up an FE analysis model of the structure and ground anyway. An approximate assessment can be made using, for example, Equation 5.6 from Annex G of Eurocode 2 Part 1-1 (CEN, 2004). K R =

(EJ )s El 3

(5.6)

where K R = relative stiffness, (EJ where K (EJ ))s is ‘the approximate value of the flexural rigidity per unit un it wid width th of the bu build ilding ing st struc ructur turee un under der con consid sider erat ation ion,, obt obtain ained ed by sum summin ming g the flexural rigidity of the foundation, of each framed member and any shear wall’, E E is is the Young’s modulus of the ground and l is is the length of the foundation. A K R value higher than 0.5 is said to be indicative of a rigid structural system. As with most assumptions taken in designing an FE model, the best way to check whether the assumption is valid is to perform FE analyses with and without the assumption and to compare the outputs from each analysis.

5.3.2 5.3 .2

How are How are co coeffi efficie cients nts of subg subgra rade de reac reactio tion n determ determined ined for beam-spring models?

This section has been included in this book on FE analysis because the outputs of FE analysis can provide a reliable source of coefficients of subgrade reaction. The coefficient of subgrade reaction k (also called modulus of subgrade reaction and bedding modulus) is derived from the bearing pressure (or subgrade reaction) on soil divi di vide ded d by th thee resu esult ltin ing g gr grou ound nd de defle flect ctio ion. n. He Henc ncee it itss un units its ar aree pr pres essu sure re/d /dis ista tanc ncee (typically MN/m3). Clearly this method is not intended to simulate soil but merely to simulate the influence of soil on the engineering behaviour of structures. Typically, the coefficient of subgrade reaction is discretised into individual springs with spring constants, fixed at one node and attached to beam or plate elements representing the structure at the other. The main advantage of this method is its simplicity, particularly lar ly in pr prog ogra rams ms ded dedica icated ted to mo more re com comple plex x ana analy lyses ses of st struc ructur tures. es. Ho Howe weve ver, r, on onee significant disadvantage is that the shear stiffness of soil is not taken into account, leading to unrealistic deflections in some cases, particularly at the edges of structures where the transfer of stress to soil outside the structure is not modelled. Often, the k value at edges is increased to help take account of this and to obtain more realistic deflections. Unfortunately, k is Unfortunately, k is sometimes mistaken for a soil property, like stiffness, as though there is a soil test available to measure k. It is not a soil parameter but rather an interaction parameter applicable to a particular combination of circumstances in the ground and in the structure. 159 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.


The only way to measure k directly is to measure the deflection of a full-scale structure with known loads, which is clearly impractical, so a second significant disadvantage of the method is the difficulty of determining k determining k.. This is because a lot of influences, including the following, need to be included in a single parameter: g

g g g

stiffness of the gro stiffness ground, und, and its variation variation with depth, appropria appropriate te for the stress stress and strain level flexural flexur al stiffness of the structure geometry geom etry of the structure structure and the soil (e.g. soil layers, layers, proximity proximity to bedrock) bedrock) loading conditions conditions (e.g. point loads, loads, distributed loads).

The most accurate way to estimate k is to perform an FE analysis of the problem that takes explicit account of all these influences and then obtain k k from from the outputs of bearing pressure and deflection on an appropriately spaced grid, as illustrated by example in Section 8.2.4. Such a grid of k values k values could then provide the input to a structural analysis program modelling the influence of the ground in this way. Further iterations of applied load and deflection may be required until the load distribution in the geotechnical and structural analyses are sufficiently close. REFERENCES

Addenbrooke TI, Ong JCW and Potts DM (2002) Finite element analysis of a compensation grouting field trial in soft clay. Proceedings of the Institution of Civil Engineers – Geotechnical Engineering 155(1): 47–58. Bourne-Webb PJ, Potts DM, Ko ¨ nig D and Rowbottom D (2011) Analysis of model sheet pile walls with plastic hinges. Ge Ge´ ´ otechnique otechnique 61(6): 487–499. Brook Br ookes es CL (20 (2016) 16) How How to Mod Model el Str Struct uctur ural al Co Concr ncrete ete Usi Using ng Fi Finit nitee Ele Elemen mentt Ana Analy lysis sis.. NAFEMS, Hamilton. CEN (2004) EN 1992-1-1 Eurocode 2: Design of concrete structures – Part 1-1: General rules and rules for buildings. CEN, Brussels. Dong YP, Burd HJ and Houlsby GT (2016) Finite-element analysis of a deep excavation case history. Ge Ge´ ´ otechnique otechnique 66(1): 1–15. Lees AS (2017) The simulation simulation of geogri geogrid-sta d-stabili bilised sed soil by finite element analy analysis. sis. Proceed ce edin ings gs of th thee 19 19th th In Inte tern rnat atio iona nall Co Conf nfer eren ence ce on So Soil il Me Mech chan anic icss an and d Ge Geot otec echn hnic ical al Engineering,, Seoul, Korea, 17–22 September. Engineering Li Z, Soga K and Wright P (2015) Behaviour of cast-iron bolted tunnels and their modelling. Tunnelling ling. Tunnelling and Underground Space Technology 50: 250–269. Mo ¨ ¨ ller SC and Vermeer PA (2005) On design analyses of NATM-tunnels. In Underground Space Use: Analysis of the Past and Lessons for the Future (Erdem (Erdem and Sol Solak ak (ed (eds.) s.)). ). Taylor & Francis Group, London, pp. 233–238. Muir Wood AM (1975) The circular tunnel in elastic ground. Ge Ge´ ´ otechnique otechnique 25(1): 115–127. Pan anet et M an and d Gu Guen enot ot A (1 (198 982) 2) Anal Analys ysis is of co conv nver erge genc ncee be behi hind nd th thee fa face ce of a tu tunn nnel el.. Proceedings Tunnelling ’82, ’82, London. Institution of Mining and Metallurgy, pp. 197–204. Rombach GA (2011) Finite-element Design of Concrete Structures, Structures, 2nd edn. ICE Publishing, London. Scha ¨ ¨ dl dlic ich h B an and d Sc Schw hwei eige gerr HF (2 (201 014) 4) A ne new w co cons nsti titu tuti tiv ve mo mode dell fo forr sh shot otcr cret ete. e. In Numerical Numer ical Metho Methods ds in Geote Geotechnic chnical al Engin Engineerin eering g (Hicks, (Hicks, Bri Brinkg nkgrev revee and Ro Rohe he (ed (eds.) s.)). ). Taylor & Francis Group, London, pp. 103–108. 160 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.


Soga K, Bolton MD, Au SKA, Komiya K, Hamelin JP, van Cotthem A, Buchet G and Michel JP (2000) (2000) Development of compensation grouting modelling and control system. In Geotechnical Aspects of Underground Construction in Soft Ground (Kusakabe, (Kusakabe, Fujita and Miyazaki (eds.)). Balkema, Rotterdam, pp. 425–430. Tschuchnigg F and Schweiger HF (2015) (2015) The embedded pile concept – Verification of an efficient effici ent tool for mode modelling lling comp complex lex deep foun foundat dations ions.. Computers Computers and Geote Geotechnic chnicss 63: 244–254. Wang F, Huang H, Soga K, Li Z, Zhang D and Tsuno K (2012) Deformation analysis of a tunnel with concrete segmental lining subjected to ground surface loading using novel joint model. Proceedings of the World Tunnel Congress 2012, Thailand, pp. 364–366. Wisser C, Augarde CE and Burd HJ (2005) Numerical modelling of compensation grouting above shallow tunnels. International Journal for Numerical and Analytical Methods in Geomechanics 29: 443–471. Zdravkovic ´ ´ L, Potts DM and St John HD (2005) Modelling of a 3D excavation in finite element analysis. Ge Ge´ ´ otechnique otechnique 55(7): 497–513.

161



Chapter 6

Can FE analysis be used with design codes? 6.1 .1..

Inttroducti In tio on

In this chapter, only the limit state design concept will be considered, since this has been adopted in the majority of geotechnical design codes. This concept uses design criteria to define limits within which structures are safe and fit for use. The ultimate limit state state (ULS) concerns safety and occurs when a structure or the ground ground suffers from a loss of stability. The serviceability limit state (SLS) concerns the proper functioning and appearance of a structure in service and occurs when a structure or the ground experiences, for example, deformations that are perceptible, cause damage (e.g. cracking) or prevent functioning of machines (e.g. lifts). A safety margin against a ULS occurring is introduced by statistical analysis, by direct selection of conservative parameters or, more commonly, by applying safety factors prescribed by a design code. The safety factor may be a single (global) value applied once in the calculation or else distributed among partial factors applied to individual parameters such as loads, material strengths (e.g. undrained shear strength) and resistances (e.g. bearing resistance). Except for Section 6.1.3, where statistical methods are considered briefly, it is assumed throughout this chapter that the more commonplace partial factor method is being used.

6.1.1 6.1 .1

Why per perfor form m geotec geotechni hnical cal des design ign with FE anal analys ysis is ins instea tead d of conventional methods?

Design cod Design codes es ar aree wri written tten pri prima marily rily wit with h con conve venti ntiona onall des design ign met metho hods ds in min mind, d, so perf pe rfor ormi ming ng de desig sign n by FE an anal aly ysi siss in acc ccor orda danc ncee wi with th a de desi sign gn co code de is no nott al alwa way ys straightforward. Indeed, the flexibility of FE analysis means that no geotechnical design code could provide a set of rules to cover all possible applications of FE analysis. Only a relatively small set of quite fundamental rules may be provided, if at all, leaving the user responsible for interpreting the code for each specific problem and making the correct decisions to ensure that designs comply with the code. Consequently, FE analysis should only be used to perform design in accordance with a design code when the additional workload can be justified. Some of the advantages of the FE method were covered in Section 1.1.1. In addition to these, some specific advantages associated with design codes include: 163 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.


g g g

simultaneous simultaneo us checking checking for limit states states in multiple multiple forms checking check ing SLS and and ULS with one one analysis analysis model model taking into account account the flexibility of structures structures and soil–structure soil–structure interaction interaction effects when checking SLS and ULS.

FE ana analy lysis sis can be app applie lied d to SL SLS S des design ign mo more re str straig aightf htforw orward ardly ly,, as des descri cribed bed in Section 6.2. ULS design is also possible by FE analysis, but there are more aspects to consider, as described in Section 6.3.

6.1.2 6.1 .2

What influe What influence ncess the the occur occurren rence ce of of a limit limit stat state e apart apart from from gro ground und strength (for ULS) and stiffness (for SLS)?

Due to the complexity of FE analysis, there are many influences on the prediction of limit states. Certainly much more than considering merely loads, ground strength (and its partial factor) for ULS design and ground stiffness for SLS design. The sensitivity of limit state predictions to these other influences should be considered, including the following.

Discretisation of geometry Stresses are calculated only at the integration points, so the distribution of stress is known only approximately. As a mesh becomes finer in critical areas, the resolution of stress values improves and the exact solution of a ULS can be approached. If the mesh is too coarse, FE analysis tends to under-predict displacements (SLS) and over-predict failu fa ilure re lo load adss (U (ULS LS). ). Th This is is a pa parti rticu cula larr pr prob oble lem m in 3D an anal aly ysis wh wher eree lo low wer or orde derr elements are normally used and a very high number of elements are required to simulate failure states states accurately in 3D compared compared with 2D. Whether a particular mesh is sufficiently fine in critical areas can only be confirmed by performing a sensitivity analysis with progressively finer meshes until there is no significant change in the predicted limit state. Alternatively, adaptive meshing provides an automated approach (see Section 1.3.2). The choice of element type can also influence the occurrence of a limit state. Some higher order elements with full integration, such as the 15-noded triangle ( Sloan, 2013) 2013 ) in 2D analyses, have been found to provide reasonably accurate failure state predictions (see Section 1.3.1). Lower order elements that were used more commonly in the past were prone to locking (i.e. an over-stiff response due to incompressible constraints and insufficient fici ent deg degree reess of fr freed eedom om)) in und undra raine ined d an analy alyses ses,, par partic ticula ularly rly in 2D axi axisy symme mmetri tricc models, and in some drained analyses with dilation, but this happened less when reduced integration was used. The choice of element type for structures (e.g. continuum or noncontinuum) as covered in Section 5.1.1 also has a significant influence on the prediction of limit states in soil–structure interaction problems.

Initial stress state The initial stress state and the stress ratio K ratio K 0 have a significant influence on the prediction of limit states in subsequent analysis stages. K 0 influences the prediction of structural forces, particularly when displacements are small and active and passive pressures have not been mobilised as well as in heavily over-consolidated soils with high K 0 values. 164 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.

Can FE analysis be used with design codes?

The earth pressure on stiff structures, such as tunnel linings, is usually higher at SLS than at ULS. Horizontal in situ stresses or K 0 are difficult to measure or estimate, as described in Sections 1.4.1 and 3.4.1. Upper and lower bound values should be considered where there is uncertainty in order to understand the influence of K 0 on key outputs and to judge which values should gov govern ern the design. Equations used to estimate K 0 (e.g. Equations 3.6, 3.8 to 3.11) generally contain the friction angle w but this does not necessarily mean that K that K 0 should be factored along with shear strength. Efforts should focus on obtaining a realistic initial stress state as well as performing a parametric study on K 0 where there is uncertainty. ′

Preceding construction stages The simulated construction activities cause changes to the stress state, stress path and stress history which, particularly with advanced constitutive models, influence the prediction of limit states in subsequent stages.

Boundary conditions Concentrated rather than distributed loads are a convenient simplification but can cause artificial local failures and unrealistically high outputs of structural forces at singularities (see Section 5.1.9). When determining failure loads or resistances by FE analysis, imposing displacements and obtaining the failure load from the output (strain control) tends to be more successful than imposing an increasing load to failure (stress control). Assumptions at soil–structure interfaces, such as perfectly smooth, perfectly rough or adopting interface elements clearly have an impact on the prediction of limit states, as they do in conventional design. Similarly, the assumptions of rigid, stiff or flexible structures (see Section 5.3.1) affect the distribution of stresses imposed on the ground and, hence, the prediction of limit states. When model boundaries are too close to the area of interest, the imposed fixities introduce boundary effects to the analysis and could cause failure loads to be over-predicted (see Section 1.2.3).

Drainage conditions An assumption of wholly drained or undrained conditions has an important effect on the prediction of limit states – effects that are conservative only in certain cases (see Section 4.2.2). In the intermediate cases of partial drainage, a coupled consolidation analysis may be required if adopting the drained or undrained assumption would be too inaccurate. Consolidation analyses allow temporal predictions of deformation as well as taking account of strength changes in soil due to consolidation. Such predictions are heavily dependent on soil permeability and drainage path lengths which are often highly uncertain parameters, so should be subject to a parametric study in order to understand more fully the likelihood of a limit state occurring. 165 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.


Constitutive Cons titutive behaviour A constitutive model defines the relationship between stress and strain in an FE model. Clearly, the accuracy of limit state prediction is dependent on the selection of an appropriate constitutive model, as covered in Chapter 2. In general, more advanced models are required requ ired for accu accura rate te defor deformat mation ion (SLS (SLS)) and stru structur ctural al forc forcee (ULS (ULS)) pre predictio diction n than ground failure (ULS) prediction, but there are exceptions. For instance, when modelling undrained behaviour in terms of effective stress (Method A – see Section 4.2.4), parameters such as stiffness have a direct influence on the prediction of excess pore pressures and hence undrained shear strength. Flow rules influence limit state predictions. An associated flow rule where the dilation and friction angles are equal can lead to dangerously over-predicted failure loads in undrained and confined problems. In most cases, a non-associated flow rule should be used and dilation set to zero to obtain realistic failure loads or to help ensure that they err on the safe side. Predictions of deformation (and whether deformations are reversible) for the verification of SLS are influenced by any yield of the ground and hence by strength-related parameters. In turn, the prediction of structural forces for ULS verification is influenced by the stiffness of both the ground and structure (see Section 5.3.1).

Analysis options The accuracy of limit state predictions is affected by choices made by the user or automatically by the program in the setting up of FE analysis models, such as the number of load loa d st steps eps,, the int integr egrat ation ion sch scheme eme and tol tolera erance nces. s. Str String ingent ent tol toler eranc ances es sho should uld be adopted to help obtain accurate predictions (see Section 1.4.3).

6.1.3 6.1 .3

How Ho w can can the the relia reliabil bility ity of desi design gnss by FE FE analy analysis sis be chec checke ked? d?

It is difficult to achieve high accuracy in geotechnical design because there are uncertainties in ground properties and site conditions, because all design methods have elements of approximation and assumption and because errors can occur. One of the fundamental roles of a design code is to introduce reliability into design, i.e. to reduce the probability of limit state occurrences to acceptable levels, taking into account these uncertainties. Design codes introduce reliability into designs primarily in two ways:

1 2

by requiring requiring the study study or selection of paramete parameters rs to obtain a prescribed prescribed degree degree of conservatism (see Section 3.4.1 for geotechnical material parameters) by requiring requiring additiona additionall reliability reliability for ULS desig design n by, for example, example, a parti partial al factor method or statistical analysis.

There are a number of different approaches to try to achieve the level of reliability in FE analysis required by a design code, as described in the following paragraphs.

Deterministic approach Single parameter values are selected according to the requirements of the design code (e.g. ‘moderately conservative’ values) and the design calculation is performed once with 166 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.


the selected parameters together with any partial factors. There are two main drawbacks with this approach. First, it provides no indication of the effect of the variability in ground properties on the reliability of the design, thereby placing a high importance on the selected value. Second, single values of partial factors provided in a design code cannot possibly fit all cases and provide a consistent level of reliability.

Probabilistic or stochastic approach Rather than selecting single values for calculation, all available data are used to define random probability distributions allowing a problem to be fully analysed statistically. This approach is called non-deterministic because random patterns of parameters rather than specific values are used in calculations. Multiple calculations can be performed (as in a Monte Carlo simulation) leading to a direct calculation of reliability (i.e. probability of fai failur lure). e). Thi Thiss ap appr proa oach ch can be app applie lied d in FE an analy alysis sis usi using ng the Ran Rando dom m Fin Finite ite Element Method (Fenton (Fenton and Griffiths, 2008) 2008 ) where random, spatially varying parameter distributions are generated. Multiple analyses are then performed and reliability determined from the outputs.

Sensitivity analysis and parametric study Sensitivity analysis involves varying the input parameters to an FE analysis in order to determine which have the most influence on the key outputs from the model. This is then often followed by a parametric study where a smaller number of the critical input parameters identified in the sensitivity analysis are varied between permissible ranges in order to determine the permissible ranges of critical outputs. These are commonly performed in FE an analy alyses ses and for form m an int interm ermedi ediat atee app appro roac ach h bet betwe ween en the det determ ermini inisti sticc an and d probabilistic approaches described previously. Rather than relying on the selection of single values without considering the effect of parameter variability, as in the deterministic approach, different values across a range are studied and the outputs used to assess reliability. Simple probabilistic methods can be introduced by considering the mean and coefficient of variation of parameters in the selection of input values for parametric stu tudi dies es an and d th then en re relia liabi bili lity ty va valu lues es de dete term rmin ined ed fr from om th thee ou outp tput uts, s, as ou outl tlin ined ed in Section 7.3.3.

6.2. 6.2. 6.2. 6. 2.1 1

Servic Serv icea eabi bili lity ty li limi mitt sta tate te (S (SLS LS)) How Ho w is is the the SL SLS S veri verifie fied d usi using ng FE ana analy lysi sis? s?

The prediction of SLSs is well suited to FE analysis because the displacement finite element method is intended primarily, as the name suggests, to predict displacements. It is also suited to FE analysis because the SLS can occur in the conditions experienced by a structure in service, whereas a ULS requires unrealistic conditions due to the low probability of its occurrence. However, it is worth making the distinction between verifying that an SLS is sufficiently unlik un likely ely to occ occur ur an and d pr predi edicti cting ng ac actua tuall beh behav aviou iourr as ac accur curat ately ely as pos possib sible. le. In the former, due to uncertainties in the properties of the ground, cautious estimates of input parameters, initial stresses and groundwater level, etc. may be adopted in an FE model. Other simplifying assumptions and decisions in setting up the model (such as drainage condit con dition ions, s, ign ignori oring ng an aniso isotr tropi opicc beh behav aviou iour, r, ado adopti pting ng non non-co -conti ntinu nuum um elem element entss for 167 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.


structures, etc.) should also err on the conservative side. Specified loads, particularly the variable (live) loads, are necessarily significantly higher than day-to-day actual loadings, even ev en wh when en unf unfac actor tored. ed. Co Conse nseque quentl ntly, y, sho should uld out outpu puts ts sho show w tha thatt the SLS wa wass not reached, there is a high degree of confidence that the SLS will not be exceeded in the real structure because of all the cautious assumptions and selections. If the outputs were then compared with site monitoring data, it should come as no surprise if the predictions were well on the safe side of the actual measurements. If a ‘most probable’ estimate of deformations were required that would be expected to be closer to measured deformations, then ‘best estimate’ rather than cautious estimates of parameters should be adopted. These include loadings, drainage conditions, groundwater level, etc., as well as constitutive model input parameters. Where verification of both ULS and SLS is required, it is possible in some instances to verify both by simulating the main construction sequence only once since partial factors in the ULS case are applied either to the outputs in output factoring or factoring or in separate ULS stages for input for input factoring, factoring, as described in Section 6.3.2. This leaves unfactored parameters (except (exc ept possibly possibly for a small factor on variable variable loads) which are suited to SLS verification verification in the main construction sequence. This is not always possible because sometimes different parameters are appropriate for SLS and ULS, e.g. a peak strength may be appropriate for realistic deformation predictions in the SLS case while a safer post-peak strength may be appropriate for the ULS case. Also, a more conservative view may be taken of some geometries, such as excavation depth, in the ULS case compared with the SLS case.

6.3. 6.3. 6.3. 6. 3.1 1

Geotechn Geotec hnica icall ult ultima imate te lim limit it st stat ate e (UL (ULS) S) How Ho w is is the the UL ULS S ver verifi ified ed usi using ng FE an anal alys ysis is? ?

Simulating a failure (or ULS) is less straightforward than for an SLS because it involves more complicated material behaviour (yielding and failure) and because an event is being simulated that should be highly unlikely, so the FE model needs to be manipulated in order to simulate an unrealistic situation. It is primarily for this second reason that verifying ULS by FE analysis is not straightforward, particularly for complex models. There are essentially two main approaches to introducing partial factors in geotechnical limit state design, as described in this section and summarised in Figure in Figure 6.1. 6.1. Combining these into a dual a dual factoring approach factoring approach is the most consistent for FE analysis, as described at the end of this section.

Input factoring (or material factoring approach) Partial factors are applied on input parameters in an FE analysis at the sources of uncertainty, i.e. on variable loads and ground strength parameters. They can also be applied on permanent (dead) structural loads, depending on the requirements of the design code. Shear strengths at ground–structure interfaces are also factored. Water levels (and hence pressures) should be set to their worst case or design levels as defined by the design code. Input factoring is suited to problems involving equilibrium of the ground (e.g. retaining wal alls ls,, em emba bank nkme ment nts, s, cu cutt sl slop opes es), ), as sh show own n in Figur Figuree 6.2 6.2,, be beca caus usee fa fact ctor orin ing g th thee 168 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.


Figure 6.1 Input and output factoring approaches INPUT

OUTPUT

Worst case water levels

YES  Any geotechnical failure?

Factored

Ground strength

Calculation

Structural and/or variable loads Ground self-weight

Factored

Structural forces

NO

☺

Already factored

Structural ULS check Unfactored

Input factoring approach INPUT

OUTPUT

Characteristic water levels Ground strength Variable (live) loads

Geotechnical loads, e.g. anchor load, pile load

Unfactored

Small factor

Other loads, including ground self-weight

Unfactored

Apply load (effect) factor

Geotechnical resistance check

Calculation

Structural forces, e.g. retaining wall bending moment

Apply load (effect) factor

Structural ULS check

Output factoring approach

self-weight of the ground (which is very difficult to do) is not required and because these failures are governed by the shear strength of the ground mass. If, having factored the ground strength parameters, geotechnical failure does not occur, the geotechnical ULS can be said to have been verified (while not forgetting the other influences on limit state prediction described in Section 6.1.2). The factors should be applied at dedicated stages during the analysis to avoid running the whole analysis with unrealistic factored input parameters (see Section 6.3.2). The ground strength can be further reduced until failure occurs in order to identify the most critical failure mechanism and obtain a factor of safety at each stage. The strength reduction can be performed by a one-step reduction for basic constitutive models or by a stepwise strength-reduction procedure, as described in Section 6.3.5. Input factoring is less suited to problems where predominantly interface rather than ground failure occurs (e.g. pile foundations and ground anchors). This is because failure is governed by the particular properties of the interface which differ from those of the undisturbed ground mass. Installation of such structures results in remoulding, mixing and changes in density of the ground immediately surrounding the structure, so the parameters of the undisturbed ground (and their partial factors) become less relevant 169 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.


Figure Figur e 6.2 Suitable methods for verifying ULS of geotechnical structures

Internal loads Ground failure

External loads Ground failure

Overall stability

External loads Interface failure

Piled foundations Spread foundations

Suited to either

Retaining walls

Ground anchors, soil nails, rock bolts, etc.

Suited to input factoring

Suited to output factoring

in local failures of these structures. Furthermore, their performance is defined in terms of a directly measured resistance, e.g. pile compressive resistance or anchor pull-out load, and resistance is often either determined by load testing or calculated by direct design methods from in situ tests. This contrasts with a retaining wall, for example, whose performance is defined in terms of the parameters of the ground around it. Input factoring is also less suited to providing factored values of structural forces with a consistent degree of conservatism (although it does provide a valuable additional check as described in Dual in Dual approach later in this section). Factoring ground strength usually transfers more stress to structural elements, causing structural forces to increase, but the relationship can be non-linear. Stiff structures with small ground displacements (e.g. a multi-propped embedded retaining wall in stiff soil) result in no yield in the ground, so factoring the shear strength of the ground can have no effect on structural forces and so introduces no safety margin. Therefore, it should be viewed as an additional check on structural forces in cases where weaker than expected ground has a particularly strong influence on structural forces, e.g. in support structures to marginally stable slopes (see Output factoring). factoring). 170 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.


Output factoring (or load (effect) and resistance factoring approach) With outp With output ut fact factoring oring,, FE ana analyse lysess are performed with essen essentially tially unfactored unfactored input parameters (except perhaps for a small factor on variable or live loads to take account of the greater uncertainty on variable loads – see later in this section) and characteristic water levels (as defined by the design code). Then partial factors are applied on load effects eff ects in the ou outpu tputt of FE ana analy lyses. ses. Pa Partia rtiall fa facto ctors rs ar aree als also o app applie lied d on re resis sistan tances ces usually obtained from a parallel calculation by either FE analysis (see Section 6.3.6) or other calculation methods, and the factored outputs are compared with the factored resistance. If the factored resistance (e.g. ground anchor pull-out resistance) is not less than the factored load (e.g. ground anchor load), then that particular ULS for that construction stage has been verified. It is suited to cases where the load effect and resistance to be compared are both well defined and largely independent of each other. For instance: g g g

pile axial axial load and compressi compressive ve or tensile tensile resistance resistance anchor, anch or, nail or rock bolt bolt load and pull-out pull-out resista resistance nce applied appl ied load on a spre spread ad foundation foundation and bearing bearing or slidi sliding ng resistance. resistance.

The last of those three examples refers more to pad foundations with simple loadings (for which an FE analysis would not normally be required). In raft foundations, soil– structur stru cturee inter interact action ion effects becom becomee signi significan ficantt and check checking ing for geote geotechnic chnical al failu failure re by output factoring is less straightforward. Also, once inclined or eccentric loads are introduced, combined horizontal and vertical failures may be predicted by FE analysis, which are more difficult to compare with calculated resistances. Output factoring is less suited to cases where an output load is not well defined (e.g. in overall stability) or where the output acting on a structure comes from the ground, e.g. the pressure from retained ground acting on a retaining wall. This is because the output interacts with the wall and, in turn, with the resistance also acting on the structure. The output and resistance are therefore not independent and the difference between them is often too small (particularly when the K 0 value is high) to demonstrate that passive resistance failure is sufficiently unlikely to occur (Lees, (Lees, 2013). 2013). Therefo Ther efore re,, for ve verif rifyin ying g geo geotec techni hnical cal ULS by FE an analy alysis sis,, out output put fa facto ctorin ring g is mor moree suited sui ted to ‘in ‘inter terfa face ce fai failur lure’ e’ typ typee st struc ructur tures es wit with h ex exter ternal nal loa loads, ds, e.g e.g.. pil pilee fou found ndat ation ion,, ground anchor, soil nail and rock bolt, where the loads applied to them are outputs from the FE analysis arising from other interactions within the model, for example: g g g g

raft foundatio foundation n supported supported on piles embedded retaining retaining wall wall supported by ground ground anchors anchors slope suppo supported rted by soil nails nails tunnel tunn el supporte supported d by rock rock bolts. bolts.

For cases where a pre-defined external load is applied directly to a structure, e.g. axial load applied to a foundation pile, the ULS check is a simple comparison between applied 171 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.


load and resistance and an FE analysis is not needed. The exception to this is where the resistance is being calculated by FE analysis, which is covered in Section 6.3.6. Output factoring provides factored values of structural forces with a consistent degree of conservatism in most cases as defined by the partial factor value. The factored values are then comp compared ared against against stru structur ctural al resi resistan stances ces as descri described bed in Sectio Section n 6.4. 6.4.1. 1. How Howeve ever, r, there are some instances where output factoring may not provide an adequate safety margin. A common example is when structures (e.g. retaining wall or ground anchors) are used to improve the margin of safety on the stability of a slope. With unfactored input parameters, the slope is stable (just) and structural force outputs would be very low or zero. Applying a partial factor to a near-zero output value would not provide an adequate safety margin. When the ground strength is factored (i.e. input factoring), the slope becomes unstable and structural forces increase to values higher than those obtained from output factoring. In this case, input factoring would provide a more appropriate output for verification of ULS and this is one advantage of the dual factoring approach described later in this section. Design codes generally have different partial factors for permanent (dead) and variable (live) loads. Since it is not possible to differentiate between permanent and variable loads in outputs, the input values of variable load should be factored by the ratio between the variable and permanent load factors. For example, the Structural Eurocodes, at the time of writing, had partial factors of 1.35 and 1.50 on permanent and variable loads, respectively. Therefore, the input values of variable load would be factored by 1.5/1.35 = 1.1 and then the outputs factored by 1.35 so that the effects of all the loads are factored in accordance with the code. A disadvantage of verifying geotechnical ULS by resistance factoring is that it requires prior selection of failure mechanisms for which resistance factors are provided in the design code. The FE method’s advantage of determining the most critical of all failure forms is therefore lost. Some programs allow a pile or anchor resistance (factored or unfactored) to be specified in the input parameters. In effect, the resistance is still determined in a parallel calculation (by FE analysis or other method) or from a load test but the comparison between the output (e.g. anchor load) and resistance (e.g. anchor pull-out resistance) is made automatically by the program. When the output load reaches the specified limiting force, any additional load would need to be redistributed elsewhere in the FE model. Such a progr pr ogram am fea featur turee is pot potent ential ially ly use useful ful for au autom tomat atic ic ve verifi rifica catio tion n of ULS of pil pilee and anchor-type structures as well as to check for potential combined failure mechanisms.

Dual approach Both the input and output factoring approaches have their advantages and disadvantages when applied applied in FE analysis. analysis. Taking only only one approac approach h risks missing a critica criticall ULS due to shortcomings in one or other of the two approaches. Therefore, both approaches should be employed to help verify that all possible ULSs are sufficiently unlikely to occur and to gain from the advantages of each approach, as summarised in Table 6.1. 6.1. 172 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.


Summaryy of the advantages advantages combi combined ned in the dual fac factorin toring g appr approach oach Table 6.1 Summar Input factoring g g

g

g

Suited to verifying ULS ULS ground failures Further strength strength reduction reduction identifies most most critical of all ground failures Failure prediction not not constrained constrained to particular failure forms Obtains more onerous structural structural forces forces in cases where weaker than expected ground has a significant effect

Output factoring g

g

g

Suited to verifyi verifying ng geotechnical geotechnical ULS of piles, anchors, soil nails, rock bolts, etc. Factored Factor ed structural structural forces obtained obtained with consistent degree of conservatism in most cases Calculations are performed with unfactored unfactored parameters (except for variable load) so should shoul d be more realistic realistic

In particular, input factoring is suited to checking for possible failures in the ground while output factoring is suited to verifying adequate resistance of foundation piles, ground anchors, soil nails, rock bolts and similar structures, as well as providing factored values of structural forces. Structural forces resulting Structural resulting from both input and output facto factoring ring should be obtained and the most onerous values from each used to verify structural resistance. This is to cover cases where weaker than expected ground would have a particularly significant effect on structural forces, such as in support structures to marginally stable slopes.

6.3.2 6.3 .2

How Ho w ar are e part partial ial fac factor torss appl applied ied in FE ana analy lysis sis? ?

A well-executed FE analysis should provide a reasonably accurate simulation of a realworld geotechnical structure. A ULS should be sufficiently unlikely to occur that to simulate it would not be an accurate representation of the real geotechnical structure. The advantage of output factoring is that the FE analysis remains realistic while the chec ch eck k on wh wheth ether er a ULS is su suffic fficie ientl ntly y unl unlik ikel ely y is pe perfo rforme rmed d onl only y on the out output puts. s. However, as described in Section 6.3.1, input factoring also needs to be performed and this is more difficult because the FE model begins to depart from reality. There are essentially two approaches to applying input factors. Factored input parameters can be used from the start and throughout every stage of the analysis, as shown in Figure in Figure 6.3, 6.3, or the main construction sequence can be simulated with unfactored input parameters and then loads factored and ground strengths reduced in adjunct analysis stages separated from the main construction sequence (see Figure (see Figure 6.4). 6.4). Input factoring from the start is the easier of the two approaches and can be applied in any software without modification. However, the whole FE analysis becomes less realistic because the loads and ground strengths have been factored. The initial stresses may not be realistic (as demonstrated by example in Section 8.4.2) and, with successive analysis stages, the FE model may depart further from reality, particularly for highly non-linear constitutive models and in situations close to failure, and not necessarily in ´ (2012) obtained a conservative direction. Potts direction. Potts and Zdravkovic ´ (2012) obtained greater errors when using 173 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.


Figure Figur e 6.3 Dual factoring approach with input factored at start

OUTPUT FACTORING

Small factor on variable load in input for all stages, characteristic water levels

Ground strength and external loads factored in input for all stages, worst case water levels

INPUT FACTORING

Initial state Any geotechnical failure?

Factor outputs for geotechnical (piles, anchors, etc.) and structural resistance checks

Construction stage 1

Structural force outputs for structural resistance check Non-critical stage, no ULS check


Any geotechnical failure?


Construction stage 3 Structural force outputs for structural resistance check

Figure Figur e 6.4 Dual factoring approach using strength reduction Ground strength unfactored, small factor on variable load, characteristic water levels

OUTPUT FACTORING

INPUT FACTORING

Initial state



Factor external loads, worst case water levels

Reduce ground strength


ULS stage

Structural force outputs for structural resistance check

Non-critical stage,

Construction stage 2 no ULS check Factor outputs for geotechnical (piles, anchors, etc.) and structural resistance checks


Factor external loads, worst case water levels Reduce ground strength


ULS stage

Structural force outputs for structural resistance check



this ap this appr proa oach ch wit with h adv advanc anced ed con const stitu itutiv tivee mod models els in bea bearin ring g cap capac acity ity ana analy lyses, ses, for example. This approach also requires the complete construction sequence to be simulated twice when undertaking dual factoring: once with unfactored input parameters for the output factoring approach and then again with factored input parameters for the input factoring approach, as shown in Figure 6.3. Strength reduction has the advantage of allowing a dual factoring approach by simulating the complete construction sequence only once (see Figure 6.4) because the input factoring is performed on adjunct stages. This leaves the main construction sequence to be simulated with unfactored parameters (except for the small factor on variable loads – see Section 6.3.1), so it may be possible to verify the SLS as well (see Section 6.2.1). Conse Co nsequ quent ently ly,, ea each ch con constr struct uction ion st stage age is arr arriv ived ed at wit with h re reali alisti sticc pa para ramet meters ers and and,, hopefully, realistic stress states. Non-critical stages, such as adding additional support to a retaining wall without excavation, do not need an adjunct strength-reduction stage. The main disadvantage of strength reduction is that a robust procedure is needed to perform this and there is no agreed, unique way of doing this. This is discussed in Section 6.3.5.

6.3.3 6.3 .3

Whatt val Wha values ues sh shoul ould d the par partial tial fa facto ctors rs ha have? ve?

Design codes provide values of partial factors but these are unlikely to have been calibrated on or intended for FE analysis. Design codes are primarily intended for conventional design methods which often have degrees of conservatism inherent within them. Codes differ in their requirements for selecting geotechnical parameters. Terms such as ‘most probable’, ‘characteristic’, ‘moderately conservative’ and ‘worst credible’ are used, each with different meanings which should be reflected both in the selection of input parameters and in the interpretation of the partial factors stated in the code. Given th Giv thee ma many ny in influ fluen ence cess on li limi mitt st sta ate pr pred edict ictio ion n by FE an anal aly ysi siss de desc scrib ribed ed in Section 6.1.2, the mere application of partial factors in an FE analysis should not be regarded as a satisfactory verification of ULS. All the influences should be taken into account. The validation exercises described in Chapter 7 should include an assessment of whether an FE analysis has achieved the expected safety margin when performing ULS design. Where a design code provides partial factors on ground strength, these are likely to be on the commonly used w , c and cu shear strength parameters. The majority of designs by FE analysis are still performed with these basic parameters but the strength derived by a constitutive model also depends on other influences (e.g. stress state, discretisation, stiffness and dilatancy). Furthermore, some advanced models have strength defined by other parameters. Judgement is clearly needed in order to select appropriate partial factors on ground strength. This can be helped by simulating tests, such as triaxial compression tests, with appropriate stress paths to assess the partial factors needed to achieve the appropriate safety margin on shear strength, as demonstrated under Construction stages in Section 8.4.2. ′

′



6.3.4 6.3 .4

Should Sho uld gr groun ound d stif stiffne fness ss be be facto factore red d as well well as str streng ength? th?

It is true that if ground strength were lower than expected, the ground stiffness would probably be lower than expected too. However, this does not necessarily mean that the ground stiffness should be factored along with ground strength. In many cases, ground stiffness does not actually have much influence on failure loads. Differences in stiffness between ground layers could have a critical influence on stress distributions and factoring stiffness may actually reduce this influence. Some design codes and guidance recommend a reduction of ground stiffness in ULS calculations in cases where ground strength is not a critical parameter, for example to obtai ob tain n st struc ructur tural al for forces ces for an emb embedd edded ed re retai tainin ning g wa wall ll in st stiff iff cla clay y in sho short-t rt-term erm situations. If employing the dual factoring approach described in Section 6.3.1, such cases should be accounted for because the output factoring provides a consistent degree of conservatism on structural forces in most cases. Rather like strength factoring, there is a danger that stiffness factoring would provide a false impression that ULSs have been adequately verified simply because the factoring was carried out. A better approach, particularly because there is generally more uncertainty on ground stiffness than shear strength, would be to perform a parametric study as described in Section 7.3.3. This would identify whether stiffness (or other parameters) has a significant effect on the occurrence of a ULS and, if so, allow an appropriate safety margin to be introduced into the design.

6.3.5 6.3 .5

How Ho w can st stre rengt ngth h red reduct uction ion be per perfor formed med? ?

As described in Section 6.3.1, input factoring of ground strength is best performed in dedicated, adjunct ULS analysis stages during the construction sequence rather than start st arting ing the ana analy lysis sis at the ini initia tiall st stat atee wit with h unr unreal ealis istic, tic, fa facto ctored red gr grou ound nd st stren rength gth.. Consequently, this requires a reduction of ground strength in the ULS analysis stages from realistic (e.g. characteristic in Eurocode) values to factored values. Strength reduction can be performed in two ways: g

g

By a one-step reductio reduction, n, i.e. subs substitut tituting ing the material material for anot another her with the same parameters except for a lower, factored shear strength. Where the stress state then violates the failure criterion, stresses are resolved in an iterative manner with the same stress point algorithm in the software used for a normal elastic-plastic analysis. A one-step reduction is usually acceptable for basic constitutive models but the stress paths must be checked to see whether they are credible (see Section 8.4.4). On reducing the strength, the outputs are checked to see whether any geotechnical failure has occurred. One disadvantage of this approach is that it does not provide a factor of safety on geotechnical failure. By a stepwise strength-reduction strength-reduction procedure procedure included included in the analysis software. software. Usually, in the first step the strength is reduced by a user-specified factor on tan w and c or or c c u and then the shear strength is successively reduced by an automatic procedure until a specified number of steps or target factor of safety ′

′



has been achieved. As with the one-step reduction, stresses that violate the failure criterion need to be redistributed. This can be performed by the same stress point algorithm used by the software for a normal elastic-plastic analysis, which works for basic failure criteria such as Mohr–Coulomb. More advanced failure criteria ´ ´ need a more complex algorithm, such as that suggested by Potts and Zdravkovic (2012). Different programs perform the stepwise reduction in different ways and users should be aware of the method and its shortcomings. One advantage of the stepwise reduction is that a target strength factor can be reached in order to obtain factored outputs and then the shear strength further reduced in order to obtain a factor of safety on ground strength for the first failure mechanism occurring during the strength reduction. Non-convergence Non-converge nce of the solution means that ground failure may have occurred, but this must be checked in the output. Similarly, successful completion of the stepwise strength reduction does not necessarily mean that ground failure has occurred and this needs to be checked. There are several ways to identify a failure mechanism in the outputs, as demonstrated in Section 8.4.4, including the following: g

g

g

Plot a graph of displacement displacement at key nodes against against strength strength factor. Failure Failure has occurred (and the factor of safety value can be determined) at the point where displacements continue to develop toward high values without further reduction in ground strength). Plot contours contours or vectors of incremental incremental displaceme displacement nt at the end of the strengthstrengthreduction analysis stage. This often allows failure mechanisms to be identified more easily than when plotting total displacements for one stage or for all previous stages. Display Disp lay the yielding yielding elements. elements. Where Where they join up to form a mech mechanis anism, m, failure failure has occurred. Note that the appearance of yielding elements surrounded by elements still deforming elastically means that a complete failure mechanism has yet to form.

Constitutive behaviour in Section 6.1.2, dilatancy in confined and As mentioned under Constitutive undrained problems can lead to a significant over-prediction of soil strength, so the dilation angle should be set to zero in such cases. However, However, Tschuchnigg et al . (2015) also noted oscillations in outputs of factor of safety and a strong influence of the dilation angle an gle ev even en in un uncon confin fined ed pr probl oblem ems, s, su such ch as slo slope pe st stab abili ility ty,, wh when en the there re wa wass a lar large ge difference between friction and dilation angles. Clearly, the dilation angle has a strong influence on factors of safety obtained from FE analyses of a range of problems, so its effect on outputs should always be considered. In particular, stepwise strength-reduction procedures in different programs may treat the reduction of dilation angle in different ways. In general, the dilation angle should be reduced with the friction angle until it reaches zero, then reduction of the friction angle only should continue.

6.3.6 6.3 .6

Can geo geotec techni hnical cal re resis sistan tances ces be calc calcula ulated ted by by FE anal analys ysis? is?

A geotechnical resistance is the capacity of the ground to withstand a load or displacementt with men without out mec mechani hanical cal fail failur ure, e, e.g e.g.. bear bearing ing res resist istanc ance, e, pass passive ive re resis sistanc tance, e, anc anchor hor pull-out resistance. 177



As described in Section 6.3.1, specific forms of geotechnical ULS (e.g. bearing resistance or anchor pull-out) can be verified by comparing factored outputs of load (e.g. anchor load) with the corresponding factored resistance. This was part of the output factoring approach. The resistance is normally calculated in a parallel calculation, either by FE analysis, other calculation methods or direct testing. It is not normally calculated in the main FE model simulating the construction sequence because of the additional complexity that this would involve. Calculating resistances by FE analysis requires a particular structure to be brought to failure in a specific way without factoring the strength parameters. This is easier in cases where the resistance is defined in terms of an externally applied load. For example, the bearing resistance of a spread foundation is defined as the applied load that causes bearing failure. Therefore, it can be determined by imposing vertical displacement in an FE analysis (see Figure 5.7) and plotting the output of displacement and corresponding calculated force, rather like the graph from a plate load test. The bearing resistance is the value of load where displacement continues to develop without further increase in load. Output factoring is most suited to structures such as piles, ground anchors, soil nails and rock bolts whose resistance is also defined in terms of an external load but which is dependent depen dent on gro ground– und–stru structur cturee interf interface ace str strengt ength h ra rather ther than gro ground und mass str strength ength.. Since resistances of such structures are heavily dependent on the interface properties, their determination is not well suited to FE analysis. This is because the remoulding, mixi mi xing ng an and d de dens nsit ity y ch chan ange gess as asso soci cia ated wi with th in inst stal alla latio tion n ef effe fect ctss ar aree no nott ty typi pica call lly y measured in order to provide input parameters for the interfaces in the FE model. The resistance is measured directly in load tests or determined by direct design methods calibrated against load tests. Consequently, the only practical way to simulate failure by FE analysis is to adjust interface properties until outputs match load test results, in which case the resistance may as well be obtained directly from the load test results. For structures where failure is caused by geotechnical (internal) loads, such as earth pressures on a retaining wall, the calculation of resistance by FE analysis is far from straightforward. Geotechnical loads cannot be arbitrarily increased like external loads because they depend on the self-weight of the ground. Perturbing forces can be applied to induce failure, as described by Smith by Smith and Gilbert (2011a, (2011a , 2011b 2011b), ), but in complex cases it may not be clear where to apply such forces. Hence, when verifying the passive resistance of retaining walls by output factoring, the passive resistance tends to be calculated by a separate method such as stress field analysis. This has the significant drawback of combining two different calculation methods in an interaction problem and in some cases it is not possible to verify the ULS (Lees, 2013).

6.3.7 6.3 .7

What partia What partiall factor factorss are are applie applied d to und undrai rained ned she shear ar str streng ength th in FE analysis?

As described in Section 4.2.4, there are different methods of modelling undrained soil behaviour. While Methods B and C have their disadvantages, such as not taking into 178 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.


account changes in undrained strength due to, for example, consolidation, applying the partial factor on undrained shear strength c strength c u recommended in a design code is relatively straightforward. Method A has some advantages, including taking into account undrained shear strength chan ch ange gess an and d its co comp mpat atib ibil ility ity wi with th ad adva vanc nced ed co cons nsti titu tuti tiv ve mo mode dels ls,, bu butt it ha hass th thee important difference that cu is calculated by the FE analysis rather than being an input parameter. As well as potential inaccuracies in the calculation of cu (as described in Section 4.2.5), this means that the partial factor on cu recommended in a design code cannot be applied directly to cu . In many constitutive models, shear strength is specified typically in terms of the effective stress parameters w and and c c . While design codes provide partial factors on these parameters, they are intended for drained behaviour where there is less un uncert certain ainty ty on she shear ar str streng ength. th. Re Recom commen mended ded pa partia rtiall fa facto ctors rs for und undra raine ined d strength tend to be higher due to the greater uncertainty. However, this higher factor should not simply be applied to the effective stress shear strength parameters because (a) there are other influences on the predicted shear strength (e.g. excess pore pressure) and (b) the factors of safety in terms of effective and total stresses work in slightly different ways, as highlighted by Tschuchnigg et al . (2015) and illustrated in Figure 6.5. 6.5. ′

′

Engineering judgement is needed to select appropriate partial factors on effective stress strength parameters when using Method A in order to obtain the safety margin intended by the design code. Then, as described in Section 4.2.5, plots of deviatoric stress in the outputs should be checked to see that mobilised strength does not exceed the expected (factored) value of cu according to Equation 4.2.

Figure 6.5 Strength reduction in terms of effective and total stress τ

Fo S =

tan ϕ′i tan ϕ′f

tan ϕ′i =

′

c i

′

c f

e s s o r e t r n s e iv c t i io i v t u c d e e e e f t h r E f f g h n e e r r s t tan ϕ′f

c u,i

Total stress strength reduction c u,f

FoS =

c u,i c u,f

′ c ′ f c i

σ′

Mohr’s circle at failure



6.3.8 6.3 .8

Can ult ultima imate te limi limitt stat states es be be veri verified fied in rock rock mas masses ses usi using ng FE analysis?

The modelling of rock masses by FE analysis was described in Sections 2.2.2 and 2.2.3. The process of verifying ULSs in soft rocks is similar to that for soils described elsewhere in this chapter. In hard rocks, the strength of the rock mass is heavily influenced by the mechanical properties, spacing, orientation and persistence of the discontinuities, particularly in low-stress conditions such as in slopes and surface excavations. Therefore, it is more difficult to take account of the uncertainty in rock mass strength simply by factoring a strength parameter. Variability in the geometry of discontinuities also needs to be considered, for instance by adopting different fracture network models in FE models with explicit discontinuity modelling (see Section 2.2.3). Hammah et al . (2007) applied FE analysis with explicit modelling of discontinuities using interface elements to the prediction of factors of safety for various rock slopes. By employing a strength-reduction approach on the Mohr–Coulomb friction properties of both the intact blocks and the interfaces, they obtained results that compared well with wi th va valu lues es ob obta tain ined ed fr from om di disc scrret etee el elem emen entt an anal aly yse sess of th thee sa same me roc ock k sl slop opes es.. Although the FE analysis could not simulate very large displacements and detachment of blocks, they found that factors of safety could be obtained with sufficient accuracy by identifying only the onset of large displacements. Since the failure criterion of the Hoek–Brown model (described in Section 2.3.3) is not Mohr–Coulomb, some modification to the yield function is necessary in order to perform for m st stren rength gth re redu ductio ction n in an equ equiva ivalen lentt wa way y to Moh Mohr–C r–Cou oulom lomb-b b-base ased d mod models els,, as described by Benz by Benz et et al . (2007). (2007).

6.4. 6.4. 6.4.1

Struct Stru ctur ural al li limi mitt sta tate tes s How are str structur uctural al limit limit sta states tes verified verified in soil soil–str –structur ucture e intera interactio ction n problems?

Factored outputs of structural forces (e.g. bending moment, shear force and axial force) are obtained for structural members from input and output factoring approaches as describ des cribed ed in Sec Sectio tion n 6.3 6.3.1. .1. Whe When n emp employ loying ing the dua duall fa facto ctorin ring g app appro roac ach, h, the mos mostt onerous value from the input and output factoring is used to verify that a structural ULS is sufficiently unlikely to occur. This value is compared with the corresponding structural resistance calculated and factored in accordance with the relevant structural design code. If it is less than or equal to the factored resistance, the particular structural ULS has been verified for that member in that construction stage. Linear elast Linear elastic ic mat material erial mode models ls pro provide vide suffic sufficiently iently accurate accurate pre predicti dictions ons of stru structur ctural al forces in most cases, as described in Section 5.2. It is less likely that predictions of stress would be accurate, particularly for reinforced concrete where uncracked sections are assumed, so it is important that structural forces rather than stresses are used to verify ULS. The sections used in geotechnical structures tend to be directly comparable with structural design codes for checking ULS, e.g. rectangular sections for reinforced concrete 180 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.


and standard sections for steel members, so verification of ULS should be straightforward in most cases. With unusual sections, it is usually necessary to divide them up into regular sections in order to obtain output compatible with structural design codes. Some FE analysis programs allow limits on structural forces to be specified for structural members which are equivalent to resistances. In effect, the program is performing the comparison between output and resistance automatically. When the output reaches the limit, any additional load needs to be redistributed. This is a useful feature for checking combined ground/structure failure mechanisms, e.g. active failure of soil combined with a pla plasti sticc hin hinge ge in an emb embedd edded ed re retai tainin ning g wa wall. ll. Sim Simila ilarly rly,, ad advan vanced ced no non-l n-line inear ar con con-stitutive models can be adopted for structural materials to simulate yielding and other features of material behaviour as summarised in Section 5.2. However, real structures have a limited capacity for yield and could even fail in a brittle way. The rotation capacity of reinforced concrete sections is limited and depends on the amount of reinforcement, while only certain steel sections have high rotation capacity, as described in Section 5.2.2 for sheet piles. Therefore, limits on plastic strains should be checked carefully rather than allowing unlimited yield. It is possible to apply factors to structural material strength parameters but note that the effective stiffness of the structure would be reduced and so structural forces may be under-estimated due to soil–structure interaction effects (see Section 5.3.1). The output factoring approach of comparing structural forces with structural resistances is really intended inten ded for linea linearr stru structur ctural al beha behaviou viourr ra rather ther than the high highly ly nonnon-linea linearr beha behaviou viourr inherent in many advanced structural constitutive models. When performing ground strength reduction as part of the input factoring approach (see Section 6.3.5), it is advantageous to incorporate structural material strength reduction concurrently in order to identify critical failure mechanisms of combined geotechnical and structural failures. In addition, structural resistance should still be verified by the output factoring approach. REFERENCES

Benz T, Schwab R, Kauther RA and Vermeer PA (2007) A Hoek–Brown criterion with intrinsic intri nsic mater material ial str strength ength fact factoriza orization. tion. Interna Internatio tional nal Jou Journa rnall of Ro Rock ck Mec Mechan hanics ics and Mining Sciences 45(2): 210–222. Fenton GA and Griffiths DV (2008) Risk Assessment in Geotechnical Engineering. Engineering . Wiley, Hoboken, NJ. Hamma Ha mmah h RE RE,, Ya Yaco coub ub TE TE,, Co Cork rkum um B, Wi Wibo bowo wo F an and d Cu Curr rran an JH (2 (200 007) 7) Analysis Analysis of blocky rock slopes with finite element shear strength reduction analysis. Proceedings of the 1s 1stt Can Canada ada-U. -U.S. S. Ro Rock ck Mec Mechan hanics ics Sym Sympos posium ium,, Van anco couv uver, er, Ca Cana nada da,, 27 27–3 –31 1 Ma May y (Eberhardt, Stead and Morrison (eds.)). CRC Press, Boca Raton, FL, pp. 329–334. Lees A (2013) Using numerical analysis with geotechnical design codes. In Modern Geotechnical Design Codes of Practice (Arnold, Fenton, Hicks, Schweckendiek and Simpson (eds.)). IOS Press, Amsterdam. 181 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.


Potts DM and Zdravkovic ´ ´ L (2012) Accounting for partial material factors in numerical analysis. Ge Ge´ ´ otechnique otechnique 62(12): 1053–1065. Sloan SW (2013) Geotechnical (2013) Geotechnical stability analysis. Ge Ge´ ´ otechnique otechnique 63(7): 531–572. Smith C and Gilbert M (2011a) Ultimate limit state design to Eurocode 7 using numerical methods, Part 1: methodology and theory. Ground Engineering 44(10) October: 25–30. Smith C and Gilbert M (2011b) Ultimate (2011b) Ultimate limit state design to Eurocode 7 using numerical methods, Part 2: proposed design procedure and application. Ground Engineering 44(11) November: 24–29. Tschuchnigg F, Schweiger HF, Sloan SW, Lyamin AV and Raissakis I (2015) Comparison of finit finite-ele e-element ment limi limitt analy analysis sis and str strengt ength h red reductio uction n techn technique iques. s. Ge Ge´ ´ otechnique otechnique 65(4): 249–257.

182



Chapter 7

How is the accuracy of outputs assessed? 7.1 .1.. 7.1.1 7.1 .1

Introd Int oduc ucti tion on Doess the Doe the accu accura racy cy of eve every ry geote geotech chnic nical al FE FE analy analysis sis nee need d to be checked?

In many fields of engineering, once an FE analysis model has been validated for one application, even for quite complex physical behaviour, it can be applied again and again in similar applications without the need for re-validation. However, once a geotechnical FE analysis model has been validated for one application, it does not become automatically valid for other similar applications. This is because each application in ground engineering is unique. Ground conditions can be very variable and engineering behaviour too complicated to be represented by a single, calibrated model. Therefore, each and every geotechnical FE analysis model needs to be validated. The process to achieve this is described in this chapter. Failing to validate an FE analysis model and providing output to a design team that is inaccurate could lead to incorrect design decisions being made. If the errors were overly pessimistic, this could lead to the unnecessary costs and delays associated with an uneconomic design. If the errors were overly optimistic, this could lead to an unserviceable structure (e.g. cracks in walls or malfunctioning machinery), a reduced margin of safety or even failure, as well as the delays and additional costs associated with overcoming these problems.

7.1.2 7.1 .2

Who is re resp spons onsibl ible e for for the the accu accura racy cy of an an FE FE mode model? l?

In any design calculation, the engineering team is responsible for the correctness and appr ap propr opria iaten teness ess of the cal calcul culat ation ion.. The sam samee app applie liess for an any y geo geotec techni hnical cal des design ign calculations performed by FE analysis. The engineers have ultimate responsibility for the calculation but should receive some support in the different tasks from other roles, as described in this section. However, the engineers still have the responsibility to ascertain whether all the necessary tasks have been completed.

Engineer or user of FE analysis software The engineer is responsible for the way a program is used. This includes verifying that a program works properly on the computer and operating system version being used (see Section 7.2.1). It also includes selecting the program options, constitutive models and input parameters and creating the geometry in order to create an FE model that provides 183 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.


the critical outputs with sufficient accuracy. The engineer must also perform a proper validation of the model and assess its reliability, as described in this chapter. The eng The engine ineer er sho should uld also re repor portt to his his/he /herr man manage agemen mentt or the cli client ent whe when n the there re is insufficient information, e.g. regarding ground conditions or a proposed structure, if this prevents an adequate FE model being created. The engineer may also take on some developer roles when creating user-defined subroutines or constitutive models – see Software developer in this section.

Engineering manager The en The engi gine neeri ering ng ma mana nage gerr ha hass over eral alll resp espon onsib sibil ilit ity y fo forr th thee re reli liab abil ilit ity y of th thee FE model and seeing that proper analysis practices have been followed. Managers without suffi su ffici cien entt ex expe perie rienc ncee in FE an anal aly ysi siss wo woul uld d ne need ed to co cons nsul ultt so some me gu guid idan ance ce in th this is area. ar ea. NAF NAFEMS EMS,, for ex examp ample, le, pr pro ovid videe doc docum ument entat ation ion inc includ luding ing gui guideb deboo ooks ks on the validation of geotechnical numerical models (Brinkgreve, ( Brinkgreve, 2013) 2013) and quality assurance procedures (e.g. Chillery, (e.g. Chillery, 2014). 2014). However, any engineering manager should be able to assess the quality of FE model outputs using his/her wider experience of geotechnical engineering. The eng The engine ineeri ering ng ma manag nager er als also o has the imp import ortan antt res respo ponsi nsibil bility ity of ens ensuri uring ng tha thatt the engineer has sufficient competency to perform particular FE analysis tasks, including backgr ba ckground ound kno knowledg wledgee of the geote geotechnic chnical al ma material terialss and pro problems blems being simul simulate ated. d. As well as maintaining existing levels of competency and keeping engineers up to date with developments, this requires mentoring and training to be arranged for engineers to gain new competences in order to undertake more FE analysis tasks.

Client The client has the responsibility for providing sufficient resources and allowing sufficient time to complete FE analysis tasks properly, in order to benefit from the potential construction cost savings brought by more advanced analysis. Additional investment in site investigations is also required to provide the more detailed ground information needed by FE analysis compared with conventional design methods.

Software developer The software developer is responsible for programming the code of the FE analysis program, checking that the code works correctly and that the theory behind the program has been correctly applied (see Section 7.2.1). He or she must also properly document the models and methods implemented in the software and make this documentation available to the user. Program Progra m users also tak takee on some dev develop eloper er resp responsib onsibility ility when user-develope user-developed d subroutines or constitutive models are implemented into a program. They need to check that the added feature works correctly and also interacts properly with other parts of the program. 184 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.

How is the accuracy of outputs assessed?

7.1.3 7.1 .3

Whatt are Wha are the the poten potentia tiall sour sources ces of erro errorr in an FE FE analy analysis sis? ?

The following is a summary of some of the potential sources of error in any FE analysis. Readers should refer to Brinkgreve (2013) for a more comprehensive description of the potential sources of error and for a very useful checklist. The power and flexibility of FE analysis brings with it a high potential for things to go wrong. The techniques to avoid many of the potential errors listed here are described in the preceding chapters. Indeed, the likelihood of error is clearly very much reduced when software users have the competency to perform the particular analysis task in hand.

Operating systems and hardware With the alm With almos ostt infi infinit nitee po possi ssible ble com combin binat ation ionss of ha hardw rdwar are, e, ope opera ratin ting g sy syste stems ms an and d other software packages running simultaneously with the FE analysis program, small differences in the calculation process that cannot be foreseen may occur. These small differences can become magnified later in the calculation to such an extent that significant errors may be introduced. This can be checked by running the same analysis on different combinations of hardware and operating systems.

Misuse of methods Errors can occur when FE analysis is applied to cases to which it is not well suited or when the user misunderstands the theory behind the software and sets up an analysis model in an incorrect way. For example, using FE analysis of a continuum with small deformation theory to model the effects of large deformations (see Section 1.4.3) or toppling mechanisms in rock masses (see Section 2.2.3) could lead to large errors in the outputs because the analysis methods are fundamentally inappropriate. Such errors can be avoided when users of software understand the methods being employed as well as the limitations of the methods.

Bugs In spite of a software developer’s best efforts to identify and solve bugs in a program, they may still exist, particularly in rarely used parts of the program or with unusual combinations of conditions. Therefore, bugs are always a potential source of error and users should beware.

Tweaks Tweaks are known features in a program that introduce error but are needed to improve the robustness of software. For example, the Mohr–Coulomb failure criterion in principal stress space is formed of a set of yield surfaces that meet at corners. The corners create numerical difficulties because two yield surfaces can be active at the same time when stress states are at a corner. Programs overcome this in different ways, such as rounding off the corners, which leads to slightly different results being obtained from different programs. Therefore, it is important to know well the analysis software being used in order to be awar aree of it itss tw twea eaks ks an and d ap appr prox oxim imat atio ions ns th tha at ma may y be beco come me si sign gnifi ifica cant nt in cer certa tain in applications. 185 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.


Calculation approximations Non-linear problems require the governing FE equations to be solved in incremental form. A number of solution strategies are available and the choice of strategy has a strong influence on the accuracy of the outputs. None of the strategies are exact but require a criterion (usually a tolerance on equilibrium error in static loading problems) defining when the solution is accepted. The specified tolerance is a compromise between accuracy and efficiency. A tight tolerance leads to higher accuracy but longer computation times, while a loose tolerance leads to lower accuracy but shorter computation times. The tolerance value should not be viewed as a direct measure of calculation accuracy because other factors play a role. For instance, an ill-conditioned stiffness matrix (due to, for example, large stiffness or permeability ratios) could have a detrimental effect on calculation accuracy compared with a conditioned stiffness matrix with the same tolerance value. Furthermore, the different stress point algorithms used to calculate stress changes in increments have a strong influence on accuracy. Tolerances that may or may not be controlled by the user are often used in other parts of FE analysis software for greater robustness, for instance to deal with zero stress states in stress-dependent stiffness models. It is important that users understand the tolerances and calculation methods used by the software in order to assess whether they achieve the required degree of accuracy for each analysis task.

Input errors FE analyses require a large amount of input data which increases the likelihood of some data simply being entered incorrectly. To help avoid such errors, the input data should be checked. Checking can be facilitated when the software contains features to tabulate input data and displays warning messages when data fall outside of credible ranges.

Constitutive model As described in Chapter 2, no constitutive model captures all aspects of ground behaviour, so one is selected that includes the important aspects but ignores others. This introduces some approximation. Most models have been calibrated only over limited stress space (e.g. triaxial, hollow cylinder), while on implementation in FE analysis they may be used in general stress space where they have not been calibrated and where some tweaks to the model may have been necessary in order to extend them to general stress space. Linear elastic models are often adopted for structural materials, which is acceptable in many cases (see Section 5.2) but is an approximation of real structural material behaviour. Fu Further rther simpl simplifyin ifying g assum assumptio ptions ns are also made at gro ground– und–stru structur cturee inter interface facess (Section 5.1.3).

Obtaining geotechnical parameters Even with the best efforts to minimise errors during sampling, testing and interpretation, they will still occur, as described in Chapter 3. The parameter validation procedures described in Section 3.4.2 should identify any significant errors but there are still further 186 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.


sources of error in the adoption of model parameters. Stress paths during testing are unlikely to match those occurring in the ground during construction, so the derived parameters may not be totally representative of field behaviour. Parameters derived from triax tri axial ial tes testt con condit dition ions, s, for ex examp ample, le, ma may y nee need d som somee man manipu ipula latio tion n for pla plane ne st stra rain in condit con dition ionss to tak takee ac accou count nt of int interm ermedi ediat atee effe effecti ctive ve st stres resss val values ues (as des descri cribed bed in Section 3.4.1), which introduces further uncertainty. There is also natural variation in ground parameters (aleatoric errors) and the locations of ground layers, but only tiny fractions of the ground are investigated. Consequently, simplified simp lified geometries are adop adopted ted for gro ground und lay layers ers and par paramete ameterr vari variati ation on withi within n ground layers tends to be ignored. This leads to the conservative selection of model paramete par ameters rs and the assum assumption ption of homo homogeneo geneous us gro ground und lay layers. ers. Ho Howev wever, er, the effect of the na natur tural al var varia iatio tion n on ke key y ou outpu tputs ts can be ass assesse essed d in pa para ramet metric ric st studi udies es (see Section 7.3.3).

Geometrical simplifications As described in Section 1.2, simplifications are introduced to the geometry to make the modelling process more manageable and efficient. Simplifications include adopting 2D axis ax isymm ymmetri etricc or pla plane ne st stra rain in ass assum umpti ption ons, s, eli elimin minat ating ing det detail ailss fr from om the geo geomet metry ry,, particularly from the periphery of the area of interest, that are considered unimportant and locating the boundaries to the FE model such that the ground far away from the area of interest can be excluded. If properly judged, such simplifications should not introduce significant error to the key outputs, as described in Section 1.2. Discretising the geometry into a mesh of elements also involves approximation but producing an efficient mesh can help to minimise the error, as described in Section 1.3.2. Adopt Ad opting ing no non-c n-con ontin tinuu uum m ele elemen ments ts for st struc ructur tures es is a com common mon sim simpli plifica ficatio tion n tha thatt intro int roduc duces es som somee err error, or, as sum summar marise ised d in Ta Table ble 5.2 5.2.. Str Struct uctur ural al con connec nectio tions ns ar aree als also o usually idealised in some way, as described in Section 5.1.7.

Construction process simplifications The first stage of any FE analysis involves establishing the initial stresses in the ground. Due to the same uncertainties in the ground parameters, this involves some approximation ma tion of gro ground und lay layer er geom geometry etry,, dens density, ity, str stress ess ra ratio tio and gro groundw undwater ater conditions. conditions. Wheree hist Wher historica oricall manman-made made act activitie ivitiess ha have ve affe affected cted in situ str stresses, esses, furth further er appr approxioximati ma tion on is in intr trod oduc uced ed in sim simul ulat atin ing g th these ese.. Ad Adva vanc nced ed co cons nsti titu tuti tiv ve mo mode dels ls re requ quir iree addit ad dition ional al ini initia tiall st stat atee pa para ramet meters, ers, suc such h as pr pre-co e-conso nsolid lidat ation ion str stress, ess, tha thatt ar aree oft often en difficult to estimate or determine. The errors introduced at the initial stage are likely to influence all the subsequent stages of the analysis, so it is very important to keep these to a minimum. Simulation Simulat ion of cons constructi truction on pro processes cesses inv involv olves es some appr approxim oximati ation on since, while rea reall construction activities are essentially continuous, simulations require such processes to be discretised into large steps in order to be manageable. Processes tend to be simplified, with installation effects, for example, often being ignored (see Section 1.4.2). 187 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.


Loadings are often idealised into point or line loads that do not exist in reality. Even distributed loads are converted to equivalent nodal loads whose accuracy depends on the density of nodes in the mesh (see Section 5.1.8).

Groundwater Groundw ater conditions Groundwater levels, both internal and external, vary in reality due to tidal, seasonal and other effects but assumptions are usually made, often placing them at maximum or minimum levels for conservatism, which introduces error. Hydrostatic conditions are regularly assumed although, in reality, they may only be close to hydrostatic. Groundwater flow and consolidation analyses are always subject to the high degree of uncertainty over the permeability of the ground. An as assu sump mptio tion n of un undr drai aine ned d be beha havi viou ourr is an ap appr prox oxim ima atio tion n be beca caus usee ev even en lo lowwpermeability soils experience partial drainage at normal loading rates, while the methods of simulating undrained behaviour (see Section 4.2.4) also involve some approximation.

7.2. 7.2. 7.2.1 7.2 .1

Assess Asse ssin ing g acc ccur ura acy Whatt is the dif Wha differ ferenc ence e betwe between en verifi verifica catio tion n and and valida validatio tion? n?

Verifica Verifi catio tion n esse essenti ntiall ally y mea means ns che checki cking ng tha thatt a pr prog ogra ram m wor works ks cor corre rectl ctly. y. It inv involv olves es checking that the underlying mathematical model has been implemented in the computer program correctly and this is normally carried out by reproducing a theoretical solution until sufficient accuracy has been achieved. It also involves identifying and solving bugs in the software code. The software developer is responsible for verification (except in the cases of user-defined sub-routines and constitutive models where the user has responsibility for verification). Nevertheless, some bugs may still remain within the program and the user should beware of these. Verification also involves checking that a program works correctly on a particular computer and operating system version and this is the responsibility of the user. This can be performed by running an FE analysis of a previous project or case study with the new computer or operating system version and comparing its output with that from a known compatible computer and operating system. Validation means checking whether an FE model represents reality sufficiently accurately. Therefore, while a mathematical model may have been implemented correctly (as checked by verification), the mathematical method may still be inappropriate for the problem being simulated (misuse of method) or incorrect input parameters may have been specified. The purpose of validation is to check whether the mathematical methods, input inp ut par param ameter eterss an and d set set-up -up of the FE mod model el ha have ve pr produ oduced ced suf suffici ficient ently ly ac accur curat atee outpu ou tputs. ts. Th This is is the res respo ponsi nsibil bility ity of the use userr an and d the techniqu techniques es for do doing ing this ar aree described in the next section.

7.2.2 7.2 .2

How Ho w ar are e out output putss che check cked ed for ac accu cura racy? cy?

The full output from a typical FE analysis is too voluminous and the calculations too complicated to be checked explicitly as in conventional calculations. Therefore, agreement needs to be reached at an early stage between project stakeholders on an acceptable level of checking and documentation, which is likely to differ from the requirements for conventional design calculations. 188 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.


This section contains a summary of the common methods used to check FE analysis outputs. They generally involve comparing outputs with expected outcomes (based on experience or published case histories), other calculation methods or ongoing site monitoring. Above all, a check of FE analysis outputs should be based on a combination of these methods rather than relying on just one. More detail and further references on each method can be found in the guidebook on validation by Brinkgreve (2013).

Initial check Signifi Sign ifica cant nt err error orss ca can n be id iden enti tifie fied d re read adil ily y fr from om a qu quic ick k gl glan ance ce at th thee ou outp tput utss by engineers with relevant experience in the structure types being simulated. Any values that are very different from what would be expected based on experience indicate that there is a significant error in the FE analysis. The following list contains outputs that are commonly viewed for each analysis stage as part of an initial check: g

g

g

g

g

g

deformed mesh, using deformed using both true and exaggera exaggerated ted scales, to check that deformations look reasonable steady-s stea dy-stat tatee and excess pore pressure pressure contour contour plot plotss to check that that the impo imposed sed groundwater flow boundary conditions and drainage conditions worked as expected vector vect or plots of princ principal ipal stress stress and strain directio directions ns to check that they are consistent with applied loads stress str ess paths for key integrati integration on points to check that they are are as exp expected ected and that constitutive model and input parameter selections were appropriate contour conto ur plots of normal and shear shear strain to check that that they seem reasonabl reasonablee and to help highlight any problem areas in the FE model plot state parameters parameters to check that advanced advanced constitutive constitutive models are performing as expected, e.g. stiffness in strain-dependent models.

On identifying an issue, further investigation of the output and input data should reveal the source of the error. With 3D models, output for locations within the model is not immediately apparent. It is important to display data on sections that step sequentially through the model in order to reveal the output in the heart of the model.

Comparison with known solutions For each of the common geotechnical structure types there are alternative analytical methods to FE analysis. They are usually less sophisticated than FE analysis, so typically provide a single output (e.g. displacement or failure load) for a single structure type (e.g. strip foundation or slip circle). Many of the methods are well known, reliable and simple to use. Therefore, they provide an alternative calculation method whose output should be reasonably close to the output from an FE analysis of the same problem. Comparing the two outputs with each other provides a means of validating the FE analysis. How close the outputs should be depends on the appropriateness of each method for the problem being analysed. 189 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.


The following is a list of commonly used alternative analytical methods: Analyticall (or closed Analytica closed-form -form)) solutions: solutions: for rela relativ tively ely simple problems problems,, a set of equations describing behaviour can be solved exactly by symbolic manipulation. These tend to be available for linear elastic problems such as the settlement beneath a rigid strip foundation or for beam bending in structures, or for steady flow in confined and unconfined porous media. These are generally more useful for verification than validation because of their limited scope. Limit equilibriu equilibrium m and stress field methods: methods: these provide provide simplified simplified predictions predictions of failure loads on pre-defined mechanisms. For example, slope stability solutions, Caquot–Kerisel equations for active and passive pressure with soil–wall friction, Prandtl’s solution for bearing capacity on cohesive soil. Design Desig n charts and empirical empirical design methods: methods: largely largely based on case study observations, a number of design charts and methods have been published for most geotechnical structures, particularly for deformations where, in the past, practical calculation methods did not exist. For example, Clough and O’Rourke (1990) published design charts for wall movements and ground surface settlements around retaining walls. Such methods are rather broad and approximate but provide an indication of the order of magnitude of outputs to be expected from FE analyses of similar problems.

g

g

g

As an ex exam ampl ple, e, th thee va vali lida dati tion on of ou outp tput utss of gr grou ound nd su surf rfa ace se settl ttlem emen entt be behi hind nd an embedded retaining wall supported at the top is shown in Figure 7.1. 7.1. The empirical profil pr ofilee wa wass ob obtai tained ned fr from om Clo Clough ugh an and d O’R O’Rou ourk rkee (19 (1990 90)) and thi thiss is com compa pared red wit with h outputs from two FE analysis models (A and B) that adopted different constitutive models for the soil. The settlement output from analysis A was in the opposite direction and plotted as a different shape to the empirical profile. Such instances suggest that the analysis had a Figure Figur e 7.1 Example validation of ground surface settlement outputs –15

FE analysis A –10

Horizontal distance behind wall: m

m –5 m : 0 t n e 0 m e l t t e S 5

10

20

30

40

Empirical profile 10

15

FE analysis B


50


serious error and some aspect, such as the selection of constitutive model, was in error. The output from analysis B showed a settlement profile with the same shape and direction as the empirical profile but with a small difference in the values. In such cases the ground and structure are probably being modelled in the correct way while some minor adjustment to the input parameters would achieve a closer match with the validation data, if that was considered necessary. In analysis A, an LEPP model was adopted for the soil which resulted in an overprediction of heave on excavating the soil in front of the wall. Therefore, the constitutive model was inappropriate as proved by this validation exercise. In analysis B, a more appropriate non-linear strain-dependent stiffness model was adopted, hence the better match with the validation data. The slight difference between the analysis output and the empirical profile was considered to be due to the effect of the wall stiffness and, since the empirical profile is only approximate, it would probably be unnecessary to try to achieve a closer match with the empirical profile in this case.

Comparison with site monitoring data Monitoring data collected during construction should be compared with the outputs from FE analyses of the same construction construction activities. This will help validate validate the FE analysis and allow changes to be made in order to improve the reliability of predictions for the remainder of construction. Such observational approaches to design are discussed further in Section 7.3.1. Note that a close match between FE analysis outputs and monitoring data should only be expected if ‘most probable’ estimates of input parameters, loadings, etc.. ha etc hav ve bee been n ado adopte pted. d. In ma many ny FE mo mode dels ls un under dertak taken en for de desig sign n pu purpo rposes ses,, mor moree conservative estimates of parameters and assumptions are usually taken which would normally result in more onerous outputs compared with the monitoring data.

Comparison with other numerical analysis software When additional software is available, it is very useful to perform the same simulation on another verified FE analysis program and/or a program using a different numerical modelling technique, such as the finite difference method, beam-spring method or limit analysis (Sloan, (Sloan, 2013), 2013 ), also on different hardware. The outputs are likely to differ due to the different models and methods employed in each program. However, if the outputs are reasonably close, perhaps within 10%, then this would provide some validation of the original FE analysis, provided that the same errors were not repeated in each analysis model.

Case histories Published case histories of similar geotechnical structures in similar ground conditions can provide a useful source of validation data. They must be well documented in order to be useful, including full details on the construction sequence and ground properties. Ideally, the case histories would include FE analysis simulations in order to learn from the methods, models and parameters used successfully in similar projects. Detailed case histor his tories ies wit with h mon monito itorin ring g da data ta can also be use useful ful sin since ce the these se can be sim simula ulated ted and outputs compared with the monitoring data in order to validate the modelling approach to a particular problem. 191 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.


7.3 .3.. 7.3.1 7.3 .1

Managi Mana ging ng err rror ors s If ina inaccu ccura racy cy ex exis ists, ts, ho how w shou should ld thi thiss be be manag managed? ed?

A judgement needs to be made about an acceptable level of accuracy. Clearly, an exact match between FE model and reality will never be achieved and nor is it necessary. For design, it is important that errors cause the critical outputs to err on the conservative side. After that, the appropriate degree of accuracy depends on the individual needs of each project. Where the potential cost savings in construction can justify additional effort on FE analysis at the design stage, then the FE analysis can be improved with more detailed study, perhaps by a specialist, and possibly additional site investigation. In other cases, a less accurate but conservative FE analysis may be sufficient to meet the needs of the project. Sensitivity analyses are required to identify the parameters with greatest influence on the key outputs, followed by parametric studies in order to assess the reliability of FE analysis outputs, as described in Sections 7.3.2 and 7.3.3. The design should be able to withstand, by adequate margins, the range of outputs arisin ari sing g fr from om the par parame ametri tricc st stud udy. y. Ad Addit dition ionall ally, y, an ob observ servat ation ional al ap appr proa oach ch can be employed where, if monitoring during the early stages of construction is found to lie within the more favourable ranges of output from the parametric study, then a more cost-effective design can be introduced during construction. This requires less control than the observational method (which can bring greater cost savings in some situations – see Section 7.3.4) because if anybody neglects to study the monitoring data, then the more pessimistic design prevails.

7.3.2 7.3 .2

What is the dif What differ ferenc ence e betw between een sen sensit sitivi ivity ty ana analy lysis sis and parametric study?

Both involve evaluating the effect of input parameters, which can include constitutive model mo del pa para ramet meters, ers, bou bounda ndary ry con condit dition ionss (e. (e.g. g. imp impose osed d loa loads ds or dis displa placem cement ents) s) or geometrical values (e.g. excavation depth), on the critical outputs required from the FE analysis. A sensitivity analysis involves analysis involves varying all or certain input parameters to determine which affect the outputs the most. A parametric study involves study involves varying input parameters between certain ranges in order to determine the dependency of key outputs on input parameter uncertainty. To study all the parameters in this way can be very time-consuming, so more attention is paid to the critical parameters as identified in the sensitivity analysis. Typically, three values of each parameter are assessed: one at each end of a range (as discussed in Section 7.3.3) and the deterministic value. On completion of the parametric studies, as well as gaining an improved understanding of the behaviour of the structure, a range of plausible responses of the structure will be obtained, allowing the reliability of the design to be assessed. 192 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.


7.3.3 7.3 .3

Over wha Over whatt rang range e of par parame ameter terss shoul should d a par parame ametri tricc stud study y be undertaken?

Where geotechnical parameters are considered to vary randomly across a site, a parametric study could be undertaken where each parameter is varied between the maximum and minimum values interpreted from the site investigation data. However, this may result in an exaggerated range of outputs because it is highly unlikely where parameters vary randomly that the maximum and minimum parameter values would ever exist in a large enough volume of ground to govern the outputs in this way. It is more appropriate to vary parameters over a narrower range, but one that still takes account of the variability of a parameter about its mean. The statistical measure of this variability is the standard deviation s which represents the range of a parameter each side of the mean within which about 68% of values would lie assuming a normal distribution. In most cases it would be appropriate to vary a parameter across a range of about 1s above above and below the mean. But ho But how w ca can n s be cal calcul culat ated? ed? Ra Rarel rely y is the there re suf suffici ficient ent da data ta av avail ailabl ablee fr from om a sit sitee investigation for it to be calculated directly from the data using a formula. So, it can be estimated using the ‘3s rule’ as illustrated in Figure 7.2 and described by Duncan (2000). Since 99.7% of values in a normal distribution lie within +3s of of the mean, for practical pra ctical purposes +3s can be considered the upper and lower bound to all the data and this is probably easier to estimate than the +s bounds. As shown in Figure 7.2, the mean, highest conceivable and lowest conceivable lines are drawn on the data.

Figure 7.2 The ‘3s rule’ rule’ and range for parametric study of arbitrary data set Arbitrary parameter 0

20

40

60

80

0

2

4 h t p e D

6

8

10

l e b v a i c e n c o t s e w L o

–3 σ

M e a n

H i g h e s t c o n c e i v a b l e

– σ + σ Suitable range for parametric study

+3 σ



Table 7.1 Typical coefficient of variation values for geotechnical parameters (Duncan, 2000) Parameter

Coefficient of variation V : %

Weight density, g Buoyant weight density, g b ′ Friction angle, w Undrained shear strength, c u ′ Undrained strength ratio, c u / s sv Compression index, C c Pre-consolidation pressure, p c Permeability k of saturated clay Permeability k of pa part rtia iallllyy sa satu turrate ted d cl clay ay

3–7 0–10 2–13 13–40 5–15 10–37 10–35 68–90 130– 13 0–24 240 0

Since all the data should lie within the outer lines, these often need to be placed further apart than might intuitively be judged based on a limited amount of data. However, any data considered to be anomalous should be ignored because these are not representative. The +s lines lines can then be added one-third of the distance between the mean and +3s lines and these represent a suitable range for parametric study in most cases. Duncan Dunc an (2 (200 000) 0) co coll llec ecte ted d pu publ blis ishe hed d va valu lues es of st stan anda dard rd de devi viat atio ion n fo forr a nu numb mber er of different geotechnical parameters expressed as coefficients of variation V which which can be used to validate derived values. A selection of the values is shown in Table 7.1. 7.1. s can be determined from V simply simply by multiplying the mean of a particular data set by V . A par parame ametri tricc st stud udy y ma may y the then n be per perfor formed med usi using ng the fol follo lowin wing g pr proce ocedur dure. e. The FE analysis is first performed with deterministic values of parameters. These can be the average (in situations where most probable predictions are required) or more conservative values if the design is in accordance with a design code. The parameters to be included in the parametric study (possibly identified from a sensitivity analysis) are then changed to the top or bottom of their range (mean +s ) such that each combination is used as input data to a separate FE analysis. All other parameters not included in the parametric study remain at their deterministic values. For example, in a simple FE analysis of an embedded wall supported at the top in a granular soil simulated with a LEPP model with a Mohr–Coulomb failure criterion and stiffn st iffness ess inc incre reasi asing ng wit with h dep depth, th, a par param ametri etricc st study udy wa wass und undert ertak aken en as fol follo lows. ws. Tw Two o ′ ′ parameters were studied: the friction angle w and Young’s modulus E of the soil for which coefficients of variation of 10% and 40%, respectively, were determined. The FE analysis was run five times with the combinations of parameters shown in Table 7.2, 7.2, starting with the deterministic values (which were the mean values in this case) and then with each parameter either at the top end of the range (mean + one standard deviation) or at the bottom end of the range (mean − one standard deviation). All other parameters remained constant at their deterministic values. 194 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.


Table 7.2 Example parametric study combinations ′

E ′ value

Name

w

value

Det.

w det

Comb. 1

w mean + s w

Comb. 2

w mean − s w

Comb. 3

w mean + s w

Comb. 4

w mean − s w

′ ′ ′ ′

E ′det ′

′

′

′

E ′mean + s E

′

E ′mean − s E

′

E ′mean − s E

′

E ′mean + s E

′

′

Outputs of retaining wall deflection and bending moment obtained from these combinations are shown in Figures 7.3 and 7.4 7.4,, respectively. From these it is immediately apparent how much the outputs change in response to the two soil parameters varying within plausible ranges of values around the mean. It facilitates greatly in assessing the reliability of the design should these soil parameters be higher or lower than the interpreted mean value. Also ad Also adde ded d to th thee be bend ndin ing g mo mome ment nt di diag agra ram m in Fi Figu gure re 7. 7.4 4 ar aree th thee ma maxi ximu mum m an and d minimum values that might be determined using a partial factoring approach. Output factoring (see Section 6.3.1) was applied to the outputs of maximum and minimum bending moment from the deterministic analysis by multiplying them by 1.35 which is thee re th reco comm mmen ende ded d pe perm rman anen entt lo load ad (e (effe ffect ct)) fa fact ctor or in Eu Euro roco code de 7 (CE CEN, N, 20 2004 04), ), fo forr example. The factored minimum value was about the same as that obtained from the parametric study, while the factored maximum value was somewhat higher. Figure 7.3 Parametric study outputs – retaining wall deflection Wall deflection: mm 0

0

10

20

30

40

2 4 m : h t p e D

6

Det. Comb.1 Comb.2 Comb.3 Comb.4

8

10 12 14



Figure Figur e 7.4 Parametric study outputs – retaining wall bending moment Wall bending moment: kNm/m –300

–200

–100

0

100

200

0

det. min. ×1.35

2

Det. Comb.1 Comb.2 Comb.3 Comb.4

4 m : h t p e D

6 8 10 12

det. max. ×1.35

14

This sug This sugges gests ts tha thatt the 1.3 1.35 5 fa facto ctorr wou would ld ha have ve bee been n ap appr prop opria riate te to tak takee ac accou count nt of uncertainty in the soil parameters in the upper part of the wall but rather high for the lower part. This highlights the advantage of performing even a simple reliability analysis compared with using the ‘one size fits all’ partial factoring approach. Howeve Howe ver, r, suc such h a sim simple ple pr proba obabil bilis istic tic app appro roac ach h con consid siders ers on only ly ra rando ndomly mly va varia riable ble parameters. If there were clear zoning of parameters with softer ground encountered in some areas more than others, then this would either need to be considered explicitly in the FE model or more advanced stochastic methods would be required that consider the spa spatia tiall va varia riabil bility ity of inp input ut pa para ramet meters ers (r (refe eferr to Fenton Fenton and Griffi Griffiths, ths, 200 2008 8 , for example).

7.3. 7. 3.4 4

What Wh at is th the e ob obse serv rvat atio iona nall me meth thod od? ?

There is more to the observational method than simply observing the implementation of a design. It is a carefully considered approach to design, well suited to numerical analysis. It was developed by Peck (1969) and is also summarised by Clayton et al . (1995). (1995). Some key elements to the method are as follows: g

g

g

g

sufficient site investiga sufficient investigation tion to determine a ground ground model and parameters, parameters, but not necessarily in detail interpreta interp retation tion of the mos mostt probable probable and the permi permissible ssible range range of gro ground und conditions numerical nume rical analyses analyses with par parametr ametric ic studies to determine most probable probable and permissible ranges of outputs selection of parameters parameters to be be monitored during construction construction



g

g

g g

documentation of most probable documentation probable and permissible range range of monitored parameters parameters as obtained from the analysis output pre-determination pre-determina tion of design or construction sequence sequence modifications modifications for every every foreseeable deviation of monitored parameters from most probable values start of of construction construction and and monitoring monitoring implementation of design changes and and construction construction sequence modifications modifications in response to actual observed behaviour.

There is a greater potential for cost savings than with the more simple observational approach (see Section 7.3.1) because pre-determined design options are selected during constr con struct uction ion in re respo sponse nse to bet better ter or wor worse se tha than n ex expec pected ted mo monit nitor ored ed per perfor forman mance. ce. However, the observational method requires strict control with clearly defined responsibilities for studying and interpreting the monitoring data since, if the data were not studied in a timely manner, a worse than expected performance may go unnoticed, leading to a serious failure. Ideally, the engineers who performed the parametric studies should also study and interpret the monitoring data since they would understand most the implications of the data on the design options. REFERENCES

Brinkgre Brinkg reve ve RBJ (20 (2013) 13) Valid Validat ating ing Num Numeri erical cal Mod Modell elling ing in Geo Geotec techni hnical cal Eng Engine ineeri ering ng.. NAFEMS, Hamilton. CEN (2004) EN 1997-1 Eurocode 7: Geotechnical design – Part 1: General rules. CEN, Brussels. Chille Chi llery ry M (20 (2014) 14) NAFEMS NAFEMS EMS,, NAFEMS Sim Simula ulatio tion n Ha Handb ndbook ook – Qu Quali ality ty Man Manage agemen mentt . NAF Hamilton. Clayton CRI, Matthews MC and Simons NE (1995) Site Investigation, Investigation, 2nd edn. Blackwell Science, Oxford. Clough GW and O’Rourke TD (1990) Constru Construction ction indu induced ced mov movements ements of insitu walls. Proceedings ASCE Conference on Design and Performance of Earth Retaining Structures, Structures , Cornell, ASCE Pub. no. 25, pp. 439–470. Duncan JM (2000) Factors of safety and reliability in geotechnical engineering. Journal of Geotechnical and Geoenvironmental Engineering 126(4): 307–316. Fenton GA and Griffiths DV (2008) Risk Assessment in Geotechnical Engineering. Engineering . Wiley, Hoboken, NJ. Peck RB (1969) Advanta Advantages ges and lim limita itatio tions ns of the ob observ servat ation ional al meth method od in app applie lied d soi soill mechanics. Ge Ge´ ´ otechnique otechnique 19(2): 171–187. Sloan SW (2013) Geotechnical stability analysis. Ge Ge´ ´ otechnique otechnique 63(7): 531–572.



Chapter 8

Examples 8.1.

Inttroduction In

Three examples are presented in this chapter to help illustrate some of the points made in the precedin preceding g cha chapte pters. rs. The re relev levan antt sect section ionss fr from om Cha Chapte pterr 1 on sett setting ing up an FE analysis model are shown in brackets (e.g. §1.2.3) in the second section of each example, where more information to justify a particular decision can be found. The examples have been kept relatively simple in order to illustrate the decision-making process, parameter determination and validation of outputs as described in the preceding chapters clearly without being burdened by the level of detail associated with real projects. They are not intended to be benchmarks or to be used for validation purposes, nor should the outputs or findings from these examples necessarily be expected to be applicable to other structures of the same type. Readers should refer to published case histories relevant to the ground conditions and structure types in their particular projects for more guidance and for sources of validation data. In an effort to keep this book entirely software-neutral, the specific software and, sometimes, the specific constitutive models employed in these examples are not mentioned. However, enough information is provided for readers to prepare similar models using their preferred software and to compare outputs with those presented here for each example.

8.2. 8.2.1

Raft fou Raft founda ndatio tion n wit with h set settlem tlement ent-r -redu educing cing pile piles s example Summary

This example concerns the construction of a raft foundation to support a multi-storey building. The ground was composed of a medium-dense, lightly over-consolidated sand and because settlements were likely to be large, consideration was given to the use of settlement-reducing piles. The particular features of this example include: g g g g g

3D model model with with a plane plane of symme symmetry try K 0 initial stress procedure double-hardening, double-hardenin g, hyperbolic constitutive constitutive model with stress-dependen stress-dependentt stiffness obtaining geotechnical geotechnical parameters parameters from pressuremeter pressuremeter and pile load load test data data drained displacement analysis 199



g

g

both continuu continuum m and non-continu non-continuum um elements elements were used for the raft and piles and the outputs compared validation using alternative alternative analysis analysis methods. methods.

8.2.2 8.2. 2 Sett Se ttin ing g up th the e FE an anal alys ysis is mo mode dell Justification for using FE analysis (§1.1.1) The alternative analysis methods (e.g. boundary element method) adopt assumptions such as a single size of pile or that the piles are distributed on a regular grid across the entire area of the raft. In this example, a small number of piles of different lengths placed near the centre of the raft needed to be considered. Such irregular geometry was more suited to FE analysis.

Aims of the model (§1.1.2) 1

2 3 4

to predict predict the settlement distribut distribution ion across across the raft raft with and without without the proposed settlement-reducing piles in order to select the appropriate option or assess alternatives to predict predict the loads in the settlement-r settlement-reduci educing ng piles in order to check their their design to provide provide appropria appropriate te coefficients coefficients of subgrade subgrade reaction reaction for the raft to be used in the structural design of the raft to predict predict bending moment moment in the ra raft ft in order to pro provide vide validat validation ion data data for the structural analysis model.

Geometrical Geom etrical simpl simplifica ifications tions (§1.2.1 and §1.2.2) This site was surrounded by roads on all four sides, as shown in Figure 8.1, 8.1, so there was no significant interaction with nearby buildings and these were omitted from the model. Similarly, interactions between the road, its associated underground services and the raft were not considered significant. Therefore, the geometrical features included in the FE analysis model could be simplified to those only of the proposed raft foundation and piles. The layout of the settlement-reducing piles is shown in Figure 8.2. 8.2. Regarding the superstructure, while its stiffness would have a significant effect on the behaviour of the raft foundation (as described in Section 5.3.1), its inclusion in the geotechnical FE analysis model was not necessary in order to take this into account. Rather, coefficients of subgrade reaction obtained from the output of the geotechnical FE model were adopted in the springs representing the soil in the structural FE model. Revised foundation loads from the structural model were then applied back into the geotechnical FE model. Although not presented in this example, in practice this iterative procedure betwe bet ween en the geo geotech technic nical al an and d st struc ructur tural al mo model delss un until til the fou found ndat ation ion loa loads ds and coefficients of subgrade reaction in both models are in approximate agreement provides an efficient means of taking account of superstructure stiffness effects on raft foundation behaviour. The core walls had a significant stiffening effect on the raft so these were included in the geotechnical FE analysis model, but rather than the full height of the core walls, they were included to one-storey height (3.5 m). The square shape of the raft foundation and inclusion of piles precluded the use of a 2D plane strain or axisymmetric assumption, so a 3D model was necessary. 200 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.

Examples

Figure 8.1 Raft foundation example geometry and loads All columns 1 m square, on 10 m grid except at core

1 MN 0 .0 m 2 MN l e ev v e el l

Flat ground level R O A D

2 MN

2 MN

1 MN 5 MN 3 m

D A O R

m 5

2 MN

1m

4 m 3 m

2 MN

m 5

5 MN

1m D A O R

6m 1 MN

2 MN 2 MN Plane of symmetry

2 MN 1 MN

Raft at ground level, 32 m square, 1.2 m-thick reinforced concrete

Core walls: 300 mm-thick reinforced concrete, 250 kN/m vertical line load on all walls

R O O AD

Figure 8.2 Plan of raft showing settlement-reducing pile locations

5m

5m 5

5m

8.5 m

6

5m Pile no. 1

2

3

4

2.5 m 2.5 m 5m

All piles 600 mm dia. Pile with toe level –15 m Pile with toe level –20 m



However, a plane of symmetry existed in the structure (see Figure 8.1) and the ground conditions were uniform, so only half of the geometry needed to be modelled. When continuum (volume) elements were used to represent the raft, the geometry of the raft in the model matched reality, i.e. the top surface of the raft was at ground level and the raft volume was embedded within the ground. However, this was not possible when using non-continuum (shell) elements. If the shell elements were placed at ground level, the raft would not benefit from the surroundin surrounding g overburden and unrealistic, local failures would result at the raft edges. Furthermore, the tops of the piles would be at the wrong level. If the shell elements were placed at formation level such that they were embedded in the ground, vertical slopes would be left in the ground around the raft that would collapse. To overcome these problems, the ground level in the model was set at formation level (−1.2 m) with a surcharge placed on the ground surface to represent the weight of the omitted ground. Such an assumption omitted the strength and stiffness of the ground above formation level, but was conservative.

Model boundaries and fixities (§1.2.3 and §1.2.4) Initially based on rules of thumb (Figure 1.7), the boundaries boundaries to the model were placed placed at 90 m (3B (3B ) from the raft. It was found that the vertical boundaries could be moved inwards to about 60 m (1.8B (1.8 B ) and 75 m (2.3B (2.3 B ) from the raft edge in the X- and Y-axis directions, respectively, as shown in in Figure 8.3(a), 8.3(a), without introducing any significant boundary effects on the key outputs (except on the plane of symmetry). The bottom boundary was fixed in all three axis directions and the vertical boundaries (including the plane of symmetry) were fixed only in the horizontal direction perpendicular to the boundary plane. When non-continuum (shell) elements were used for the raft, its rotation at the plane of symmetry perpendicular to the boundary was fixed in order to simulate the rotational restraint from the raft on the other side of the plane of symmetry excluded from the model.

Finite element mesh (§1.3) Second order 10-node tetrahedral elements with three degrees of freedom per node were used to model the ground and structures (when solid elements were used). When non When non-co -conti ntinuu nuum m elem element entss we were re use used d to re repr prese esent nt the st struc ructur tures, es, seco second nd ord order er 6-node triangular shell elements were used to represent the raft foundation and core walls. These elements had five degrees of freedom per node (axial displacement (1), transverse displacement (2) and rotation (2)). Since the raft was quite thick (1.2 m), it was probably around the limit of what could normally be modelled reasonably accurately with wit h she shell ll ele elemen ments, ts, hen hence ce wh why y bot both h con contin tinuu uum m and non non-co -conti ntinuu nuum m ele elemen ments ts we were re adopted in this case. The piles were modelled with special 3-node beam elements with six degrees of freedom per node (three translational and three rotational), as described in Section 5.1.4. Since the core walls were relatively thin (0.3 m thickness), these were modelled using shell elements in all the analyses and continuum elements were not used. Where shell elements 202 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.

Examples

Figure 8.3 (a) FE mesh with solid elements for raft; (b) Close-up of FE mesh with solid elements for raft 150 m

90 m

Side fixed in Y direction

No fixity on top surface

Side fixed in X direction 90 m Side fixed in Y direction

Side fixed in X direction

Z Y

(a)

Base fixed in X, Y and Z directions X

Predefined 1 m wide prism through full length of raft

(b)

Column locations with node at centre and elements defining 1 m2 area for distributed load

Shell elements on plane of symmetry had material properties of half-thickness wall (150 mm) and half line load (125 kN/m) Shell elements for core walls extended into solid raft elements to simulate fixed connections

Z Y X

were used for the raft, a fully fixed connection with the core walls was automatically formed. Where solid elements were used for the raft, a pinned connection would be formed with the core walls composed of shell elements. Since concentric, vertical loads were applied to the core walls, a pinned connection would probably be adequate in this case, but in order to be consistent with the shell–shell connection in the other case, the core wall shell elements were extended 0.5 m into the solid raft elements in order to simulate a fully fixed connection, as described in Section 5.1.7. 203 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.


Figure 8.3 (c) FE mesh with solid elements elements for raft and piles ( fro front nt elements hidden); hidden); (d) Close-up of FE mesh with shell elements for raft

(c) Surcharge applied to top surface around raft to represent weight of ground omitted from mesh

Column locations with node at centre and elements defining 1 m2 area for distributed load

Z Shell elements on plane of symmetry had material properties of half-thickness wall (150 mm) and half line load (125 kN/m) (d)

Shell elements at plane of symmetry fixed against rotation

Shell elements and top surface at –1.2 m level Y X

Interface elements were adopted between the piles and ground when solid elements were used use d for the pil piles, es, whi while le spe specia ciall int interf erfac acee ele elemen ments ts we were re use used d bet betwe ween en the pil piles es and ground when special beam elements were used. A perfectly rough interface (no interface elements) was assumed between the raft and the ground because no significant slippage or uplift was expected. The meshes were generated as shown in Figure 8.3 with smaller elements located where steep stress and strain gradients were expected. During simulations of pile load tests, it was found that outputs were particularly sensitive to mesh refinement when using either solid or special beam elements to represent the piles. Hence, the mesh was made particularly fine immediately around each pile, as shown (with the elements in front removed) in Figure 8.3(c). A further FE analysis was performed with a finer mesh and it was found 204 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.

Examples

that the critical outputs (raft settlement and pile load) were not affected significantly. Therefore, the adopted mesh was considered acceptable.

Initial stresses (§1.4.1) The initial stresses were specified with vertical effective stress based on a uniform ground density of 20 kN/m 3, horizontal ground surface and hydrostatic pore water pressure with a groundwater level at −1.2 m (coincident with formation level of raft) and zero pore pressure above this level. A K 0 value of 0.5 was derived by simulation of pressuremeter tests as described in Section 8.2.3.

Construction stages (§1.4.2) No installation effects were considered because the weight of the raft was approximately equal to the weight of the ground excavated during its construction while the effects of pilee ins pil instal talla latio tion n we were re tak taken en int into o ac accou count nt in the ba backck-an analy alysis sis of the pil pilee loa load d tes tests. ts. Drained soil conditions were assumed throughout.

1

2

3

4

Establish in situ Establish situ stres stresses ses As described earlier, with ground level at 0.0 m and groundwater level at −1.2 m with hydrostatic conditions. Lower Low er ground to formation formation level level (shell raft raft elements elements only) The ground level was lowered from 0.0 m to −1.2 m level and a uniform surcharge of 24 kPa applied at the same time to represent the weight of the removed ground. This stage prepared the geometry for those cases where shell elements were used to represent the raft. Install Inst all raft raft foundation foundation and core core walls walls In the case of solid elements, the constitutive model for the elements in the raft volume was changed from the soil model to the concrete model. This resulted in a moderate increase in weight due to the increased density (from 20 to 24 kN/m 3). In the case of shell elements, the shell elements were added to the top surface of the model. The surcharge representing the ground was removed from the raft area and the weight of the raft was included in the material properties of the raft. Alternatively, the surcharge could have been increased to 28.8 kPa and a zero density specified for the shell elements. The core walls were added as shell elements in all the analyses. They had zero specified weight because the weight of the core walls was included in the line loads. The displacements were reset to zero at the beginning of this stage so that outputs of displacement from this stage onwards were those due solely to the self-weight of the raft and piles (if any) and the applied loads. Install Inst all piles (if used) In the case of solid elements, the method was the same as for the raft. In the case of special beam elements, the beam elements were added to the model along the pile axes with no removal of ground elements. An additional weight of 1.13 kN per metre pile length was included to take account of the additional pile weight over and above the ground weight occupying the pile volume in the FE model. 205



5

Apply struc structural tural loads The loads shown in Figure 8.1 were applied concurrently in one step. This was considered a reasonable assumption because the multi-storey structure was to be constructed in complete floors such that the loading on the foundations would increase approximately uniformly. Were one side to be constructed before the other, consideration would have been given to different loadings during the construction phases. Since the core walls were represented by shell elements, the loading could only be applied as a line load (and not an area load), but this was considered a reasonable assumption since the core walls were only 0.3 m thick in reality. However, the columns were 1.0 m square and applying the column loads as point loads might have been unrealistic. In order to assess the validity of such an assumption, analyses were performed with point loads and area loads and the outputs compared. Each 1.0 m 2 area was represented by one side of four tetrahedral elements, so the area load was converted into 13 equivalent nodal loads – significantly more than a single nodal load – as described in Section 5.1.8.

Calculation options (§1.4.3) The Modified Newton–Raphson solution scheme was adopted with arc length control. Automatic step-sizing was utilised and the maximum equilibrium error was set at 1%. The small deformation (Total Lagrangian) formulation was adopted in all analysis stages.

8.2.3 8.2.3 Obtain Obt aining ing par parame ameter terss and and cons constit tituti utive ve mod model el fea featur tures es Constitutive model selection The soil at this site was a medium-dense, lightly over-consolidated sand. The important aspects of its behaviour in this case were hardening hardening behaviour under under deviatoric stress and compress comp ression ion and a nonnon-linea linearr str stress–s ess–strai train n resp response onse duri during ng str stress ess chan changes ges from low, near-surface stresses to high stresses under the loaded foundation. No significant anisotropy was expected in the soil. Therefore, a double-hardening model with hyperbolic stress–strain relationship and stress-dependent stiffness was selected. A linear elastic model was selected for the reinforced concrete structures because stresses in these elements were expected to be relatively low and within the linear range of stress– strain behaviour. No consideration of concrete cracking was considered necessary and the properties of the reinforcement were smeared across the section.

Obtaining parameters In situ rather than laboratory parameter testing was the only option due to the difficulty of ob obta tain inin ing g un undi dist stur urbe bed d sa samp mple less fr from om sa sand nd.. Th Thee se self lf-b -bor orin ing g pr pres essu surrem emet eter er (a (ass describ des cribed ed in Sec Sectio tion n 3.3 3.3.4) .4) pr prov ovide ided d the mos mostt ac accur curat atee mea means ns of ob obtai tainin ning g sti stiffn ffness ess parameters, and it was also used to obtain shear strength parameters. A sample pressuremeter plot from the sand at 7 m depth is shown in Figure 8.4 from which wh ich she shear ar st stren rength gth and st stiffn iffness ess par param ameter eterss we were re der deriv ived. ed. Th Thee she shear ar st stre rengt ngth h wa wass derived from both the loading and unloading portions of the pressuremeter curve. The loading portion of the curve was replotted on natural logarithm scales, as shown in 206 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.

Examples

Figure 8.4 Sample pressuremeter test data 500

400 Unloading

Loading Unload–reload loop

a P 300 k : ′ p

e r u s s e r 200 P

100

0 0.00

Test data FEA

0.02

0.04

0.06

0.08

0.10

0.12

0.14

Cavity strain εc

Figure 8.5. 8.5. The curve reached a constant maximum slope S = 0.415 which was used to ′ derive values of peak internal friction angle w peak and dilatancy angle c angle c according according to the Hughes et Hughes et al . (1977) method (1977) method and with Rowe’s stress dilatancy law. w ′peak = 38 and c = 5 were obtained from Equations 8.1 and 8.2, respectively. 8

8

Figure 8.5 Derivation of shear strength from loading portion 8

6 S

a P k : 4 ′

1

p

n l

2

0 –8

–7

–6

–5 ln εc

–4

–3

–2



A constant volume value w ′cv for the sand was required in these equations which was ′ c was = 34 ). obtained from direct shear tests in a laboratory (w ( w cv ). c was later set to zero in the FE analysis of the raft foundation due to the risk of over-predicting soil strength in confined problems such as piled foundations. 8

sin w ′ =

S 1 + (S − 1) sin w ′cv

(8.1)

c = S + (S − 1) sin w sin c sin sin w ′cv

(8.2)

The unloading portion of the curve (ignoring the last horizontal leg of the curve) was plotted as ln p ln p ′ against the natural log of the difference between the current strain and maximum maxi mum str strain ain (−ln(1max − 1c )) )),, as shown shown in Figure Figure 8.6 8.6.. Th Thee plo plott app appro roac ached hed an asymptote with slope S d of 2.60 and a w ′peak of 35 was obtained from Equation 8.3 according to Houlsby to Houlsby et et al . (1986). (1986). This is comparable with the value obtained from the loading portion. The value can be refined during simulation of the pressuremeter test as described later. 8

′

sin w =



′

1 + sin w cv sin w cv + S d ′

     1 + sin w ′cv sin w cv + S d ′

−

2

−1

(8.3)

The elastic ground stiffness can be determined from unload–reload loops in the pressuremeter test and one of these is shown in close-up in Figure Figure 8.7 8.7.. Th Thee av aver erage age she shear ar modulus G of this loop is calculated simply by drawing a chord through the apexes of the loop, the slope of which equals 2G 2 G. In this case, G case, G = 56 MPa was obtained which was Figure Figur e 8.6 Derivation of shear strength from unloading portion 8

6

a P k : 4 ′

S d

p

n l

1 2

0

2

3

4

5

6

7

8

–ln (ε (εmax – εc)


Examples

Figure 8.7 Unload–reload loop and derivation of elastic shear modulus

G

550 500 a P k : 450 ′ p

e r u s s 400 e r P

2G

350 1 300 0.0960

0.0965

0.0970

0.0975

0.0980

Cavity strain εc

appropr appr opria iate te for the st stra rain in lev level el of the unl unload oad–r –relo eload ad loo loop p (ap (appr proxi oxima mately tely 0.0 0.04%) 4%) whereas slightly higher strain levels were expected in the raft analysis. The full straindependency of stiffness was determined from the unload–reload loop, as demonstrated in the following paragraph. To determine the strain-dependency of stiffness, the reloading portion of the loop was plotted on natural log axes of p′ against shear strain g according according to Equation 8.4, with local values (i.e. p′ and 1c reset to zero at the start of the reload portion) as shown in Figure 8.8. 8.8. A straight line was obtained with a slope b of of 0.576. The intercept of this line at ln local dA/A = 0 was h = 8.76, or 10.8 MPa when the log was reversed. g = area ratio

dA

A

=1−

1

  1 + 1c

2

(8.4)

The secant and tangent pressuremeter shear moduli were calculated from Equations 8.5 and 8.6, respectively, as described by Whittle by Whittle (1999). (1999). The plots of both moduli are shown in Figure in Figure 8.9. 8.9. Practically, the minimum strain for elastic stiffness from a pressuremeter is about g = 0.01% depending on the resolution of the instrument and the maximum is about g = 1% depending on the soil because plastic strains begin to develop. The stiffness at smaller strains could be obtained by seismic testing (see Section 3.3.5). For an estimated average strain level beneath the raft of 0.2%, a pressuremeter tangent elastic shear modulus G modulus G pt of 44 MPa was obtained. Muir Wood (1990) showed (1990) showed that (in clay) the p tangent modulus (G ( G t ) obtained in the pressuremeter test is equal to the secant modulus G s obtained from laboratory testing. For this test in sand, G s was assumed approximately equivalent to G pt and then the value adjusted as necessary during pressuremeter test simulations. G ps = hbg b − 1

(8.5)

G pt =hb 2g b − 1

(8.6) 209



Figure Figur e 8.8 Derivation of strain-dependent elastic shear modulus

G

5

a P k : ′

β

p

l 4 a c o l n l

1 Line intercept η intercept η

3 –11

–10

–9

–8

–7

ln local εc

The same strength and stiffness parameters were derived from the other pressuremeter tests (not presented here) in order to select appropriate characteristic values for input into the model. In order to account for the stress-dependency of stiffness, the effective stress at the start of each unload–reload loop was estimated using Equation 3.3. In the ′ unload–reload loop analysed in Figure 8.7, p′u was 477 kPa and s ′v0 and s h0 were esti′ mated to be 100 and 50 kPa, respectively, giving s h = 135 kPa. The stress-dependency ′ of stiffness was determined in terms of the minor principal effective stress s v0 which in this case was 100 kPa because the horizontal stress had become the major principal stress. Figure Figur e 8.9 Deca Decayy of elast elastic ic stiffness with strain 300

200 a P M :

Secant pressuremeter shear modulus Gsp

G

100

Tangent pressuremeter shear modulus Gtp 0 0.01

0 .1 Shear strain γ strain γ:: %

1


Examples

Figure 8.10 Stress-dependency of elastic unload–reload stiffness 150

100

a P M : r u

E

50

Model Test data

0 0

50

100

150

200

250

σ′3: kPa

The elastic unload–reload Young’s modulus E ur ur was calculated from the standard elastic relationships provided in Appendix 3.1 (assuming a Poisson’s ratio of 0.2) from all the pressuremeter tests and plotted against the corresponding s ′3 values in Figure in Figure 8.10. 8.10. The stress-dependency was expressed as an exponential function (Equation 8.7) as used by Duncan and Chang (1970) and in some other constitutive models. E ur = K ur pa

s ′3 pa

n



(8.7)

where pa is atmospheric pressure (100 kPa). A close fit with the data was achieved with K ur ur = 900 and n = 0.5. From this relationship it was possible to estimate similar parameters for the primary loading stiffness as described in the next paragraph. These parameters were then checked and revised in test simulations as described later. In order to account for the non-linear behaviour of the sand on primary loading, the Kondner Kon dner (196 (1963) 3) hyperb hyperboli olicc str stress ess–s –str train ain rel relat ation ionshi ship p as des describ cribed ed by Dun Duncan can an and d Chang Ch ang (1970) (1970) wa wass ado adopte pted d as sho shown wn in Figure Figure 8.11 8.11.. Th Thee pri primar mary y loa loadin ding g st stiffn iffness ess can be expressed as the initial tangent modulus E i or the secant modulus E 50 50 at 50% of the deviatoric stress at failure (q (qf ). In many soils, E i is about 0.6E 0.6 E ur and E E 50 ur and 50 is about one-third onethird of E ur Ther erefo efore, re, usi using ng sim simila ilarr equ equat ation ionss to Equ Equat ation ion 8.7 for the st stres resssur . Th dependency of stiffness, as shown in Equations 8.8 and 8.9, K i = 545 or K or K 50 50 = 300 were adopted while n was kept at 0.5 because this tends to remain the same for the different stiffness moduli. E i = K i pa

s ′3 pa

n

   

E 50 = K 50 pa

s ′3 pa

(8.8)

n

(8.9)

Rf is the ratio between q between qf and the asymptote to the hyperbolic curve which was estimated as 0.9 as typically used for most soils. 211 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.


Figure 8.11 Hyperbolic stress–strain relationship (Kondner, 1963) q qa qf

Asymptote Failure E 50

E i

1 1 0.5qf E ur

1

εa

There are uncertainties inherent in all the parameter derivations from the pressuremeter described in the preceding paragraphs. The pressuremeter test is well suited to simulation due to its axisymmetry and the full pressuremeter test was simulated (using the method described in Figure 3.12) in a 2D axisymmetric FE analysis. The w ′ value was adjusted to 36 in order to improve the match between the test simulation outputs and the test data as shown in Figure 8.4. c 8.4. c was was kept at 5 during this test simulation but taken as zero in the FE analysis of the raft foundation and piles due to the likelihood of over-predicting soil strength when c . 0 in confined problems such as piled foundations. The initial state parameters, which were difficult to determine directly from the test data, were also estimated and adjusted until a good match with the test data was achieved. It was found that a K 0 value of 0.5 was appropriate and the pre-consolidation stress (defining the location of the cap yield surface for compression hardening) was assumed equal to the in situ stress, i.e. normally consolidated conditions. 8

8

8

′ The peak shear strength w peak was adopted in order to obtain more realistic predictions of deform def ormat ation ion.. Ho Howe weve ver, r, if ult ultima imate te lim limit it st stat ates es nee needed ded to be pr predi edicted cted,, it wou would ld be appropriate to adopt a lower, post-peak shear strength. Furthermore, vertical pressuremeters provide provide str strength ength and stiff stiffness ness par paramete ameters rs in the horiz horizonta ontall dire direction ction whereas stre st ress ss ch chan ange gess in th thee raf aftt pr prob oble lem m wer eree or orie ient nta ate ted d pr pred edom omin inan antl tly y in th thee ve vert rtic ical al direction. Nevertheless, the degree of anisotropy in the sand was judged to be low, so the pr pressu essurem remete eter-d r-deri erive ved d par parame ameters ters we were re con consid sider ered ed app appro ropri priat atee for thi thiss ra raft ft problem.

A You Young’s ng’s modu modulus lus E GPa a wa wass ad adop opte ted d fo forr th thee re rein info forc rced ed co conc ncre rete te wi with th a E of 30 GP 3 Poisson’s ratio of 0.15. The weight density of the concrete was taken as 24 kN/m . 212 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.

Examples

Figure 8.12 Back-analysis of pile load test 350 300 N k : d a o l l a i x a d e i l p p A

250 200 Modified beam 150

Solid elements

100

Test data

50 0 0

1 2 3 4 Pile head settlement: mm

5

With the soil and concrete parameters set, the only simple way to parameterise ground– pile interaction behaviour was in the properties of the interface between them. Interface elements with a Coulomb friction criterion were installed between the pile and soil solid elements. The friction properties of the interface were determined by back-analysis of load tests on similar pile types and sizes to those proposed to be included in the raft analysis model. The output from one such back-analysis on a 0.6 m diameter, 10 m long pile is shown with the test data in Figure 8.12. 8.12. A reasonable match with the test data was achieved with a shear strength at the interface of 80% of the internal shear strength of the sand. In the case of the special beam elements for the piles, the variation of shaft friction along the pile and the end bearing were specified separately. This was slightly more complicated than with the solid elements but a reasonable match with the test data was also achieved as shown in Figure 8.12.

8.2.4

Outputs

The outputs of vertical displacement in the raft at the end of the final construction stage are presented in Figure in Figure 8.13 for the cases with and without piles and with either continuum or non-continuum structural elements (a positive value denotes settlement). Comparing the outputs without piles, it is apparent that when shell elements were used for the raft, about 10–15% higher settlement was predicted and the maximum deflected slope was higher. This was possibly due to the additional support provided to the solid elements by the interaction with the ground at the sides and by the moment-reducing effect of interface friction (see Section 5.1.2). As expected, the inclusion of piles resulted in smaller settlements with a significant reduction in the deflected slope (or differential settlement) which is the main function of settlement-reducing piles. The piles exhibited a stiffer response as solid elements than they did as special beam elements, in spite of the parameters for both being obtained from the same pile load test. The outputs of bearing pressure (vertical total stress at formation level) and pile load in the last construction stage are shown in Figure 8.14 and Table 8.1, 8.1, respectively (pile 213 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.


Figure 8.13 Outpu Outputt of vertical displacemen displacementt in raft in final constructio construction n stag stage: e: (a) raft only, solid elements; (b) raft with piles, solid elements 16 19 22 25 28 31 34 37 40 mm

(a) 16 19 22 25 28 31 34 37 40 mm

(b)

numbers are shown in Figure 8.2). While the bearing pressure plots without piles are similar, except for more stress concentrations at the edges of the raft composed of shell elements, the plots with piles show clearly that a higher bearing pressure was predicted with the shell elements compared with the solid elements, with a greater share of the load being taken by the piles when using solid elements, as was also evident in the pile loads (12% of the total foundation load taken by the piles with solid elements compared with 8% with shell and special beam elements). In this respect, the example illustrates well the effect of decisions taken at the analysis planning stage on the outputs obtained at the end. Obtaining outputs of bending moment, axial force and other structural forces from the solid elements is less straightforward than from the shell elements (see Table 5.2). The axial forces at the tops of the piles shown in Table 8.1 were obtained by integrating the vertical normal stress over the cross-sectional area of each pile. In this case, the area 214 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.

Examples

Figure 8.13 (c) raft only, shell elements; (d) raft with piles, shell and special beam elements 16 19 22 25 28 31 34 37 40 mm

(c) 16 19 22 25 28 31 34 37 40 mm

(d)

to be integrated was well defined, but in the case of the raft, rather than integrate over the entire cross-sectional area (which would be of limited use in design), sections needed to be pre-defined in the mesh. As shown in Figure 8.3(b), a 1.0 m wide prism was predefined in the mesh running in the X-axis direction, such that bending moment per metre width could be obtained for the raft for any section along that prism. The bending moment was obtained by integrating the normal stresses at integration points multiplied by corresponding lever arms about the neutral axis for sections along the prism. Some progra pro grams ms perfo perform rm these calcu calculat lations ions auto automa maticall tically y which signi significan ficantly tly simpl simplifies ifies the process. The resulting bending moment is shown in Figure 8.15 along 8.15 along with the direct output from the equivalent shell element analysis along the same section line (a positive value denotes sagging moment). The plots were very similar, except the shell element output was more spiky at the column locations (X = 61, 71 and 91 m) while the solid element output was more spiky at the connections with the core walls (X = 76 to 86 m) 215 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.


Figure 8.14 Outpu Outputt of bearing pressure pressure at formation formation level beneath raft: (a) ra raft ft only, solid elements; (b) raft with piles, solid elements

40

65

90 (a)

115

140

165

190 kPa

(b)

where the shell elements were extended into the solid elements. The addition of settlementreducing piles appeared to cause a greater reduction in bending moment in the core wall area in the solid element case compared with the shell element case. The bending moment outputs presented in Figure 8.15 were from analyses where the column loads were distributed over the 1 m 2 area of the columns. Analyses were also performed with point loads to represent the column loads and the bending moment outputs from each case are compared in Figure in Figure 8.16. 8.16. Clearly, the output in between the column locat loc ation ionss wa wass un unaf affect fected ed by the loa loadin ding g ass assum umpti ption on but wa wass sig signifi nifican cantly tly af affec fected ted immediately around each column location. With solid raft elements, the local maxima unde un derr po poin intt lo load adss wer eree ab abou outt 50 50% % hi high gher er at th thee sm smal alll co colu lumn mn lo load adss (X = 61 an and d 91 m) an and d ab abou outt 10 10% % hi high gher er at th thee la larg rgee co colu lumn mn lo load ad (X = 71 m). Wi With th she shell ll ra raft ft elements the effect was even greater at about 80% and 30% higher at the small and large loads, respectively. Therefore, in this example, the use of point loads resulted in an 216 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.

Examples

Figure 8.14 (c) raft only, shell elements; (d) raft with piles, shell and special beam elements

40

65

90 (c)

115

140

165

190 kPa

(d)

Outputt of axial load taken by settle settlement-r ment-reduci educing ng piles Table 8.1 Outpu Axial load at top of pile: kN Pile no.

Solid elements

1 2 3 4 5 6

686 1117 1132 745 681 686

Total

5047 (12%)

Special beam elements 430 635 717 457 482 482 3203 (8%)



Figure 8.15 Outputs of bending moment in raft along section on X-axis; (a) raft only; (b) raft with piles 2000

m 1500 / m N k : 1000 t n e m o m 500 g n i d n e B 0

60

Shell elements Solid elements

70

X: m

80

90

–500 (a)

2000


60

Shell elements Solid elements

70

X: m

80

90

–500 (b)

unrealistic over-prediction of local bending moment at the column locations, particularly when using shell elements for the raft. A key advantage of using non-continuum elements for structures is the direct output of structural forces. The distribution of bending moment across the entire raft could be plotted straightforwardly, as shown in Figure 8.17 for distributed column loads. M x denotes bending orientated along the X-axis for rotation about the Y-axis while M y deno de note tess be bend ndin ing g or orie ient ntat ated ed al alon ong g th thee YY-ax axis is fo forr rot ota ati tion on ab abou outt th thee XX-ax axis is an and d a positi po sitive ve val value ue den denote otess sag saggin ging g mo momen ment. t. It wa wass ap appar parent ent tha thatt the inc inclus lusion ion of pil piles es redu re duce ced d th thee be bend ndin ing g mo mome ment nt ge gene nera rall lly y in th thee ce cent ntre re of th thee ra raft ft wh wher eree th they ey wer eree located. 218 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.

Examples

Figure 8.16 Outputs of bending moment in raft along section on X-axis with point and distributed loads: (a) raft only, shell elements; (b) raft only, solid elements 2000


60

Point loads Area loads

70

X: m

80

90

–500 (a)

2000


60

Point loads Area loads

70

X: m

80

90

–500 (b)

The outputs of vertical displacement and bearing pressure shown in Figures 8.13 and 8.14 were processed to derive coefficients of subgrade reaction k which could be used in other stru structur ctural al anal analysi ysiss mode models ls wher wheree gro ground– und–stru structur cturee inter interact action ion effects wer weree simulated approximately by spring elements. k is simply the bearing pressure divided by the settlement, but an element of approximation is introduced into the calculation because displacement output is provided at the nodes while stress output is provided at integration points and they are not coincident. Consequently, direct output is not normally available in an FE analysis program and the user needs to process the data. In this example, the raft was divided into a grid 1 m wide at the edge and 3 m square internally as shown in Figure 8.18. 8.18. By inspecting the output data, approximate average values of settlement and bearing pressure for each square were obtained. From these, k 219 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.


Figure 8.17 Bending moment output from shell elements: (a)

M x ,

raft only; (b)

M x , raft with piles

2000

1600

1200

(a)

800

400

0

–400 kNm/m

(b)

was determined as shown in Figure 8.18. The values obtained were quite consistent acro ac ross ss ea each ch ra raft, ft, inc increa reasin sing g gen gener erall ally y to towa ward rd the edg edges es wh wher eree set settle tlemen mentt wa wass lo lowe wer. r. Higher values were obtained when solid elements were used for the raft because lower settlements were predicted in these cases. In the cases with piles, spring stiffness values for each pile are also shown which were obtained simply by dividing the pile axial force (Table 8.1) by the vertical displacement (Figure 8.13). Springs representing the settlement-reducing piles would need to be added to any beam-spring type model in addition to the springs representing the ground mass.

8.2. 8. 2.5 5

Val alid ida ati tio on

In order to validate the critical outputs, they were compared with the results of alternativ na tivee an analy alysis sis met metho hods. ds. Whe When n com compar paring ing dif differ ferent ent ana analy lysis sis met method hods, s, man many y of the assumptions inherent in each are different so an exact match between outputs should not be expected. The goal was to achieve outputs that were reasonably close in order to have greater confidence that the FE analysis model represented reality sufficiently accurately. 220 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.

Examples

Figure 8.17 (c)

M y,

raft only; (d)

M y,

raft with piles

2000

1600

1200

(c)

800

400

0

–400 kNm/m

(d)

First, the one-dimensional settlement under the centre of the raft was calculated assuming a uniformly distributed load over the raft area. The ground beneath the raft was divided into a number of layers with a one-dimensional stiffness E ′0 assigned to each layer and the Boussinesq solution was used to calculate stress changes at the centre of each layer. The compression of each layer was summed in order to obtain a settlement value. The total applied load (including self-weight of the raft) distributed uniformly over the raft area gave a surcharge q surcharge q = 60.5 kPa. To take account of the increasing stress and decreasing strain with depth, the E 0′ value was assumed to increase from 20 MPa near the surface to 140 MPa at 64 m (2B (2 B ) depth which was assumed to be the depth of influence of the foundation. From these values a maximum settlement of 44 mm was obtained. This compared very favourably with the values of 35 mm and 39 mm obtained in the FE analysis with solid and shell elements, respectively, representing the raft. Since the Boussinesq type solution would be expected to be conservative, obtaining somewhat lower values in the FE analysis was considered reasonable. 221 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.


Figure 8.18 Coefficients of subgrade reaction reaction and pile spring stiffness values derived from outputs: (a) raft only, solid elements; (b) raft only, shell elements 3.3

3 .1

2 .9

3.1

3.2

2 .9

2 .9

3 .1

2.9

2 .9

3.1

2.9

2.9

3.1

2.9

2.7

2.8

2.8

2.7

2.8

2.8

2.8

3.0

3.2

2.8

2.9

2.7

2.6

2.6

2.6

2.5

2.5

2.6

2.7

2.8

3.0

3.0

2.8

2.6

2.6

2.5

2.5

2.5

2.5

2.5

2.6

2.7

3.0

2.8

2.8

2.6

2.5

2.5

2.4

2.4

2.4

2.5

2.5

2.6

2.9

2.9

2.8

2.6

2.4

2.4

2.4

2.4

2.4

2.4

2.4

2.6

2.9

(a) 4.1

3 .7

3 .4

3.3

3 .1

3 .2

3 .2

3 .1

3 .2

3 .5

3 .7

4.3

3.6

2.8

2.6

2.4

2.4

2.4

2.4

2.4

2.5

2.6

2.9

3.8

3.4

2.6

2.3

2.3

2.3

2.2

2.2

2.3

2.3

2.6

3.4

3.4

2.5

2.2

2.2

2.2

2.2

2.1

2.2

2.2

2.3

2.5

3.3

3.3

2.5

2.2

2.2

2.1

2.2

2.2

2.2

2.2

2.3

2.5

3.0

3.3

2.5

2.3

2.1

2.2

2.2

2.2

2.1

2.2

2.2

2.4

2.9

2.2

(b)

Y

Coefficients of subgrade reaction (MN/m3)

X

In order to take account of the raft stiffness, a second elastic analysis solution was studied. Results from a uniformly loaded square raft in frictionless contact with a homogeneous isotropic half-space were used, as derived by Fraser and Wardle (1976) and described in Hemsley in Hemsley (1998). (1998). For the case with the settlement-reducing piles, there was no readily available, simple, alternative analysis method. While piled raft analysis methods exist, they tend to assume piles are distributed across the whole area of the raft, rather than just in the central area as in this case. Therefore, the same elastic raft analysis method was used but with a crude 222 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.

Examples

Figure 8.18 (c) raft with piles, solid elements; (d) raft with piles, shell and special beam elements 3.3

3.2

3.0

3 .1

3.1

3.0

3.1

3.1

3 .0

2 .9

3 .2

3.2

3.2

3.0

2.8

2.7

2.7

2.5

2.6

2.6

2.7

2.8

3.1

3.2

3.0

2.9

2.7

2.8

2.5

2.4

2.4

2.5

2.6

2.6

2.8

3.0

2.9

2.8

2.6

2.4

2.4

2.4

2.3

2.4

2.5

2.5

2.6

3.0

2.4

2.4

2.4

2.5

2.8

26

2.8

2.6

2.5

2.5

2.4

2.4

2.3 42

2.3

42 2.3

27

2.8

2.6

2.4

2.4

26

27

2.2

2.3

2.3

2.3

2.5

2.7

(c)

3.9

3.5

3.2

3 .2

3.0

3.0

3.0

3.0

3 .1

3 .2

3 .4

3.8

3.4

2.8

2.6

2.5

2.3

2.4

2.4

2.4

2.4

2.5

2.9

3.5

3.2

2.6

2.4

2.2

2.2

2.1

2.1

2.2

2.2

2.2

2.6

3.2

3.0

2.5

2.2

2.2

2.1

2.0

2.0

2.1

2.1

2.2

2.6

3.2

2.0

2.2

2.3

2.5

2.9

2.3

2.5

2.8

15

2.9

2.4

2.2

2.1

2.11

2.2

2.1 21

2.0

19 2.0

14

2.9

2.4

2.1

2.0

15

2.0

(d)

14

2.0

2.1 Y

Coefficients of subgrade reaction (MN/m3) Pile spring stiffness (kN/mm) X

average stiffness of the soil and piles taken for the half-space (the 12 piles occupied 0.08% of the ground volume under the raft to a depth of 64 m (2 B )). The ra raft ft dimen dimensions sions wer weree B = L = 32 m, the thickness t1 = 1.2 m, You Young’s ng’s modulus modulus E 1 = 30 000 MP MPa a and Po Poisso isson’s n’s ra ratio tio n1 = 0. 0.15 15.. Th Thee un unifo iform rm su surc rcha harg rgee was q = 60.5 kPa. For the half-space, a single Young’s modulus needed to be selected taking into account the stress state and strain level in the soil. This value was judged to be E 2 = 30 MPa. 223 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.


Compariso parison n betw between een elastic analysis analysis and FE analy analysis sis outputs for ra raft ft (without piles) Table 8.2 Com Dw AB :

Elastic analysis

mm

Dw AC:

mm

∗

M

: kNm/m

12

23

620

FE analysis (solid structural elements)

9

17

500 (away from columns)

FE analysis (non-continuum structural elements)

10

18


In the case with piles, 0.08% of the soil was assumed to be composed of concrete which raised the E 2 value to 54 MPa. The Poisson’s ratio in both cases was taken as n2 = 0.2. These values gave relative raft stiffness values of K r = 0.052 in the raft only case and 0.029 0.0 29 for the ra raft ft wit with h pil piles. es. The ela elast stic ic ana analy lysis sis sol soluti utions ons pr prov ovide ided d ou outpu tputs ts of the differential vertical displacement between the centre and mid-side (D ( DwAB ) and between the centre and corner (D (DwAC) as well as the maximum bending moment M ∗ . The results from fr om thi thiss ela elasti sticc ana analy lysis sis ar aree com compar pared ed wit with h the FE an analy alysis sis ou outpu tputs ts in Tables Tables 8.2 and 8.3 8.3.. As before, the FE analysis outputs compared very favourably with the elastic analysis results. The values from the FE analyses were slightly lower, which was expected. The bending moment outputs shown were the maximum values away from the column loads because the elastic analysis assumed a uniformly distributed load so the values at the column loads would not be expected to be close to the values obtained from the elastic analysis. The comparisons between the outputs of the FE analyses and the two elastic analyses showed a reasonably close match and gave more confidence in the accuracy of the FE analysis model. Further techniques of validation should be undertaken, as described in Section 7.2.2, in order to improve confidence in the outputs further.

Table 8.3 Com Compariso parison n betw between een elastic analysis analysis and FE analy analysis sis outputs for ra raft ft with settl settlementementreducing piles Dw AB :

mm

Dw AC:

mm

∗

M

: kNm/m

Elastic analysis

8

15

415

FE analysis (solid structural elements)

5

8


FE analysis (non-continuum structural elements)

6

11



Examples

8.3. 8.3. 8.3.1

Shaft exc Sha xca ava vati tion on exa xamp mple le Summary

This example concerns the excavation of a shaft in firm over-consolidated clay supported by a sheet pile wall. The sheet pile wall will be supported by hoop forces in a capping beam and by passive resistance from the ground inside the shaft. The groundwater level in the shaft will be permanently lowered. A buried sewer that is sensitive to deflection exists near to the shaft, requiring an accurate prediction of excavation-induced ground movement. The particular features of this example include: g g g g

g g g g g

2D axisymmetric model K 0 initial stress procedure double-hardening, double-hardenin g, stress- and strain-dep strain-dependent endent stiffness constitutive constitutive model obtaining drained drained geotechnical parameters parameters from advanced, advanced, undrained laborato laboratory ry triaxial tests groundwater ground water flow flow analysis analysis and hydrau hydraulic lic failure undrained undrain ed ‘A’ conditions followed followed by consolidation consolidation and drained conditions conditions structural non-continuum elements with anisotropic anisotropic stiffness properties properties dual factoring factoring ULS check with stepwise stepwise strength strength reduction reduction validation validatio n using monitoring data data from a similar shaft shaft excavation. excavation.

8.3.2 8.3. 2 Setti Se tting ng up th the e FE an anal alys ysis is mo mode dell Justification for using FE analysis (§1.1.1) The alternative analysis methods, such as beam-spring models, do not consider 2D axisymmetric conditions (e.g. the contribution of hoop forces and orthotropic structures) or else consider them in an approximate way. In particular, the requirement to predict excavation-induced ground movement around the shaft was justification for using FE analysis. While empirical methods of predicting ground surface settlement behind retaining walls exist, these tend to be for straight walls rather than shafts. Furthermore, by using an advanced constitutive model, there was scope for obtaining a more economical design.

Aims of the model (§1.1.2) 1 2 3

to predict predict the ground movem movement ent at the location location of a buri buried ed sewer sewer near the proposed shaft to perform perform a ULS design to select the appropria appropriate te sheet pile section and and embedment depth to predict groundwa groundwater ter head due to permanent groundwa groundwater ter lowering and check check for hydraulic failure.

Geometrical Geom etrical simpl simplifica ifications tions (§1.2.1 and §1.2.2) This site had no significant structural features except for the proposed shaft and the sewer nearby, as shown in Figure 8.19. 8.19. Including the sewer in the model would require 225 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.


a ful fulll 3D geo geomet metry ry.. Ho Howe weve ver, r, by ign ignori oring ng the con contri tribu butio tion n of the sew sewer er to gr grou ound nd stiffn st iffness ess (wh (which ich wa wass con conser serva vativ tive), e), it wa wass po possib ssible le to ob obtai tain n ap appr proxi oxima mate te gr grou ound nd deflections along the line of the sewer using a 2D axisymmetric model, as described in Section 8.3.4. Furthermore Furtherm ore,, the gro ground und cond condition itionss wer weree quite uniform and with no incli inclined ned str strata ata which also allowed a 2D axisymmetric geometrical assumption to be adopted. Although a concrete base to the shaft and some infrastructure would be constructed later, no input to the design of these was required from this FE analysis. Therefore, for conservatism in the shaft wall design, these structures and their associated loads were omitted from the model.

Model boundaries and fixities (§1.2.3 and §1.2.4) Initially based on rules of thumb (Figure 1.7), the remote vertical boundary to the model was placed 45 m (3B (3 B ) from the sheet pile wall and the bottom boundary 30 m (2B (2 B ) below the excavation level. It was found that the remote vertical boundary could be moved inwards to about 43 m (2.8B (2.8 B ) from the sheet pile wall and the bottom boundary to 25 m (1.7B (1.7B ) below the excavation level, as shown in Figure 8.20, 8.20, without introducing any significant boundary effects on the key outputs (except on the plane of symmetry). These locations were governed by the groundwater flow analysis rather than the displacement analysis. The bottom boundary was fixed in both axis directions and the vertical boundaries (including the axis of symmetry) were fixed only in the horizontal direction. The hydraulic boundary conditions are described under Construction stages. stages.

Figure 8.19 Shaft excavation example geometry and groundwater levels 1 m square rc capping beam

Water levels: Worst case 0 m Characteristic –1 m

ground level 0 m

3m

Sewer –1 m

Excavation level –5 m Lowered to –6 m inside shaft

Wall toe level –10 m

Sheet pile wall

Soil: firm over-consolidated clay 15 m diameter


Examples

Figure 8.20 FE mesh for shaft excavation example Axis of symmetry

10 m

7.5 m

0m

–5 m

–10 m

S i d e f i x e d i n

n o i t c e r i d X n i d e x i f e d i S

X d i r e c t i o n

Base fixed in X and Y directions

–30 m

50 m Y

X

Finite element mesh (§1.3) Cubic strain 15-node triangular elements with three degrees of freedom per node were used us ed to mo mode dell th thee gr grou ound nd.. Th Thee us usee of su such ch hi high gher er or orde derr el elem emen ents ts is pa part rtic icul ular arly ly important for the prediction of failure states in axisymmetric models as described in Section 1.3.1. The sheet pile wall and capping beam were modelled with 5-node line elements forming 2D shell elements with three degrees of freedom (two translational and one rotational) per node. The elem element entss we were re loc locat ated ed at the centrelin centrelinee ra radiu diuss of the shaft shaft wa wall. ll. Th Thee benefits of using continuum elements (see Section 5.1.2) were considered minor for a thin sheet pile wall compared with the benefits of their easier use. Interface elements were used between the shell elements and the area elements for the ground to take account of the reduced friction between the wall and ground as well as to create a closed hydraulic boundary condition for the groundwater flow analysis. The mesh was generated as shown in Figure 8.20 with smaller elements created near the wall where steep stress and strain gradients were expected. 227 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.


A further FE analysis was performed with a finer mesh and it was found that the critical outputs outp uts wer weree not aff affected ected sign significan ificantly. tly. Therefore, Therefore, the ado adopted pted mesh was consi considere dered d acceptable.

Initial stresses (§1.4.1) The initial stresses were specified with vertical effective stress based on a uniform ground density of 20 kN/m 3 and a horizontal ground surface. Pore water pressures were assumed hydrostatic below a groundwater level at 1 m depth and zero above this level. A K 0 value of 1.0 (o (obta btaine ined d as des descri cribed bed in Sec Sectio tion n 8. 8.3.3 3.3)) was use used d to cal calcul cula ate ho horiz rizon ontal tal eff effect ectiv ivee st stre ress. ss.

Construction stages (§1.4.2) The installation effects on the clay soil associated with the proposed sheet pile wall were expected to be minor so were not considered in the FE analysis. Undrained ‘A’ conditions (see Section 4.2.4) were simulated during construction followed by dissipation of excess pore pressures in a consolidation analysis to simulate long-term conditions. Drained conditions were assumed in the subsequent ULS stages since the drained long-term condition was known to be the most onerous case. The importance of following the correct stress path (i.e. undrained construction then consolidation to long-term drained conditions) rather than analysing only drained conditions was highlighted in Section 4.2.2. The sequencing of the construction stages is illustrated in Figure in Figure 8.21, while 8.21, while each stage is described in detail as follows:

1

2

3

4

5

Establish in situ Establish situ stres stresses ses As described earlier, with ground level at 0.0 m and groundwater level at −1.0 m with hydrostatic conditions. Installl sheet pile wall Instal wall and capping capping beam (undrained) (undrained) The shell elements of the sheet piled wall and capping beam as well as the interface elements were activated. Excavate shaft (undrained) The ground elements and attached interface elements within the shaft to 5.0 m depth were deactivated. The steady-state pore water pressure remained unchanged but no external water pressures were applied within the shaft (in effect, the groundwater within the shaft was removed with the soil). Apply surch surcharge arge (undra (undrained) ined) A uniformly distributed load of 10 kPa was applied at the ground surface on a 10 m wide strip immediately behind the wall to represent construction traffic. Long-term, Long-te rm, drained conditions conditions (groundwater (groundwater flow then consolidation consolidation analysis) analysis) It was proposed to install permanent dewatering measures in the shaft to lower the groundwater level to 1.0 m below the base of the shaft. Therefore, a groundwater flow analysis was performed in order to establish the new steadystate pore pressures with the hydraulic boundary conditions shown in Figure 8.22 (with the characteristic water level of −1 m on the right-hand boundary). Then a consolidation analysis was performed until the excess pore pressures


Examples

Figure 8.21 Sequence of analysis stages 1. Establish in situ stresses

2. Install sheet pile wall

3. Excavate shaft

4. Apply surcharge

5. Long-term conditions ULS output factoring

ULS input factoring

6. Factor surcharge and over-dig

7. Higher groundwater level, factor surcharge and over-dig

Factor structural force outputs

8. Ground strength reduction by factor required by code

9. Continue ground strength reduction to failure

6

7

resulting from the excavation, surcharge and new steady-state pore pressure dissipated to less than 1 kPa. The outputs from this stage represented the long-term, drained conditions following undrained construction conditions and were expected to be more onerous in terms of ground movement, wall stability and wall structural forces since this was primarily an unloading problem (refer to Section 4.2.2). ULS output facto factoring ring (draine (drained) d) Continuing from Stage 5, the surcharge was increased by the ratio of partial factors variable to permanent as given in the design code (1.5/1.35 ≈ 1.1 in this example) to 11.1 kPa. The excavation depth was increased to 5.5 m by deactivating an additional layer of elements and the attached interface elements to allow for over-dig as required by the design code. The steady-state pore pressure remained the same. Outputs of structural force in the sheet pile wall elements were factored for structural ULS checks as described in Section 6.3.1. ULS input factoring factoring (groundwater (groundwater flow analys analysis is then drained drained analysis) Continuing from Stage 5, a groundwater flow analysis was performed with the 229



groundwater level outside the shaft raised by 1 m to the ground surface which was judged to be the worst case water level. The groundw groundwater ater level within the shaft remained at −6.0 m level because this was controlled by the dewatering measures. Since the clay had low permeability, it was not considered necessary to consider the case of a temporary failure of the dewatering measures. This generated the ULS steady-state pore pressure. The surcharge was increased by the partial factor required by the design code (1.3 in this case) to 13 kPa. The excavation depth was increased to 5.5 m by deactivating an additional layer of elements and the attached interface elements to allow for over-dig as required by the design code. This stage prepared the analysis for strength reduction in the subsequent stages. ULS input factoring factoring – strength strength reduction reduction by required required partial partial factor Continuing from Stage 7, the shear strength of the ground was reduced in a stepwise fashion until the partial factor on drained strength was reached. For this process, the constitutive model was simplified to a LEPP model with Mohr– Coulomb failure criterion in order to use the stepwise strength reduction feature of the FE analysis program. The outputs were then checked for any geotechnical failure while the outputs of structural force in the sheet pile wall from this stage were compared with those from Stage 6 and the more onerous of the two was used in the structural ULS check, as described in Section 6.3.1. ULS input input factoring factoring – continued continued strength strength reduction reduction Strength reduction was continued in the same way until equilibrium could not be established and the most critical ground failure mechanism was identified. The sheet pile wall constitutive model remained linear elastic without a strength criterion, but it would have been an option to adopt an elastic-plastic model to study the effect of reduced bending resistance in the sheet piles.

8

9

Figure 8.22 Hydraulic boundary conditions Open boundary

–6 m

) y y r t r a e d n m u m o b y s d f o e s s i o l x a C (

y r a d n u o b d e s o l C

Specified head on righthand boundary, worst case and characteristic levels

Specified head on left-hand boundary

0m –1 m

O p e n b o u n d a r y

Closed boundary


Examples

Calculation options (§1.4.3) The Modified Newton–Raphson solution scheme was adopted with arc length control. Automatic step-sizing was utilised for both displacement and consolidation analysis and the maximum equilibrium error was set at 1%. The small deformation (Total Lagrangian) formulation was adopted in all analysis stages.

8.3.3 8.3.3 Obtain Obt aining ing par parame ameters ters and con const stitu itutiv tive e mode modell feat featur ures es Constitutive model selection The clay was over-consolidated so was likely to exhibit initially elastic behaviour in deviatoric loading followed by yielding behaviour when its stress state reached the yield surface. Large stress changes would be caused by the excavation of the shaft, so a stressdependent soil stiffness was important in order not to over-estimate the heave of the base of the excavation. While the critical case for deformations and ULS design was likely to be the long-term drained case in this unloading problem, construction needed to be simulated in undrained conditions followed by dissipation of excess pore pressures in order to follow a more accurate stress path (as explained in Section 4.2.2). This required a nonnon-linea linearr stiff stiffness ness cons constituti titutive ve mode modell capa capable ble of reas reasonab onably ly acc accura urate te exce excess ss por poree pressure prediction using the ‘Undrained A’ approach. Furthermore, accurate prediction of excavation-induced ground movements around the shaft required a model with straindependent stiffness. No significant anisotropy was expected in the clay, so an isotropic, double-hardening model with hyperbolic stress–strain relationship and strain-dependent stiffness was selected. Stresses within the steel sheet piles and reinforced concrete capping beam were expected to be relatively low and within the linear elastic range of these materials. Therefore, linear elastic models were adopted for these materials. However, orthotropic elasticity was introduced in the steel sheet pile model due to the geometrical anisotropy of the profiled sheets (see Section 5.1.6).

Obtaining parameters High-quality samples of the clay from various locations within the zone of influence of the proposed shaft were tested in stress path triaxial cells with local strain measurement and bender elements at the top and bottom of each specimen. This allowed the full stra st rainin-dep depend endenc ency y of st stiff iffnes nesss to be der deriv ived. ed. Ea Each ch spe specim cimen en wa wass sa satur turat ated ed and the then n reconsolidated to their in situ stress state along their most recent stress path, as described under Reconsolidation under Reconsolidation stage in stage in Section 3.3.1. Then each specimen was unloaded axially in extension until shear failure which matched approximately the stress path in this unloading problem. The difficulty with obtaining parameters for the constitutive model was that the model para pa ramet meters ers we were re defi defined ned in ter terms ms of dr drain ained ed tria triaxia xiall beh behav aviou iourr wh where ereas as per perfor formin ming g drained triaxial tests on low-permeability clays would have been very time-consuming and expensive. Therefore, undrained shear stages were employed and the model parameters ete rs we were re ob obtai tained ned by ba backck-an analy alysis sis of the und undra raine ined d tes tests. ts. K 0-consolidation tests (equivalent to oedometer tests) were also undertaken using a stress path triaxial cell on specimens of reduced height in order to reduce consolidation times. 231 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.


Figure 8.23 Simulation of triaxial extension test 100

80

60 a P k :

Lab test data

q

Single-point algorithm

40

20

0 0

–1

–2

–3 εa: % εa:

0

–1

–2

–4

–5

–6

%

–3

–4

–5

–6

0

–5 Lab test data –10 a P k :

Single-point algorithm

e

u

–15

–20

–25

Some FE analysis programs have single-point algorithms for simulating simple laboratory tests and these were used to back-analyse the triaxial extension and K 0-consolidation tests. The outputs from these are compared with the test results for one set of specimens in Figures 8.23 and 8.23 and 8.24 8.24.. The specimens were obtained from where the in situ vertical effective stress was 100 kPa, with an assumed K 0 value of 1.0 and vertical pre-consolidation stress of 1100 kPa. The single-point algorithm outputs were obtained using the model 232 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.

Examples

Figure 8.24 Simulation of

K 0-consolidation

test σ′v:

0

50

kPa

100

150

200

0 Lab test data Single-point algorithm

0.1

0.2 % : v ε

0.3

0.4

0.5

parameters shown in Table 8.4 and as defined in Section 8.2.3 for the raft foundation example. The K oed ( K 0 ) comoed parameter refers to the soil stiffness in one-dimensional (K pression. A lower, post-peak shear strength was adopted since this FE model would be used for ULS verifications. For more accurate pre-failure deformation predictions, a peak strength could have been used, but the strength softening post-peak would not have been recreated with the adopted constitutive model. The stress-dependency of stiffness was determined by comparing the stiffness values obtained from different tests at different reconsolidation stresses. The in situ stress ratio K ratio K 0 is an important initial state parameter in retaining wall design and a value of 1.0 was estimated from suction measurements undertaken on high-quality samples as soon as they had been brought to the surface. The vertical pre-consolidation stress was estimated to be 1000 kPa higher than the existing in situ stress, which was found to give reasonably accurate predictions in all the test simulations. From the same triaxial extension test studied earlier in this example, a plot of secant shear modulus G modulus G s against shear strain is shown in Figure 8.25. 8.25 . The precision of the local strain instrumentation allowed shear strain to be measured from a value of about 0.01%. A very small strain (below about 0.002%) G 0 of 150 MPa was obtained from the bender element test on the reconsolidated specimen with an average normal effective stress p ′ of 100 kPa. Table 8.4 Model parameters for the clay soil ′

or

K ur

n

n

K i

K 50

700

0 .9

0.2

420 or 230

K oed

Rf

w ′

′

180

0 .9

23

8

C

c

g sat sat

g unsat unsat

5 kPa

0

20 kN/m3

18 kN/m3

8



Figure 8.25 Measured non-linear stiffness and Benz et al . model 160 G0 = 150 MPa

140 120 Lab test data Benz et al . model

100

a P M 80 : s G

60 40 20 γ0.7 = 0.012% 0 0.0001

0.001

0.01

0.1

1

10

γ: % γ:

Two strain-dependent stiffness models are commonly used, namely those of Benz of Benz et et al . (2009) and Jardine et al . (1986), (1986), and the derivation of parameters for these is shown in Figures 8.25 and 8.26, 8.26, respectively. For the Benz et al . model, g 0.7 0.7 = 0.012% was obtained with G with G0 = 150 MPa. The stress-dependency was taken into account in the same way as for the large strain stiffness values using the n n value value of 0.9. For the Jardine et al . model, the parameters shown in Table in Table 8.5 were 8.5 were obtained. Figure 8.26 Measured non-linear stiffness and Jardine et al . model 1600 1400 Lab test data Jardine et al . model

1200 1000 ′

p / s 800

G

600 400 200 0 0.0001

0.001

0.01

0.1

1

10

γ:: % γ


Examples

Table 8.5 Jardine et al . model parameters A

B

C

a

g

1100

1200

8.0 × 10 − 5%

1 .2

0 .6

Table 8.6 Sheet pile section properties d

t

t

I

A

M

d

280 28 0 mm 7. 7.5 5 mm 10 83 830 0 cm4 /m 103.3 cm2 /m 81.1 kg/m

The PU 8R section was proposed in the design for the sheet piles, and its section properties as provided by the manufacturer are shown in Table 8.6. 8.6. Shell model parameters aree spe ar specifi cified ed for an equ equiva ivalen lentt she shell ll of uni unifor form m thi thickn ckness ess bu butt wh when en re repr prese esenti nting ng a profiled structure, such as a sheet pile wall, it is not possible to define parameters that represent both the bending and axial stiffness accurately. Consequently, priority was given to the bending stiffness of the sheet piles in the vertical orientation. Furthermore, the ben bendin ding g and equ equiva ivalen lentt ax axial ial sti stiffn ffness ess in the ou out-o t-of-p f-plan lanee (ho (hoop) op) dir direct ection ion we were re orders of magnitude lower than in the vertical direction due to the profiled section which needed to be taken into account in order not to over-predict hoop stresses (the effect of this over-prediction is demonstrated in the output to this example). However, in order to avoid an ill-conditioned stiffness matrix, the difference between the bending stiffness in the vertical and horizontal directions in the model parameters was restricted to 20 times. The Poisson’s ratio was set to zero in order to further reduce the generation of hoop stres st resses. ses. The mo model del par parame ameters ters ar aree sho shown wn in Table Table 8.7 8.7.. A1 refers refers to the equ equiva ivalen lentt sectional area in the vertical direction calculated to satisfy Equation 5.4. A 2 refers to the equi eq uiva vale lent nt se secti ction onal al ar area ea in th thee ho hori rizo zont ntal al (h (hoo oop) p) di dire rect ctio ion n ca calc lcul ula ated to sa satis tisfy fy Equation 5.4 with I 2 = 0.05 0.05I I 1 . The reinforced concrete capping beam was represented by shell elements with isotropic linear lin ear ela elasti sticc ma mater terial ial pr prope operti rties es bec becaus ausee its 1 m squ squar aree sect section ion wa wass un unifo iform rm in the out-of-plane direction such that it could sustain hoop stresses effectively. A Young’s modulus E E of of 30 GPa was adopted with a Poisson’s ratio of 0.15. The weight density of the concrete was taken as 24 kN/m 3, giving a net weight of 4 kN/m 3 over the soil that occupied the actual area of the capping beam in the model. Table 8.7 Sheet pile model parameters for shell elements E

I 1

d

A1

A2

w

n

207 GP GPa a

1.0 .08 8 × 10 − 4 m4 /m

0.28 m

0.0165 m2 /m

8.26 × 10 − 4 m2 /m

0.8 kN/m/m

0



Interface elements with a Coulomb friction criterion were installed between the shell elements and the area elements of the soil. The shear strength at the interface was taken as 70% of the internal shear strength of the clay but the outputs were not particularly sensitive to variations in this value.

8.3.4

Output

The output of sheet pile wall horiz horizonta ontall deflec deflection tion in the long-term case follo following wing dissipation of excess pore pressures in construction Stage 5 is shown in Figure 8.27. 8.27. Only minor deflection occurred at the top of the wall due to the support of the capping beam and the maximum value of 7.6 mm occurred approximately at the excavation level. The output from an analysis with isotopic properties for the sheet pile wall is also shown. Clearl Cle arly, y, suc such h an ass assump umptio tion n sig signifi nifican cantly tly un under der-es -estim timat ates es defl deflect ection ion bec becau ause se lar large, ge, unrealistic hoop forces were generated in the shell elements of the wall. The predicted variation of horizontal (radial) and vertical deflection of the ground at 1 m depth with distance from the sheet pile wall is shown in Figure 8.28. 8.28. The shapes of the profiles pro files correspond correspond with thos thosee of the wa wall ll deflec deflection, tion, with less deflection adjacent adjacent to the wall where the capping beam provided support, increasing to a maximum at about 2 m from the wall due to the deflection of the wall at lower elevations. The vertical deflection reduced to an insignificant level of less than 0.5 mm at a distance of 10 m (equivalent to two times the excavation depth) from the wall which agreed with typical rules of thumb for the zone of influence of a retaining wall. By calculating the radial distance from the wall corresponding with the linear distance a along the buried sewer at 1 m depth as illustrated in Figure in Figure 8.29, 8.29, it was possible to derive excavation-induced deflection plots for the sewer as also shown in Figure 8.29. This involved some approximation seeing that the stiffness of the sewer was ignored, and also Figure 8.27 Output of sheet pile wall horizontal deflection

–2

0

Wall deflection: mm 2 4

6

8

0

–2

m : n o i t a v e l E

–4

–6

–8

Orthotropic wall Isotropic wall

–10


Examples

Figure 8.28 Output of ground movement at 1 m depth behind sheet pile wall Radial distance from wall: m 0.0

0

10

20

30

40

0.5 1.0

m m : 1.5 n o i t c e l f 2.0 e D

2.5

Vertical deflection Horizontal (radial) deflection

3.0 3.5

because the horizontal displacement u displacement u x direction varied from being perpendicular to the sewer at a = 0 to a more longitudinal direction as a increased or decreased away from zero. Nevertheless, these plots could be used to estimate approximate distortions in the sewer, and to compare them with allowable values. The ULS values of sheet pile wall bending moment are shown in Figure 8.30. 8.30. The output factoring (OF) values were obtained by multiplying the output from Stage 6 by a load effect factor of 1.35 (required by the design code) and the input factoring (IF) values were we re obt obtai ained ned dir direct ectly ly fro from m the out output put of Sta Stage ge 8. The maximum maximum abs absolu olute te val value ue of 66 kNm/m occurred at the connection with the capping beam at −1 m elevation and happened to be about the same in both the OF and IF cases. This maximum value should be used to check whether the sheet pile wall has adequate bending moment resistance. Figure 8.29 Derived output of ground movement along line of sewer

–30 ux ux

Shaft axis

ux

Analysis planes a

7.5 m 3 m Sewer

PLAN VIEW

–20

Distance along sewer a: m –10 0 10 0.0 m m : n o i t c e l f e d r e w e S

20

30

0.5 1.0 1.5 2.0 2.5 3.0

Vertical deflection Horizontal deflection ux



Figure 8.30 Factored output of bending moment in sheet pile wall Wall bending moment: kNm/m –50 0 50 0

–100

100

–2


–4

–6

–8

Orthotropic wall OF Orthotropic wall IF Isotropic wall OF Isotropic wall IF

–10

The output from an identical FE model with isotropic sheet pile wall properties is also shown and clearly the bending moment was significantly under-predicted due to the hoop forces generated in the shell elements. In Stage 8 with factored ground strength, no failure was apparent in the output. The strength reduction was continued in Stage 9 and the plot of wall toe deflection against strength factor in Figure 8.31 shows that failure was predicted at a strength factor of about 2.1. The plot of incremental displacement vectors from the end of Stage 9 in Figure 8.32 shows that the predicted critical failure mechanism was a bearing failure beneath the toe of the sheet pile wall.

Figure 8.31 Output of wall toe deflection during strength reduction 2.2 r 2.0 o t c a f

1.8

h t g n e r 1.6 t s d n 1.4 u o r G1.2

1.0 0

200 400 600 Wall toe deflection: mm

800


Examples

Figure 8.32 Vectors of incremental displacement at end of strength-reduction stage

In order to check that the undrained shear strength of the clay was not over-predicted during the undrained analysis stages, a contour plot of mobilised shear strength in terms of half the deviatoric stress in Stage 4 was plotted adjacent to the measured profile of in situ undrained shear strength (which was not expected to change significantly in the short-term construction case) as shown in Figure 8.33. 8.33 . The maximum mobilised shear strength of about 40 kPa beneath the toe of the sheet pile wall was well below the approximate measured shear strength of 130 kPa at this depth, so the FE model did not appear to be over-predicting undrained shear strength with the Undrained A approach (see Section 4.2.5). Hydraulic gradients will be generated by the groundwater lowering measures, so it was necessary to check that hydraulic failure beneath the base of the shaft was sufficiently unlikely to occur. Since the weight density of the saturated saturated ground and groundw groundwater ater were about 20 and 10 kN/m 3, respectively, the critical hydraulic gradient for hydraulic failure was 1.0. Depending on the requirements of the design code, an adequate safety margin (a safety factor of 1.5, say) would require a hydraulic gradient of less than 0.67 ( = 1/1.5). The plot of steady-state pore pressure with the raised groundwater level in Stage 7 in 239 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.


a P k 0 4

5 3

0 3

5 2

0 2

5 1

0 1

n o i t a g i t s e v n i e t i s

5

3

σ

– 2 1

σ

m o r f d e n i m r e t e d e l fi o r p u

c

h t i w d e r a p m o c s s e r t s c i r o t a i v e d

0 5 2 n o i t a g i t s e v n i e t i s m o r f e l i f o r p

0 0 2 0 a 5 P 1 k : u c 0 0 1

×

5 . 0 f o t u p t u O 3 3 . 8

0 5

u

c

0 0

5 –

0 1 –

5 1 –

0 2 –


e r u g i F


0

Examples

Figure 8.34 Steady-state pore pressure with −10 m wall toe level

20 kPa

40 kPa

60 kPa Hydraulic m gradient, 1 . i ≈ 0.94 3

80 kPa

100 kPa

120 kPa

Figure 8.34 shows Figure shows a ma maxim ximum um hyd hydra rauli ulicc gr grad adien ientt of abo about ut 0.9 0.94 4 whi which ch wa wass clea clearly rly unacceptable. Note that the proximity to hydraulic failure was not immediately apparent from the output in Stage 7, nor during the subsequent strength reduction which showed adequate safety against geotechnical failure. Verifying adequate safety against hydraulic failure requires a specific, separate check of the groundwater flow analysis outputs. A further groundwater flow analysis was performed with a deeper wall toe at −13 m elevation and the outputs of steady-state pore pressure are shown in Figure 8.35. 8.35 . The maximum hydraulic gradient was about 0.62, which was acceptable. Therefore, the sheet pile wall needed to be installed to a toe level of −13 m in order to ensure adequate safety against hydraulic failure. The FE analysis should then be re-run with the modified toe level in order to obtain updated outputs of deflection, bending moment, etc. (not presented here).

8.3. 8. 3.5 5

Val aliida dati tio on

Validat Valid ation ion in thi thiss cas casee wa wass ra rathe therr dif difficu ficult lt bec becaus ause, e, alt altho hough ugh alt altern ernat ativ ivee an analy alysis sis methods, empirical design methods and extensive case study data exist for straight-sided excavations, there is a paucity of similar sources of validation data for circular shafts. As mentioned in the previous section, the output of ground settlement behind the retaining wall in Figure 8.28 appeared credible since its shape matched inversely with the wall deflection profile in Figure 8.27 in accordance with the behaviour observed in many case studies of straight-sided excavations, e.g. Clough and O’Rourke (1990). (1990) . 241 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.


Figure 8.35 Steady-state pore pressure with −13 m wall toe level

20 kPa

40 kPa

60 kPa

Hydraulic gradient, i ≈ 0.62

80 kPa

m 7 . 3

100 kPa

120 kPa

Moderate uncertainty still existed in the predicted values of ground movement and hence sew se wer de defle flect ctio ion n du duee to th thee ab abse senc ncee of an any y ca case se st stud udy y da data ta or al alte tern rna ati tiv ve an anal aly ysis methods which should be taken into account in assessing the reliability of the sewer deflection predictions. Consideration could be given to an observational approach involving monitoring of ground settlements with contingency measures if ground movements were found to be larger than expected. To help validate validate the outputs from the sheet pile wall, inclinometer inclinometer readings from a similar shaft in similar ground and groundwater conditions were available. The excavation depth of this shaft was was 8 m instead of 5 m and the diameter was 18 m instead instead of 15 m. There was also a waling beam at 4 m below ground level in addition to the capping beam to provide additional support to the sheet piles. Although the geometry differed somewhat, if the deflection of the sheet pile wall could be predicted using a modified version of the FE analysis model, then there would be increased confidence in the predictions of the similar original FE analysis model. Inclinometer data only during construction were available, so only the short-term undrained conditions were simulated. The FE analysis output and monitoring data are compared in Figure 8.36 with the toe of the inclinometer set to match the analysis output. The agreement between the plots was very good which gave confidence in the prediction of sheet pile wall behaviour in undrained conditions. A judgement would then have to be made on the reliability of the predictions, particularly in the drained case where no case study data were found. An observational approach or design changes might be options if the margin of safety was considered too small. 242 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.

Examples

Figure 8.36 FEA output and monitoring data of sheet pile wall deflection for a similar shaft Wall horizontal deflection: mm –1

0

1

2

3

4

5

6

7

0


–5

–10 FEA output Monitoring –15

As indicated in Figure 8.31, the margin of safety on geotechnical failure obtained from the strength reduction was quite high, but what about on hydraulic failure?

8.4. 8.4. 8.4.1

Embankm Emban kmen entt co cons nstr truct uction ion ex exam ampl ple e Summary

This example concerns the construction of a highway embankment on soft, normally consolidated clay. The design of the embankment relied on the soft clay gaining strength due to consolidation under the growing weight of the embankment during construction, so the construction sequence needed to be timed and monitored carefully. The particular features of this example include: g g g g g g g g g g

2D plane plane str strain ain model gravity grav ity switch-on switch-on initial stress procedure procedure coupled consolidation analysis double-hardenin doubl e-hardening, g, stress-dependent stress-dependent stiffness constitutiv constitutivee model for embankment embankment fill Modified Modi fied Cam Clay Clay (MCC) model for clay clay foundation foundation obtaining MCC model model parameters parameters from triaxial tests modelling mode lling vertica verticall wick drains drains in a plan planee strain strain model large deformation deformation (Updated (Updated Lagrangian) Lagrangian) formulation formulation ULS check by one-step strength strength reduction validation validatio n using basic analysis analysis methods.

8.4.2 8.4. 2 Setti Se tting ng up th the e FE an anal alys ysis is mo mode dell Justification for using FE analysis (§1.1.1) An important element of the design of this embankment was was the improvement improvement of the clay foundation’s strength due to consolidation under the weight of the embankment during constr con struct uction ion.. Th Thee con consol solida idatio tion n tim timee re requi quire red d for the fou found ndat ation ion to ga gain in suf suffici ficient ent 243 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.


strength for continued embankment construction impacted directly on the construction programme. Excess pore pressures due to both normal and shear stress changes in the foundation needed to be predicted, along with their dissipation during construction and in any intermediate consolidation periods. Such complex behaviour can only really be predicted using advanced constitutive models implemented using a numerical analysis technique such as FE analysis. In addition, the sloping geometry of the clay foundation layer further precluded the use of simpler analysis methods.

Aims of the model (§1.1.2) 1 2

to predict predict the timing of the emba embankme nkment nt constructio construction n sequence sequence necessary to ensure adequate stability to predict predict the settlement of the embankment embankment crest crest on completion completion of conso consolida lidation. tion.

Geometrical Geom etrical simpl simplifica ifications tions (§1.2.1 and §1.2.2) There were no significant man-made structural features at the site except for the embankment itself. As shown in Figure in Figure 8.37, 8.37, the proposed embankment had a long, prismatic geometry that was well suited to the plane strain assumption. The structural geology was also quite uniform in the same long axis as the embankment, which allowed a 2D plane strain assumption to be adopted. From the information available, both the ground surface and clay layer appeared to be inclined incli ned at approxima approximately tely uniform uniform slopes of abou aboutt 1 : 10 which were were adopted in the FE model. The sloping ground level and layers meant that no axis of symmetry was present to allow only part of the plane strain section geometry to be analysed. If the ground layers were horizontal then an axis of symmetry would have existed down the centre of th thee em emba bank nkme ment nt an and d on only ly ha half lf th thee ge geom omet etry ry wo woul uld d ha hav ve ne need eded ed to ha have ve be been en analysed. The embankment was assumed to have a horizontal crest coincident with the highest proposed carriageway level at +4.0 m level level and side-sl side-slope opess of 1 : 3 on the downdownslope side and 1 : 2.85 on the upslope side. Minor details details such as carria carriagew geway ay cambering cambering and the shoulder were excluded from the FE model. The embankment would be constructed in a progressive fashion in layers of about 0.3 m thickness. To include the progression would require a 3D analysis but it was considered sufficiently accurate to assume infinitely long and full-width layers in a 2D plane strain analysis. Adding 0.3 m-thick layers in the FE model would also have been unnecessarily complicated, so it was considered sufficiently accurate to construct the embankment in three layers of up to 2 m thickness and this was checked by also running an analysis with thinner layers and comparing the key outputs. Since each layer was constructed in a coupled consolidation analysis over a time period, the gradual build-up of each 2 mthick layer was taken into account in the FE analysis to some extent. Vertical band drains on a 2.3 m triangular grid were to be installed to the full depth of the clay layer across the embankment footprint, as shown in Figure 8.37, to hasten consolidation times. However, installing a vertical drain in a 2D plane strain analysis 244 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.

Examples

Figure 8.37 Geometry of embankment construction example

8m

10

8m

1

Embankment fill placed in 3 layers in FE model

Soft clay

10

+0.5 m

+4.0 m +2.0 m +0.0 m

–16.5 m

Vertical drains on 2.3 m triangular grid

1

Dense sand

would simulate a continuous trench drain running in the out-of-plane direction which would cause consolidation times to be under-predicted and potentially lead to an unsafe design. To account for this, the horizontal permeability of the clay between the vertical drains in the plane strain model was reduced, as described by Hird Hird et et al . (1995). (1995). The volume of soil drained by each vertical drain is equivalent to a cylinder of diameter 1.05 times the drain spacing on an equilateral triangular grid (or 1.13 times the spacing on a square grid), giving an equivalent radius R of the drained zone for each drain of 1.2 m. The vertical drains were installed in the mesh at 2R 2 R (= 2.4 m) centres, as shown in Figure in Figure 8.38. 8.38. The horizontal permeability in the drained zone of the clay layer was then corrected according to Equation 8.10 to kpl = 1.8 × 10 − 9 m/s, assuming flow rates into the drains were sufficiently high for drain resistance to be neglected and ignoring the effects of installation-induced smear on soil permeability. kpl kax

=

2

        

R kax rs 3 – 3 ln ln + rs ks rw 4

(8.10)

(Hird et (Hird ., 1995) et al ., 245



where r w is the radius of the drain (the band drains had an effective value of 30 mm), r s where r the radius of the shear zone (assumed equal to rw, i.e. smear ignored).

Model boundaries and fixities (§1.2.3 and §1.2.4) No rules of thumb on locating boundaries were known for an unusual geometry such as this so, after some trial and error, it was found that the vertical boundaries to the mesh needed to be placed about 40 to 50 m from the embankment toe in order to eliminate significant boundary effects. A greater distance was needed on the downslope side, as shown in Figure 8.38, because more deformation of the clay foundation occurred on this side. The bottom boundary could be placed closer because the dense sand layer was significantly stiffer than the soft clay. The bottom boundary was fixed in both axis directions and the vertical boundaries were fixed only in the horizontal direction.

Finite element mesh (§1.3) Cubic strain 15-node triangular elements with three degrees of freedom per node were used throughout the mesh. The mesh was generated as shown in Figure 8.38 with smaller elements created at the embankment foundation where steep stress, pore pressure and strain gradients were expected. A further FE analysis was performed with a finer mesh and it was found that the critical outputs were not affected significantly. Therefore, the adopted mesh was considered acceptable.

Initial stresses (§1.4.1) Since the ground surface and soil layers were inclined, the initial stress profile was not uniform across the model and so could not be specified simply with pore pressures, ground gro und densi density ty and K 0 values. values. The ini initia tiall st stre resses sses we were re gen genera erated ted by ac activ tivat ating ing the self-weight of the ground (‘gravity switch-on’). Steady sta Steady state te po pore re pr press essur ures es we were re gen genera erated ted in a gr grou oundw ndwat ater er flo flow w an analy alysis sis wit with h the groundwater level at each side of the mesh assumed at 1 m below ground level, vertical boundaries were assumed open and the bottom boundary closed. Pore pressures were assumed zero above the groundwater level. The clay density was taken as 18 kN/m 3 above groundwater level while the saturated clay and sand densities were assumed to be 20 and 18 kN/m 3, respectively. In order to establish in situ stresses, a basic LEPP Mohr–Coulomb model was adopted for both strata (and remained so throughout all stages for the sand layer). In accordance with Equation 1.2, Poisson’s ratios of 0.38 and 0.33 were adopted for the clay and sand to obtain target K 0 values of 0.6 and 0.5, respectively. The self-weight of the soil layers were activated in drained conditions. While attempting to obtain a realistic initial stress state in subsequent analysis stages with the MCC model, it was found that the initial Poisson’s ratio for the clay in the LEPP model needed to be changed to 0.34. At the end of the analysis stage activating the ground’s self-weight, no plastic points were evident in the output. 246 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.

Examples

2m

21 m Side fixed in X direction

t u s t t p n u m i o 9 o 4 p s n u o o i t u a n r i g t e n o t n c i r o d f n a d e s t e c d l o e N e s A e d o N

m Z . t p . t n I 8 1 B e d C o e N d o N

s n o i t c e r i d Y d n a X n i d e x i f e s a B

s n i a r D

m 6 1 e l p m a x m e 6 n o i t c u r t s n o c t n e m k n a m b 1 m 4 e r o f h s e m E F 8 3 . 8 e r u g i F

y a l c t f o S

d n a s e s n e D

X

Y

n o i t c e r i d X n i d e x i f e d i S m 8 2

m 8



Following establishment of initial stresses with the basic constitutive models, the clay model was changed to the MCC model. Some further manipulation was necessary to obtain what was considered to be a realistic initial stress state and yield surface. This can often be achieved with temporarily enhanced ground densities or by applying vertical stress to the ground surface and then releasing it to simulate pre-loading due to deposition or groundwater lowering. Due to the variable thickness of the clay layer in this example, it was found that more credible results were obtained by applying a uniform 0.3 m vertical displacement downwards to the ground surface and then releasing it. This wass perfo wa performed rmed in dra drained ined cond condition itionss and ach achiev ieved ed the str stress ess and over-consoli over-consolida dation tion ratio (OCR) profiles in the centre of the mesh shown in Figure 8.39, 8.39, which were considered to be realistic based on the site investigation information available. The analysis outputs were quite sensitive to the initial stress profile so a parametric study (not shown here) was undertaken to assess the reliability of the outputs. As described under Construction under Construction stages in stages in this section, strength reduction was performed in ded dedica icated ted ULS stages stages in ord order er to che check ck the st stab abilit ility y of the emb emban ankme kment nt dur during ing construction. Another option, as described in Section 6.3.2, is to adopt factored strength parameters for the ground from the start and throughout all analysis stages. The drawback of this approach is that the stress state and predicted behaviour of the model can become unrealistic. To illustrate this, the same procedure described previously to establish the initial stress state was repeated but with the shear strength factored, as described later in this section, from the start. The output of the initial stress state is shown in Figure 8.40 for comparison with that shown in Figure 8.39. Clearly, a very different initia ini tiall str stress ess st stat atee wa wass ob obtai tained ned in the sof softt cla clay y and and,, if the ana analy lysis sis we were re con contin tinued ued through its subsequent stages, very different and probably inaccurate outputs would be obtained throughout.

Figure 8.39 Initial

K 0 and

0.0 0

OCR profiles established in soil layers in FE analysis

0.5

K 0 or OCR 1.0 1.5

2.0

2.5

–5 –10

m : n o i t –15 a v e l E

Soft clay

Dense sand

–20 –25

K 0 OCR

–30


Examples

Figure 8.40 Initial parameters)

K 0 and

0.0 0

OCR profiles established in soil layers in FE analysis (factored strength

0.5

K 0 or OCR 1.0 1.5

2.0

2.5

–5 –10

m : n o i t –15 a v e l E

Soft clay

Dense sand

–20 –25

K 0 OCR

–30

Construction stages (§1.4.2) The following stages were run in the FE analysis which, apart from the drained stages to establish the initial stresses, were all coupled consolidation stages. The ULS stages were consolidation stages with zero time which were equivalent to ‘Undrained A’ analyses using the active pore pressure distribution from the most recent consolidation stage. The sequencing of the construction stages is illustrated in Figure 8.41 while 8.41 while each stage is described in detail as follows:

1

Steady-state pore pressures Since the groundwater regime was non-hydrostatic, steady-state pore pressures were established by groundwater flow analysis with groundwater level set at 1 m below ground level at left and right model boundaries. 2 Gravity switch-on Activate in situ ground elements in drained conditions with basic LEPP models. 3 MCC model for clay Change constitutive model for clay stratum to MCC model. 4 Pre-load Apply 0.3 m vertical displacement to ground surface under drained conditions to simulate pre-loading due to deposition or groundwater lowering. 5 Release pre-load Release displacement under drained conditions to create OCR profile. 6 Inst Install all drains drains (0 days) days) The vertical drains as shown in Figure 8.38 were installed which set excess pore pressure as zero along their length (the steady-state pore pressure remained unchanged). The horizontal permeability of the clay between and within 1.2 m of the drains was reduced to the corrected value as described under Geometrical simplifications in this section. This and all following stages were performed as coupled consolidation analyses with the time interval indicated in brackets. 249 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.


Figure 8.41 Sequence of analysis stages 1. Steady-state pore pressure

2. Gravity switch-on

3. MCC model for clay

4. Pre-load

5. Release pre-load

6. Install drains

7. Embankment layer 1

Establishing initial state

Construction stages

8. ULS strength reduction

9. Intermediate consolidation



12. Intermediate consolidation



15. Complete consolidation

7

8

Install embankment Install embankment layer layer 1 (5 days) Activate layer 1 embankment fill elements. In this stage, a construction period of 5 days to install layer 1 was assumed. Nodal displacements were reset to zero at the beginning of this stage. In the analyses where these were used, the large deformation (Updated Lagrangian) formulation and updated water pressures were activated from this stage onwards. Layer Lay er 1 ULS strength strength reduction reduction (0 days) days)∗ Ground shear strength in all strata reduced in one step.


Examples

9

10

11 12

13

14 15

Intermediatee consolidation Intermediat consolidation (90 days) days) Continuing from Stage 7, dissipation of excess pore pressures was allowed for 90 days with no other changes in the model. Install Inst all embankment embankment layer layer 2 (5 day days) s) Activate layer 2 embankment fill elements. A construction period of 5 days was assumed. Layer Lay er 2 ULS strength strength reduction reduction (0 days) days)∗ Ground shear strength in all strata reduced in one step. Intermediat Interm ediatee consolidation consolidation (80 days) days) Continuing from Stage 10, dissipation of excess pore pressures was allowed for 80 days with no other changes in the model. Install Inst all embankment embankment layer layer 3 (5 day days) s) Activate layer 3 embankment fill elements. A construction period of 5 days was assumed. Layer Lay er 3 ULS strength strength reduction reduction (0 days) days)∗ Ground shear strength in all strata reduced in one step. Complete Com plete consolidation consolidation (to 1 kPa kPa maximum excess pore pressure) pressure) Continuing from Stage 13, dissipation of excess pore pressures was allowed until the maximum value anywhere in the clay stratum was 1 kPa.

∗

The groundwater level was left unchanged in the ULS stages because significant level changes were highly unlikely in a low-permeability soil over short construction timescales. Input factoring (strength reduction) was performed. Output factoring was not necessary because only ground stability was being considered. Regarding strength reduction, several issues needed to be overcome:

1

2

3

The strength strength factors factors required required by design codes usually usually differ between drained drained and undrained strengths whereas this example involved drained embankment fill and a partially drained (consolidating) clay foundation. So, each required a different strength factor that could be applied either individually to each layer or else a uniform, stepwise strength reduction could have been performed and then the output viewed to identify through which soil layers the failure mechanism passed and which strength factor requirement was appropriate. Design Desig n codes usually usually provide provide partial partial factors factors to be applied applied to w ′, c′ and cu whereas the MCC model does not use these parameters directly to define the failure criterion. Furthermore, the FE analysis software did not have a rigorous strengthreduction routine for constitutive models with failure criteria other than those defined by w ′, c′ and cu . The prediction prediction of shear strength strength in undrained undrained or parti partially ally drained drained conditions conditions (using the ‘Undrained A’ approach) is not straightforward and is heavily dependent on the prediction of excess pore pressure. Simply applying partial factors from design codes is not an adequate approach.

To help overcome Issues 2 and 3, a large number of CAU triaxial compression test simulations were performed using a single stress point algorithm covering the range of 251 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.


stress states in the clay foundation. Initially, the simulations were performed with the characteristic model parameters described in Section 8.4.3 (including M = 1.07), then the simulations were repeated with the M parameter parameter reduced until the output of shear strength from each test simulation was reduced by the factor required in the design code (in this case 1.4 for undrained strength). It was found that the computed undrained shear strength was reduced by at least a factor of 1.4 in most cases when M was was 0.72. However, the factor was significantly less when K when K 0 and OCR were low, but such stress states were located deeper in the clay foundation, below where failure mechanisms were expected to occur. No stepwise strength-reduction procedure was available for the MCC model in the FE analysis software, so the shear strength parameters were reduced to their factored values directly in the model parameters in the ULS analysis stages leaving the program’s stress point algorithm to correct any stress states violating a yield surface. Stress path outputs were checked as described in Section 8.4.4. This method of strength reduction also overcame Issue 1 because the shear strengths of each soil layer could be reduced by different factors according to the requirement on drained (for embankment fill and dense sand layer) and undrained (for soft clay layer) shear strength.

Calculation options (§1.4.3) The Modified Newton–Raphson solution scheme was adopted with arc length control. Automatic step-sizing was utilised for both displacement and consolidation analysis and the maximum equilibrium error was set at 1%. The sma The small ll def deform ormat ation ion (T (Tota otall Lag Lagra rang ngian ian)) for formul mulat ation ion wa wass ado adopte pted d in all ana analy lysis sis stages in the first runs of the analysis. Up to 1.2 m settlement of the embankment was predicted which was large enough to cause the small deformation assumption where the original mesh geometry is retained in calculations to become unrealistic. In particular, highly deforming ground below or near the groundwater level would, in reality, experience significant steady-state pore pressure change due to its displacement relative to the stationary groundwater level. Therefore, an additional analysis run was performed with Updated Lagrangian formulation adopted from Stage 7 onwards and then the outputs were compared as described in Section 8.4.4.

8.4.3 8.4.3 Obtain Obt aining ing par parame ameter terss and and cons constit tituti utive ve mod model el fea featur tures es Constitutive model selection Normally and lightly over-consolidated clays are highly compressible which is behaviour particularly suited to the MCC model (described under Isotropic under Isotropic hardening single surface plasticity in plasticity in Section 2.3.2) which has stress-dependent primary and unload–reload stiffness defined in terms of logarithmic isotropic compression curves. The MCC model also has a yield surface defined in terms of a pre-consolidation stress in order to simulate the effects eff ects of ov over-c er-cons onsoli olida datio tion n tha thatt ex exis isted ted nea nearr the gr grou ound nd sur surfa face. ce. Thi Thiss mo model del wa wass implemented with a Mohr–Coulomb failure surface for more robust strength prediction. Signifi Sign ifica cant nt an anis isot otro ropy py of sh shea earr st stre reng ngth th an and d st stif iffn fness ess is co comm mmon on in li ligh ghtl tly y overconsolidated clays, particularly those of low plasticity, which can have a significant effect 252 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.

Examples

on the analysis of embankment construction (see (see Zdravkovic ´ ´ et al ., ., 2002). 2002). In the soft clay in this example, no significant anisotropy was recorded in the site investigation and it was considered acceptable to adopt an isotropic MCC model. Due to the large range of stresses in the embankment fill material from near-zero at plac pl aceme ement nt to hig high h st stre resse ssess as sub subse seque quent nt la laye yers rs ar aree pla place ced d on to top, p, an and d be becau cause se the material is essentially normally consolidated when placed, highly non-linear elastic and work hardening behaviour should be expected. Therefore, a double-hardening, hyperbolic bol ic mode modell was ado adopted pted wit with h st stre ress-d ss-depen ependent dent sti stiffn ffness. ess. A simp simple le Moh Mohr–C r–Coul oulomb omb fail fa ilur uree cr crit iter erion ion wa wass ado adopte pted d bec becau ause se th thee st stre reng ngth th of th thee so soft ft cla clay y fou founda ndati tion on wa wass expected to govern the stability of the embankment during construction. The in situ, dense sand layer was expected to undergo insignificant stress and strain changes, so a simple LEPP Mohr–Coulomb model was adopted for the dense sand.

Obtaining parameters The parameters l , k and N for for the MCC model of the soft clay were obtained from isotropic consolidation tests on specimens of the soft clay in a triaxial cell as shown in the 8.42. Even on a log plot, both the plot of specific volume v against ln p ′ in Figure 8.42. unload–reload and normal compression lines are curved rather than linear, so tangents to these lines at stress levels appropriate for the analysis were drawn for deriving the parameters. There was an unusually large difference between l and k , probably due to the clay being structured. For the derivation of M , triaxial test stress paths may be plotted in q – p′ space and the slope of the critical state line determined directly. Alternatively, as in this example, M ′ can be derived from the critical state friction angle w cs . The stress paths from four CAU ′ triaxial tests (see Section 3.3.1) are shown in t – s space in Figure in Figure 8.43 [s′ = (s 1′ +s 3′ )/2 and t = (s ′1 − s 3′ )/2]. A best fit line was drawn from the origin through the end points Isotropic opic consolidation consolidation test data and derivation derivation of MCC model para parameters meters Figure 8.42 Isotr N 0 = 4.31 3.0

2.6

κ = 0.028

v

2.2 λ = 0.37

1.8 0

2

4

6

8

ln p′: kPa



Figure 8.43 CAU triaxial test stress paths and derivation of

M

30

20 a P k : t

10

sin ϕ′ sin ϕ′cs Reconsolidation stress path

1 0 0

10

20

30 s

40

50

60

′: kPa

of the stress paths (where they became stationary and where the pore pressure had stabil′ ′ ised). The slope sin w sin w cs was 0.453, giving w cs line M = 27 . The slope of the critical state line M can then be determined using one of the following equations: 8

6sin w ′cs triaxial compression: M = 3 − sin w ′cs

(8.11)

6sin w ′cs triaxial extension: M = 3 + sin w ′cs

(8.12)

In this example, for triaxial compression, M = 1.07. For a plane strain analysis the M value could be raised slightly to take account of the intermediate principal stress (as described in Section 3.4.1) but for conservatism in the ULS check, the M value value remained unchanged in this example. The permeability of the clay was derived from the isotropic consolidation tests and in situ constant head tests. Vertical and horizontal permeabilities of 1 × 10 − 9 and 1 × 10 − 8 m/s, respectively, were obtained. Higher horizontal permeability occurred due to the presence of silty laminations in the clay. The variation of permeability in both directions with changes in void ratio e was taken into account in the model according to Equation 8.13



log

k k0

=

De

1.2

(8.13)

where k and k0 are the active and initial permeabilities, respectively. As in most cases, there was considerable uncertainty in the permeability values so a parametric study (not presented here) of permeability to assess the reliability of predictions influenced by permeability would need to be made. 254 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.

Examples

Table 8.8 Derived model parameters for the soft clay M

l

k

N

g sat sat

g unsat unsat

k v

k h

1.07

0.37

0.028

4.31

20 kN/m3

18 kN/m3

1 × 10 − 9 m/s

1 × 10 − 8 m/s

Table 8.9 Model parameters for the embankment fill ′

′

′

K ur

n

n

K i or K 50

K oed

Rf

w

750

0 .5

0.2

455 or 250

250

0 .9

35

8

c

c

g

0.1 kPa

0

16 kN/m3

8

Table 8.10 Model parameters for the dense sand ′

′

E

n

200 MPa

0 .2 5

w ′

′

36

8

c

c

g

0.1 kPa

0

18 kN/m3

8

In summary, the parameters shown in Table 8.8 were derived for the soft clay in this example. Obtaining parameters for proposed earthworks structures is obviously difficult because they are yet to be installed. If the material to be used can be obtained, it is possible to prepare trial structures in situ or trial specimens in the laboratory for parameter testing. Similar materials may have been used in similar, existing structures that can be tested or back-analysed to obtain parameters, or case study data may exist. For example, Duncan et al . (19 (1980) 80) sum summar marised ised pa para ramet meters ers for a ra range nge of fill ma mater terial ialss for the hy hyper perbo bolic lic Duncan and Chang (1970) model. In this example, the embankment fill parameters were obtained based on experience of using the same material in other earthworks structures. The model parameters are shown in Table in Table 8.9 and 8.9 and the symbols have the meanings described in Section 8.2.3. The model parameters adopted for the LEPP Mohr–Coulomb model for the dense sand are shown in Table in Table 8.10. 8.10. A high linear stiffness was adopted to reflect the high confining stress and expected low strains in this layer.

8.4.4

Output

The vertical settlement of the three nodes selected for continuous output output (see Figure 8.38) is shown plotted against time in Figure in Figure 8.44. 8.44. The settlement resulting from the placement of each layer of the embankment is clearly visible. Node C experienced only minor settlement due to the placement of Layer 1 because it was located at the edge of Layer 1 where the wedge of fill material was very thin. Note that the displacement output for Node B at the crest of the embankment needed to be corrected back to zero because it already had 255 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.


Figure 8.44 Output of settlement (small deformation formulation)

0

200

Time: days 400

600

800

0.0 Node A Node B Node C

0.2 m 0.4 : t n e m 0.6 e l t t e S 0.8

1.0 1.2

Layer 1

Layer Layer 2 3

106 mm settlement caused by the settlement of lower layers associated with it when it was activated on placing Layer 3. The total construction time calculated for all excess pore pressures to fall below 1 kPa was 812 days. The same output is shown in Figure 8.45 but from the analysis with large deformation (Updated Lagrangian) Lagrangian) formulation and updated water pressures. Significantly less les s settlement and shorter consolidation times (605 days total construction time for excess pore pressures to dissipate to below 1 kPa) were predicted, so using this large deformation formulation could potentially lead to a more economic design. The ULS checks were also more favourable with the large deformation formulation, but experience shows that Figure 8.45 Output of settlement (large deformation formulation) –0.2 0

200

Time: days 40 0 400

600

800

0.0 0.2 m : t n 0.4 e m e l t t 0.6 e S

Node A Node B Node C

0.8 1.0 1.2 Layer 1

Layer Layer 2 3


Examples

Figure 8.46 Output of Node A displacement due to strength reduction 100 % : n 80 o i s s e r g 60 o r p e g a 40 t s s i s y l a 20 n A

Layer 1 Layer 2 Layer 3

0 0

10

20

30

40

Node A displacement change: mm

reasonably accurate failure predictions are often obtained with the small deformation (Total Lagrangian) formulation in spite of the large deformations. Therefore, the large deform def ormat ation ion for formu mula latio tion n sho should uld no nott nor norma mally lly be re relied lied upo upon n for ULS ve verifi rifica catio tions ns and in this example the ULS check was performed using the more conservative small deformation formulation. One of the main aims of this analysis was to predict the intermediate consolidation times necessary in order to construct the embankment with adequate stability. To check this, the shear strength of both the embankment fill and the clay foundation were factored by changing the model parameters in Stages 8, 11 and 14 immediately after placing each layer of the embankment. The maximum displacement of Node A during these stages was plotted against the progression of each stage as shown in Figure 8.46. 8.46. The reduction tio n in she shear ar str streng ength th af after ter pla placin cing g ea each ch la laye yerr wa wass com comple pleted ted wit witho hout ut any onset onset of failure apparent at Node A, suggesting that the intermediate consolidation times were adequate. Rather than rely on the displacement output of a single node, it is of course important to view the displacement of the whole model when the shear strength has been reduced, to check for failure mechanisms elsewhere. Total displacements are useful but shear failures may be masked by the deformations occurring prior to failure. It is a good idea to plot shear strains or, in this case, vectors of displacement in the last increment of the strengthred educ ucti tion on st stag ages es,, as sh sho own fo forr St Stag ages es 11 an and d 14 in Figures Figures 8.47 8.47(a (a)) an and d (b (b). ). Wh When en reducing strength, failure is most likely at the end of the stage when the strength is at its minimum value, so plotting the change in displacement in the last increment of the stage is more likely to highlight any failure mechanisms and omits the displacements occurring earl ea rlie ierr in th thee st stag age. e. Th Thee re rela lati tiv ve ma magn gnit itud udee of th thee in incr crem emen enta tall di disp spla lacem cemen ents ts (a (ass represented by the size of the arrows) at different nodes is more important than the actual magnitude of each nodal displacement in this case. Failure mechanisms are often indicated by a sudden change in magnitude across a shear plane. In these cases, no failure was evident. However, in the plots shown in Figures 8.47(c) and (d) at the same 257 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.


Figure 8.47 Vectors of incremental displacement at end of strength-reduction stages: (a) Layer 2 – no failure; (b) Layer 3 – no failure; (c) Layer 2 (inadequate consolidation time) – failure; (d) Layer 3 (inadequate consolidation time) – failure

(a)

(b)

(c)

(d)


Examples

Figure 8.48 Outpu Outputt of Node A displ displacem acement ent due to str strength ength reduction reduction (with inadequate inadequate consolidation times) 100 % : n 80 o i s s e r g 60 o r p e g a 40 t s s i s y l a 20 n A

Layer 1 Layer 2 Layer 3

0 0

10 20 30 Node A displacement change: mm

40

stagess bu stage butt wit with h ina inadeq dequa uate te int interm ermedi ediat atee con consol solida idatio tion n tim times, es, the sud sudden den cha chang ngee in displacement magnitude across each shear plane is clearly evident. The Node A displacements from the same analysis with inadequate intermediate consolidation times are also plotted in Figure 8.48 for comparison with those in Figure 8.46. The intermediate consolidation times were reduced to 60 and 50 days following placement of Layers 1 and 2, respectively. The strength-reduction stage following placement of Layer 2 reached 94% completion before failing to converge, while the stage following placement of Layer 3 reached only 90% completion. Particularly when using advanced constitutive models, stress paths should be checked for their response during strength reduction. This is because strength reduction can have unex un expe pect cted ed an and d po pote tent ntia iall lly y no nonn-co cons nser erva vati tiv ve ef effe fect ctss on co comp mple lex x mo mode dels ls.. In th this is example, because a robust strength-reduction procedure was not available in the software for the MCC model, the shear strength was factored in one step and the software’s stress point algorithm left to correct any stress states lying outside a yield surface. The stress path of deviatoric stress q stress q against average normal effective stress p stress p ′ at Integration Point Z (Figure 8.38) from the analysis with adequate intermediate consolidation times is shown in Figure in Figure 8.49. 8.49. The stress path during the ULS stages, shown in black, appeared to be credible for the decrease of strength with no unexpected stress path excursions which gave more confidence in the strength-reduction operation. Stress paths at other integration points in the vicinity of credible failure mechanisms in the clay foundation were also checked in a similar fashion.

8.4. 8. 4.5 5

Val aliida dati tio on

The total settlement of the original ground surface beneath the centre of the embankment was estimated using Terzaghi’s one-dimensional consolidation theory. Since the average thickness (16 m) of the clay layer was significantly less than the width of the embankment (40 m), the stress change in the clay was assumed uniform and equal to the surface surcharge imposed by the embankment. 259 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.


Figure 8.49 Stress path at Integration Point Z 25

Consolidation

20

Layer 3 construction Layer 2 construction

15 a P k :

Consolidation

q

10

Layer 1 construction

Consolidation

5

0 0

10

20

30

40

50

p : kPa ′

With a coefficient of compressibility m compressibility m v of 0.9 MPa − 1, which was considered appropriate for the stress level in the centre of the clay layer, a consolidation settlement of 0.46 m under Layers 1 and 2 (average surcharge 32 kPa) was obtained. Under all three layers, the average surcharge of 60 kPa gave a consolidation settlement of 0.86 m. Typically in sof softt cla clays ys,, con consol solida idatio tion n sett settlem lement ent ac accou counts nts for ab about out 90% of tot total al sett settlem lement ent.. Taking this into account, the total settlement became about 0.51 m and 0.96 m under Layer 1 + 2 and all three layers, respectively. The settlement output for Node C at the original ground surface beneath the centre of the embankment shown in Figure 8.45 with large deformation formulation reached 0.80 m under all three layers. Extrapolating the curve after placement of Layer 2 suggests that the total settlement would have been about 0.40 m. These values are about 80% of those obtained from one-dimensional consolidation theory which, given the assumptions and conservatism of this simplified method, would be about the level expected, and so provided more confidence in the outputs of settlement from the FE model. In fact, the values obtained from one-dimensional consolidation theory were more similar to the outputs of settlement from the FE analysis with small deformation formulation (Figure 8.44). By making the reasonable assumption that drainage from the clay layer occurred only in the horizontal direction toward the vertical drains, it was possible to estimate the consolidation time of the clay foundation simply from the Kjellman (1948) equation (1948) equation for consolidation by radial drainage into vertical drains (Equation 8.14). D2 D 1 t= ln − 0.75 ln  d 8ch 1 − U





(8.14)

where t = time, ch = horizontal coefficient of consolidation, D = zone of influence of  = average degree of consolidation. drain, d = equivalent diameter of drain and U 260 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.

Examples

For the arrangement of vertical drains in this example, D was 2.4 m and d was was 0.06 m. An average c average ch value of 15 m 2/year was estimated based on laboratory test data. For 95% consolidation, Equation 8.14 gave a consolidation time t of 154 days. Assuming full consol con solida idatio tion n wa wass ac achie hieve ved d whe when n ex exces cesss po pore re pr press essur uree dis dissip sipat ated ed bel below ow 1 kP kPa, a, the then n 95% consolidation occurred at a settlement of Node C of 0.76 m which corresponded with a total consolidation time of about 320 days (Figure 8.45). Taking into account the intermediate consolidation periods, it was estimated from Figure 8.45 that a theoretical instantaneous loading by all three embankment layers would have taken about 180 days to re reac ach h 95% con consol solida idatio tion. n. Sin Since ce the FE mod model el too took k int into o ac accou count nt the dec decre reasi asing ng permeability of the clay due to consolidation, it was considered reasonable that the FE model predicted a slightly longer consolidation period. Although these validation exercises gave more confidence in the accuracy of the FE analysis outputs, continuous site monitoring of embankment settlement should still be recommended due to the residual uncertainty and the safety implications of judging the necessary consolidation times between placing layers of fill. Site monitoring could also bring the economic benefit of hastening construction times if consolidation of the clay foundation were recorded to occur more rapidly than predicted by the FE model. REFERENCES

Bellotti R, Ghionna V, Jamiolkowski M, Robertson PK and Peterson RW (1989) Interpreta pr etatio tion n of mod moduli uli fro from m sel self-b f-bori oring ng pr press essur ureme emeter ter tes tests ts in san sand. d. Ge Ge´ ´ otechnique otechnique 39(2): 269–292. Benz Be nz T, Ve Verm rmee eerr PA an and d Sc Schw hwab ab R (20 (2009 09)) A small small-str -strain ain ov overla erlay y model model.. International Journal for Numerical Methods in Geomechanics 33(1): 25–44. Clough GW and O’Rourke TD (1990) Constru Construction ction indu induced ced mov movements ements of insitu walls. Proceedings ASCE Conference on Design and Performance of Earth Retaining Structures, Structures , Cornell, ASCE Pub. no. 25, pp. 439–470. Duncan JM and Chang YC (1970) Nonlinear analysis of stress and strain in soils. ASCE SM5 96: 1629–1653. Duncan JM, Byrne PM, Wang KS and Mabry P (1980) Strength, Strength, stress–st stress–strain rain and bulk modulus parameters for finite element analysis of stresses and movements in soil masses. masses . Geotechnical Engineering Research Report No. UCB/GT/80-01, University of California, Berkeley, CA. Fraser RA and Wardle LJ (1976) Numerical analysis of rectangular rafts on layered foundations. Ge Ge´ ´ otechnique otechnique 26(4): 613–630. Hems He msle ley y JA (1 (199 998) 8) Elast Elastic ic Ana Analy lysis sis of Ra Raft ft Fo Found undat ation ionss, 1s 1stt ed edn. n. Th Thom omas as Tel elfo ford rd,, London. Hird Hir d CC, Pyrah Pyrah IC, Russell Russell D and Cinicio Cinicioglu glu F (19 (1995) 95) Modelli Modelling ng the effe effect ct of ve verti rtical cal drain dr ainss in two two-di -dimen mensio sional nal fini finite te ele elemen mentt ana analy lyses ses of emb embank ankmen ments ts on sof softt gro ground und.. Canadian Geotechnical Journal 32(5) 32(5): 795–807. Houlsby GT, Wroth CP and Clarke BG (1986) Analysis of the unloading of a pressuremeter in sand. Proceedings of the 2nd International Symposium on Pressuremeter and its Marine Applications. Applications. ASTM, SPT950, 245–262. Hughes JMO, Wroth CP and Windle D (1977) Pressuremeter tests in sands. Ge Ge´ ´ otechnique otechnique 27(4): 455–477. 261 Downloaded by [ University College London] on [16/11/16]. Copyright © ICE Publishing, all rights reserved.


Jardine RJ, Potts DM, Fourie AB and Burland JB (1986) Studies of the influence of nonlinear stres stress–str s–strain ain chara characteristics cteristics in soil-st soil-structure ructure interaction. Ge Ge´ ´ otechnique otechnique 36(3): 377– 396. Kjellman W (1948) Consolidation (1948) Consolidation of fine-grained soils by drain wells. Transactions ASCE 113: Contribution to the discussion. Kondner RL (1963) Hyperbolic (1963) Hyperbolic stress–strain response: cohesive soils. ASCE SM1 82: 115– 143. Muir Wood D (1990) Stain-dependent (1990) Stain-dependent moduli and pressuremeter tests. Ge Ge´ ´ otechnique otechnique 40(3): 509–512. Whittle RW (1999) Using non-linear elasticity to obtain the engineering properties of clay – a new solution for the self boring pressu pressuremeter. remeter. Ground Engineering 32(5): 30–34. Zdravk Zdr avkovi ovicc´ L, Pot Potts ts DM and High Hightt DW (2002 (2002)) The effect of strength anisotropy on the behaviour of embankments on soft ground. Ge Ge´ ´ otechnique otechnique 52(6): 447–457.

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Geotechnical Finite Element Analysis ISBN 978-0-7277-6087-6 978-0-7277-6087-6 ICE Publishing: All rights reserved http://dx.doi.org/10.1680/g http://dx.doi.org/10.1680/gfea.60876.263 fea.60876.263

Index Page locators in italics refer to figures separate from the corresponding text. 2D axisymmetric models, concepts 9, 11 2D continuum elements elements 127, 130–136 2D vs. 3D models 7 2D plane strain models 7–10, 143–146 3D continuum elements elements 127, 130–132 3s rule 193–194 accuracy 183–197 assessment 188–191 initial checks 189 limit states 164–166 modelling elements 131 observational method 196–197 output comparisons 189–191 parameter parameter acquisition acquisition 85–87, 91–98, 186 parametric studies 192–196 reliability testing 166–167 responsibilities 183–184 sensitivity sensitivity analysis 167, 192 sources of error 185–188 acquisition see s ee parameter acquisition active pore pressure 108 analysis analysis planning planning 1–7 aims 2 design integration 5–7 information gathering 2–4 software packages 4–5 utility 1–2 analytical analytical solutions solutions 190 anisotropy consolidation tests 69 derivation 88–89 geometrical 148–150 linear elasticity 38, 48 rocks 48 soils 33, 34 , 38, 46–47 stiffness tests 68–69, 80–81

approximations and error potential 186 assessment of accuracy 188–191 associated flow 42 ‘at rest’ pore pressure 106–107 axes of symmetry 12, 16 axisymmetric simulations 2D 9, 11 oedometer tests 92–93 plate load tests 93, 94 pressuremeter tests 93, 95 9 5 structural representations 143–144, 146–148 triaxial cells 92–93 bar elements 125–128 beam elements 126 1 26, 128 bearing resistance 178 bender elements for triaxial cells 64–66 bending moment outputs raft and pile example 214–220 shaft excavation example 237, 238 BIM see building information modelling block sampling, soils 59 bonding, soils 32–33, 97–98 boundaries defining 12–15 examples 202, 203–204, 226, 246 24 6, 247 fixities 15–16, 202, 203–204, 226, 246 , 247 threshold positions 14 boundary boundary effects 12-16, 165 bubble models 45–46 building information modelling (BIM) 4 bulk modulus 115 calculation options, setup 24–26 cap hardening 44–45 case histories 191 case studies 96


Index

characteristic value derivation 86–87, 91 city centres 12 clays embankments 112, 243–261 flat plate dilatometer tests 74, 82 over-consolidated 116–117, 225–243 partially saturated 106 permeability tests 74, 84–85, 253–255 piezocone penetration test 74, 81–82 plate load tests 74 pre-consolidated stress 97 pressuremeter tests 77–79 seismic testing 74, 80–81 shaft excavation procedure 225–243 shear strength 98 standard penetration tests 82 tunnelling models 143 closed-form solutions 190 closed hydraulic boundaries 119 ‘closed’ piezometers 63 coefficient coefficient of variation variation 194 cofferdam hydraulic boundaries 121 cohesion, derivation 91 comparisons for validation case histories 191 known solutions 189–191 numerical analysis methods 191 site monitoring data 191 concrete 24, 51 , 142–143, 152–155, 181 cone pressuremeters (CPMT) 77 consolidation coupled analysis 122–123 embankment construction example 243–261 groundwater effects 107–111, 115, 120–123 limit states 165 low-permeability soils 120–122 oedometer tests 72 parameters 27 pore pressure 107–111 Terzaghi’s one-dimensional theory 259–261 triaxial cell tests 66–68 constant head tests 83, 84 constant rate of strain (CRS) oedometers 72 constitutive models 29–53 anisotropy 33–34, 38, 46–47 applications 48–51 appropriateness 30–31, 49–51 choice in setup 26 concepts 29 creep 34, 47

destructuration 46 elasticity 36–39 embankment construction 252–253 errors errors 186 failure surfaces 42–43 flow rules 42 ground behaviour 31–36, 42–43 hypoplasticity 47 limit states 166 plasticity 39–47 principal stresses 33 raft and pile example 206 rocks 35–37, 47–48 selection effects 29–30 soil behaviour behaviour 31–34 types 36–48 yield surfaces 39–42 construction stages consolidation analysis 120–123 dual factoring 173–175 embankment construction 249–252 groundwater analyses 109–113 limit states 165 raft and pile example 205–206 setup 21–24 shaft excavation example 228–231 simplification errors 187–188 strength reduction 174–179, 181 continuum elements connection 150–151 propertie propertiess 127, 130–136 raft and pile model 202 continuum continuum models, rocks 36–37, 47–48 conventional methodologies versus FE analysis analysis 1–2 Coulomb Coulomb friction criterion 135, 213 coupled consolidation analysis 122–123, 165 coupled groundwater flows 118 CPMT see cone pressuremeters CPTu see s ee piezocone penetration tests cracking, concrete 154 creep 34, 47, 68, 75, 154–155 CRS see constant rate of strain cubic strain elements, concepts 17 cut slopes, constitutive models 49 deep tunnels 15 degrees degrees of freedom freedom 125–130, 131–132 derivation anisotropy 88–89


Index

characteristic values 86–87, 91 cohesion 91 dilation angles 89 initial state parameters 89–90 intermediate principal stress 87–88 parameters 85–91 permeability 85 Poisson’s ratio 90–91 pre-consolidation stress 89–90 stress ratio 89–90, 97 useful equations 97–99 design analysis analysis planning planning 5–7 integration 6 design charts 190 design codes 163–182 rock limit states 180 serviceability limit states 163, 164, 167–168 structural limit states 180–181 ultimate limit states 163, 164, 168–181 destructuration 46, 60 deterministic reliability testing 166–167 deviatoric stress 41, 112–113 diaphragm piezometers 63 diaphragm wall installation 22–23 dilation angles 89, 107–108, 166, 177 direct shear tests 72 discontinuities in rocks 35–36, 37 , 47–48 discrete crack approach 154 discretisa discretisation tion of geometry geometry 164, 187 displacement piles constitutive models 50 installation effects 23 output factoring 171–172 distributed load modelling 152 disturbance of soil samples 60–61 DMT see flat plate dilatometers double corebarrels 60 double surface plasticity 44–45 drainage embankment construction 245–247, 249, 260–261 low-permeability soils 112 model setup 27, 120, 244–247 drained, definition 109 drained analysis assumptions 109–113 concepts 109 limit states 165–166 methodology 113

parameter derivation from undrained tests 90 parameters 27 drains drains 120, 244–247, 244–247, 245–246 driven piles constitutive models 50 installation effects 23 output factoring 171–172 Drucker–Prager failure surface 43 dry, granular soils, modelling 105 dual factoring 172–175 dynamic analysis 51 earth walls 137–138 effective stress groundwater effects 114–115, 116–117 sample disturbance 60 strength reduction 179, 238–239 elasticity anisotropic linear 38, 48 constitutive models 36–39 equations 98 linear 38, 155–156, 180–184 materials 152–156, 180–181 non-linear 38–39 structural limit states 180–181 elements choice choice 130 connections 150–151 construction stages 21–24 continuum-type 127 1 27 , 130–136, 150–151 end bearing 131, 134 , 150–151 hierarchy 17 initial stresses 18–21 interface-type 132–136, 204, 227, 236 limit states 164 non-continuum 125–134, 150–151 partial partial factors factors 181 raft and pile model 202–205 self-weight self-weight 131, 133 size 131 stiffness 130, 135 types 125–132 embankments constitutive models 50 construction stages 249–251 example procedure 243–261 hydraulic boundary conditions 121 model model setup 243–252, 245–246 model validation 259–261


Index

embankments (continued ) outputs 255–259 parameter acquisition 252–255 pore pressure 112 strength reduction 248–252, 257–259 test procedures 57 unconfined groundwater flows 118–119 empirical empirical design methods methods 190 empirical relationships, stress ratio 97 end bearing 131, 134 , 150–151 engineering managers 184 engineer’s responsibility 183 equilibrium errors 25 errors management 192–197 potential sources 185–188 Euler–Bernoulli theory 128 evaporation 120 excavations constitutive models 49 hydraulic boundary conditions 121 pore pressure 107 , 112, 121 shaft modelling example 225–243 test procedures 57 excess pore pressure 107–108, 112–113, 243–247 existing structures 3–4 explicit discontinuity modelling 48 extension 33–34 external external load input factoring factoring 168–170 extraction boundaries 119–120 extrusion, 2D plane strain models 9–10 failure surfaces, constitutive models 42–43 falling head tests 83 finite element analysis aims 2 utility 1–2, 163–164 fixed connections 150–151 fixities 15–16, 202, 203–204, 226, 246 , 247 flat plate dilatometers (DMT) 74, 82 flows groundwater analysis 118–120 embankment construction 245–247, 249, 260–261 limit states 166 meshing 17 permeabilit permeability y 85, 253–255 shaft excavation example 239–241 Sichardt’s empirical formula 15

fluid concrete 155 fluid support 23 MERGE foundations constitutive models 50–51 example procedure 199–224 hydraulic boundary conditions 121 input factoring 168–170 modelling 138–139 output factoring 170, 171–173, 174 fractured rocks 35–36, 37 , 47–48, 58, 82 free-draining boundaries 119 friction angles 166 friction friction hardening hardening 45 see also shear hardening full connections 150–151 full-flow penetrometers 81–82 gathering information 2–4 Geological Strength Index (GSI) 48 geometry 2D axisymmetric assumptions 9, 11 2D plane strain assumption 7–10, 143–146 anisotropy 33–34, 148–150 boundaries examples 202, 203–204, 226, 246 24 6, 247 fixities 15–16 location 12–15 discretisation 164, 187 elements 125–136 principal stress 33 rock behaviour 36–37 setup 7–16 simplification detail needed 9, 11–12, 13 embankment construction example 244–247, 246 2 46 errors 187 raft foundation with piles example 200–202 shaft excavation example 225–226 structures 125–152 geosynthetics, modelling 137–138 geotechnical parameters errors errors 186 resistance calculations 177–178 see also parameter acquisition gravelly ground, in situ testing 94–95 gravity gra vity switch-on 20–21, 247 greenfield greenfield conditions conditions 19, 66–68 ground anchors 136–137, 137–138


Index

ground behaviour modelling modelling 31–36, 130–136, 130–136, 156–160 strength factoring 174–179, 181, 238–239 ultimate limit states 163, 164, 168–181 ground improvement 22, 140–141 ground information requirements 3 groundwater 105–123 assumptions 109–113 boundary conditions 119–121 bulk modulus 115 consolidation analysis 115, 120–123 coupled flows 118 drainage 27, 112, 120, 244–247 drained analysis 90, 113 effective stress analysis 114–115, 116–117 errors 188 flows analysis 118–120 embankment construction 245–247, 249, 260–261 limit states 166 meshing meshing 17 permeability 85 shaft excavation example 239–241 Sichardt’s Sichardt’s empirical empirical formula formula 15 initial initial stresses stresses 19 large deformations 26 level 19, 27, 83, 84, 120 limit states 165–166 model rigour 27 pore pressure terms 106–108 pressure measurement 62–64 rates of loading 109–113 sample disturbance 60–61 shaft excavation example 239–241 sources/sinks 120 steady-s steady-state tate flows 19, 106–107, 106–107, 118, 239–241 total stress analysis 115 transient flows 118 unconfined flows 118–119 undrained undrained analysis 113–117, 231–235, 231–235, 239 see also pore water pressure grouting grouting 23–24, 140–141, 140–141, 152–155 GSI see Geological Strength Index hexahedral elements 17 hierarchy hierarchy of element element types 17 highly variable ground testing 94–95 high-pressure dilatometers (HPD) 76 historical information requirements 3

Hoek–Brown model 48 horizontal effective stress 19–20 HPD see high-pressure dilatometers HS Small model 39 hydraulic boundary conditions 119–121, 121 , 230 hydraulic hydraulic triaxial cells 64–66 hyperbolic stress–strain relationships, sand 211, 212 hypoplasticity 47 impermeable boundaries 119 implicit discontinuity modelling 47–48 infiltration boundaries 119–120 information gathering 2–4 see also parameter acquisition infrastructure, information requirements 3–4 initial checks for accuracy 189 initial stresses derivation 89–90 direct specification 20 embankment construction example 247 gravity gra vity switch-on 20–21, 247 limit states 164–165 raft and pile example 205 setup 18–21 shaft excavation example 228 input factoring factoring 168–171, 168–171, 173–174 in situ testing 72–85 concepts 72–73 flat plate dilatometers 74, 82 gravelly ground 94–95 highly variable ground 94–95 parameters obtainable 74 permeabilit permeability y 74, 83–85 piezocone penetration 74, 81–82 plate load tests 74, 82, 93, 94 9 4 pressuremeters 73–80, 75 raft and pile example 206–213 rock rock characte characterisat risation ion 58, 74, 80–82, 84–85 seismic testing 74, 80–81 soil characterisation 56–57 standard penetration tests 74, 82 installation effects 22–24 installation of piezometers 63–64 interface elements 132–136, 204, 227, 236 interface friction 130, 133 interface stresses 132–136 intermediate principal stress 33, 87–88 intrusive investigation 3–4


Index

isotropic consolidation 69 isotropic isotropic hardening 40, 44–45 isotropic linear elastic perfectly plastic models 48 isotropic softening 40 iterations, parameter setup 24 Jaky’s equation 97 K 0 see stress ratio key outputs aims 2 presentation 5–7 see also outputs kinematic kinematic hardening 40, 45–46 Kirchhoff theory 129 known solution comparisons 189–191 Kondner hyperbolic stress–strain relationship 211, 212 2 12

laboratory testing 57–58, 64–72 planning 56–57 direct shear measurement 72 oedometers oedometers 72, 92–93 parameters obtainable 70–71 simulations 92–95 triaxial cells 64–69 Lade model 39, 46 Lagrangian formulations 26 landfill 57 large deformations 25–26 LEPP see linearly elastic, perfectly plastic models level of geometric detail requirements 9, 11–12 level of groundwater 19, 27, 83, 84, 120 limit equilibrium methods 190 limit pressures 73, 75 limit states accuracy 164–166 dual factoring 172–175 input factoring factoring 168–171, 168–171, 173–174 output factoring 169, 170, 171–175 rock 180 serviceability 163, 164, 167–168 strength reduction 174–177, 179, 238–239 ultimate 163, 164, 168–181 linear elasticity concrete and grouting 152–154 soils 38 steel structures 155–156 structural limit states 180–181

linearly elastic, perfectly plastic (LEPP) models 20, 48, 194–196, 247 linear strain elements 17 linear structures 2D plane strain models 9 modelling elements 17, 126–127 , 132 linings of tunnels 143 load (effect) and resistance factoring (LRFA) 171–175 loadings coupled consolidation analysis 122–123 existing structures 4 see also construction stages; initial stresses; structures load reduction method, tunnels 142 local strain measurement 64–66 Lode’s angle 41 low-permeability soils 107 , 112, 120–122 LRFA see load (effect) and resistance factoring Lugeon test 83, 84 managing errors 192–197 Marchetti dilatometers 82 material factoring approach (MFA) 168–171, 173–174 materials, modelling 152–156, 181 Modified Cam Clay (MCC) models 39, 44, 47, 247–248, 247–248, 253–255 mean effective stress, soils 41 membrane elements 126 , 128 Me ´ ´ nard pressuremeters (MPM) 76 meshing meshing 17–18, 25–26 embankment construction example 246 24 6 raft and pile foundation example 202–205 shaft excavation example 227–228 MFA see s ee material factoring approach Mindlin theory 129 MIT-E3 model 47 modelling 2D axisymmetric 9, 11, 143-144, 146-148 boundaries 12–16 choice of elements 130 connections 150–151 constitutive model, choice in setup 26 construction stages 21–24 continuum continuum elements elements 127, 130–136, 150–151 distributed loads 152 dual factoring 172–175 earth/soil earth/soil walls 137–138


Index

embankment construction example 243–261 fixities 15–16, 202, 203–204, 226, 246 , 247 geometrical anisotropy 148–150 geometry 125–152 geosynthetics 137–138 ground improvement 22, 140–141 ground–structure interactions 130–136, 156–160 groundwater parameters 27 initial stresses 18–21 input factoring factoring 168–171, 168–171, 173–174 installation effects 22–24 iterations 24 materials 152–156 noncontinuum elements 125–134, 150–151 output factoring 169, 170, 171–175, 178 parametric studies 192–196 partial partial factor factor input 173–177 piles 140 plane strain methods 7–10, 143–146 raft with piles example 199–224 reliability testing 166–167 retaining wall supports 136–137, 137–138 rigour 185–187 sensitivity sensitivity analysis 167, 192 shaft excavation example 225–243 singularities 152 soil tests 92–93 spread foundations 138–139 strength reduction 174–177, 179, 238–239 structural limit states 180–181 submerged surfaces 27 threshold boundary positions 14 tunnels 141–143 Mohr–Coulomb failure criterion 43 , 47–48, 177, 179 monitoring monitoring data, case studies studies 96 MPM see Me ´ ´ nard pressuremeters multipoint piezometers 64 multi-surface plasticity models 45–46 NGI-ADP model 47 no-flow boundaries 119 non-associated flow 42 non-continuum elements 125–134, 150–151, 202–204 non-equilibrium pore pressure 107–108 non-linear non-linear elasticity elasticity of soils 38–39 non-linear models of concrete 154–155 normally consolidated soils, stress ratio 97

numerical analysis methods 191 observational method 6, 196–197 oedometer oedometerss 72, 92–93 open hydraulic boundaries 119 ‘open’ piezometers 62–63 output factoring 169, 170, 171–175, 178, 237–238 outputs accuracy assessment 188–191 aims 2 element readouts 132 embankment construction 255–259 presentation 5–7 raft and pile example 213–220 shaft excavation example 236–241 sources of error 185–188 over-consolidation clays 116–117, 225–243 excess pore pressure 112–113, 116–117 stress ratio 97 packer test 83, 84 parameter acquisition 55–103 accura accuracy cy 85–87, 91–98, 186 anisotropy 88–89 case studies 96 characteristic values 86–87, 91 common laboratory tests 70–71 databases 96 derivation 85–91 dilation angle 89 direct shear test 72 drained parameters from undrained tests 90 embankment example 252–255 errors 186 groundwater pressure 62–64 initial state parameters 89–90 intermediate principal stress 87–88 laboratory tests 64–72 oedometer tests 72 other sources sources 94–96 permeabilit permeability y 74, 83–85, 253–255 planning 55–58 plausibility plausibility checks 93 Poisson’s ratio derivation 90–91 pressuremeter tests 73–80, 75 axisymmetric simulations 93–95 clay 74, 77–79 cone-type 77


Index

parameter acquisition (continued ) pre-bored 76 rock 74, 80 sands 74, 79–80 self-boring 76–77 quality analysis of samples 61–62 raft and pile example 206–213 sampling sampling 59–62 shaft excavation example 231–236 site characterisation 55–58 site-specific empirical relationships 95–96 in situ tests 72–85 concepts 72–73 flat plate dilatometers 74, 82 obtainable parameters 74 permeabilit permeability y 74, 83–85 piezocone penetration 74, 81–82 plate load tests 74, 82, 93, 94 pressuremeters 73–80 seismic testing 74, 80–81 standard penetration tests 74, 82 test simulations 92–95 triaxial cells 64–71, 231–235 anisotropic consolidation 69 isotropic consolidation 69 permeability 69 reconsolidation 66–68, 231–234 shear stage 68 stiffness measurement 68–69 stress paths 64–66 useful equations 97–99 validation 91–98 parametric studies 167 partial factors dual approach 172–175 input factoring factoring 168–171, 168–171, 173–174 materials 181 models 173–177 output factoring 169, 170, 171–175, 178 undrained shear strength 178–179 values 175 partially drained conditions 105–106, 121–122 PBP see pre-bored pressuremeters perfect plasticity 40, 44 permeability elements 132 in situ testing 74, 83–85 triaxial cell tests 69, 253–255 value derivation derivation 85

permeable boundaries 119 piezocone penetration tests (CPTu) 74, 81–82 piles constitutive models 50 installation effects 22–23 modelling 140, 155 output factoring 171–172 raft foundation example 199–224 setup 140, 155, 200–206, 201, 203–204 pinned connections 150–151 plane strain models 7–10, 143–146 planning, parameter acquisition 55–58 plasticity idealised behaviours 40 materials materials 155–156, 181 soils anisotropic strength 46–47 constitutive models 39–47 creep 34, 47, 68, 75 destructuration 46 failure surfaces 42–43 flow rules 42 hypoplasticity 47 stress-dependent strength 46 types 40, 44–46 yield surfaces 39–42 steel structures 155–156 plate elements, properties 127 1 27 , 128–130 plate load tests (PLT) 74, 82, 93, 94 plausibility checks, parameter acquisition 93 PLT see plate load tests Poisson’s ratio, derivation 90–91 pore pressure consolidation 107–111, 120–123 embankments 112 excavations 107 , 112 groundwater simulation 120 initial stresses stresses 19 large deformations 26 limit states 166 measurement 62–64 parameters 27 terms 106–108 see also groundwater pre-bored pressuremeters (PBP) 76 precipitation 120 pre-consolidation stress derivation 89–90 prescribed heads 120 presentation of key outputs 5–7


Index

pressuremeter tests (PMT) 73–80, 75 7 5 axisymmetric simulations 93, 95 clay 74, 77–79 cone-type 77 pre-bored 76 raft and pile example 206–213 rock 74, 80 sands 74, 79–80 self-boring 76–77 pressure wave velocity 81 principal stresses 33 , 39–43 probabilistic reliability testing 167 proposed structures, requirements 4 pumping tests 83, 84 quadrilateral elements 17 quality evaluation, samples 61–62 raft foundations with settlement-reducing piles example 199–224 constitutive model selection 206 outputs 213–220 parameter acquisition 206–213 setup 200–206, 201 , 203–204 validation 220–224 rafts, modelling 155 rates of loading 109–113 reconsolidation 66–68, 231–234 reinforcement bars 154 reliability testing 166–167 resistance calculations 177–178 responsibility for model accuracy 183–184 retaining walls constitutive models 49 input factoring 168–170 output factoring 171–172 supports 136–137, 137–138 surface settlement 190–191 rigour of models 185–187 rising head tests 83 robustness 185–186 rocks constitutive models 35–37, 47–48 direct shear tests 72 discontinuity modelling 35–36, 37 , 47–48 Mohr–Coulomb failure criterion 47–48 permeability 74, 84, 85 plate load tests 74, 82 pressuremeter tests 80 seismic tests 81

site characterisation 58 standard penetration tests 82 triaxial cell tests 64 ultimate limit states 180 roller connections 150–151 rotary coring 60, 61 rounded sands, shear strength 98 samples acquisition 59–62 disturbance 60–61 quality evaluation 61–62 testing 64–85 sands flat plate dilatometer tests 74, 82 Kondner hyperbolic stress–strain relationship 211, 212 permeability tests 74, 84 piezocone penetration test 74, 81–82 plate load tests 74 pressuremeter tests 79–80 raft with pile example 199–224 seismic testing 74, 80–81 shear strength 82, 98 standard penetration tests 82 saturation modelling 105–106 SBP see self-boring pressuremeters SCL see sprayed concrete lining SCPT see seismic cone penetration tests secondary compression tests 68 see also creep seepage boundaries 119 seismic cone penetration tests (SCPT) 81 seismic testing 74, 80–81 selection of constitutive models 29–53 self-boring permeameters 83, 84–85 self-boring pressuremeters (SBP) 76–77, 206–213 self-weight, modelling elements 131, 133 sensitivity sensitivity analysis 167, 192 serviceability limit states (SLS) 163, 164, 167–168 settlement, test procedures 57 settlement-reducing piles with raft foundations 199–224 constitutive model selection 206 outputs 213–220 parameter acquisition 206–213 setup 200–206, 201, 203–204 validation 220–224


Index

setup 1–28 analysis analysis planning planning 1–7 analysis analysis stages stages 18–26 boundaries 12–16 calculation options 24–26 constitutive model choice 26 construction stages 21–24 embankment construction 243–252, 245–246 fixities 15–16, 202, 203–204, 226, 246 24 6, 247 geometry 7–16 ground improvement 22 groundwater parameters 27 initial stresses 18–21 installation effects 22–24 large deformations 25–26 meshing 17–18, 25–26 raft foundation with piles 200–206, 201 , 203–204 shaft excavation 225–231, 226–227 submerged surfaces 27 shaft excavation, example procedure 225–243 model model setup 225–231, 226–227 outputs 236–241 parameter acquisition 231–236 validation 241–243 shear hardening 45 see also friction hardening shear stage, triaxial cells 68 shear strength changes 31 clays 98 direct tests 72 full-flow penetrometers 81–82 interfaces 135 partial partial factors factors 178–179 pressuremeter tests 75 raft and pile example 206–208 rocks 64, 72 sands 82, 98 standard penetration tests 82 undrained 116–117, 178–179 shear wave velocity 62, 65 sheet pile walls 156, 237, 238 shell elements 127 , 128–130, 202–204, 235 Sichardt’s Sichardt’s empirical empirical formula formula 15 3s rule 193–194 simple connections 150–151 simplification geometry

detail needed 9, 11–12, 13 embankment construction example 244–247, 246 errors 187 raft foundation with piles example 200–202 shaft excavation example 225–226 simulations see modelling single surface plasticity plasticity 44 singularities, definition 152 sinks, groundwater 120 site characterisation 55–58, 94–95 site monitoring data 191, 196–197 site-specific empirical relationships 95–96 slabs 24, 51 slip elements elements 132–136 slopes 49 SLS see serviceability limit states smeared crack approach 154 soft soils, full-flow penetrometers 81–82 software packages 4–5, 184 soils anisotropy 33, 34 , 38, 46–47, 65 bonding and structure 32–33 bulk modulus 115 constitutive models 31–34, 36–47 creep 34, 47, 68, 75 databases 96 destructuration 46, 60 deviatoric stress 41, 112–113 elasticity elasticity 36–39, 98 failure surfaces 42–43 hypoplasticity 47 intermediate principal stress 33 isotropic isotropic hardening 40, 44–45 isotropic softening 40 kinematic kinematic hardening 40, 45–46 linear elasticity elasticity 38 mean effective stress 41 non-linear elasticity 38–39 perfect plasticity 40, 44 permeability 69, 74, 83–85, 132 plasticity 34, 39–47, 68, 75 sampling sampling 59–62 seismic seismic tests tests 80–81 site characterisation 55–57 stiffness 32, 38–39, 65, 68–69, 97–98, 176 strain-dependency 33–34, 39, 209–210, 234–235 strength 31–32, 46–47


Index

stress-dependency 32, 39, 211, 234 stress-path dependency 32, 38–39 structure modelling 32–33 test simulations 92–93 triaxial cell tests 64–71 undrained shear strength 116–117, 178–179, 206–208 yield surfaces 39–42 see also individual soil types . . . soil–structure interaction modelling modelling 130–136, 130–136, 156–160 strength reduction 174–177, 179, 238–239 structural limit states 180–181 ultimate limit states 163, 164, 168–181 reinforced soil walls, modelling 137–138 solution schemes 24–25 sources errors 185–188 groundwater 120 sprayed concrete lining (SCL) tunnels 142–143, 155 spread foundations bearing resistance 178 constitutive models 51 input factoring 168–170 modelling 138–139, 139 output factoring 171–172 spring elements elements 125, 126 SPT see standard penetration test standard penetration test (SPT) 74, 82 steady-state pore pressure 19, 106–107, 118, 239–241 steel structures 155–156 step-by-step method, tunnel models 142 step size, parameter setup 24 stiffness factoring 176 parameter validation 97–98 plate load tests 82 in situ seismic testing 74, 80–81 strain-dependency 33–34, 39, 209–210, 234–235 stress-dependency 32, 39, 211, 234 stress-path dependency 32, 38–39 structural connections 151 triaxial cell tests 65–66, 68–69 stochastic approaches, reliability testing 167 strain 2D modelling 7–11, 143–146 measurement 64–68

strain-dependency 33–34, 39, 209–210, 234–235 strain strain hardening hardening 40, 44–45 strain softening 40 stratum boundaries 12 strength rocks 35, 47–48 soils 31–32, 46–47 strength reduction 174–177, 179, 181 embankment construction example 248–252, 257–259 shaft excavation example 238–239 stress-dependency rocks 35, 47–48, 58 soils 32, 39, 46, 211, 234 stress field methods 190 stress paths stiffness stiffness 32, 38–39 test procedures 57 triaxial cell tests 57, 64–66, 253–254 stress ratio (K 0 ) 19–21, 44, 89–90, 97, 164–165 structural connections 150–151 structural limit states 180–181 structures 125–161 axisymmetric models 143–144, 146–148 connections 150–151 distributed load modelling 152 elements 125–136 geometrical anisotropy 148–150 geometry 125–152 ground interactions 130–136, 156–160 materials 152–156 plane strain models 7–10, 143–146 subgrade reaction 159–160, 220–224 submerged surfaces 27 suction measurement 61–62 supports supports to retaining retaining walls 136–137, 136–137, 137–138 tensile strength rocks 35, 47–48 soils 31–32 Terzaghi’s one-dimensional consolidation theory 259–261 tetrahedral elements 17, 202 threshold position boundary effects 14 Timoshenko theory 128 tolerances 186 total pore pressure 107, 108, 120 total stress 115, 179 transient groundwater flows 118


Index

transient pore pressure 107, 108 Tresca failure surface 43 triangular triangular elements 17, 202, 203–204, 227 triaxial cells 64–71, 65 anisotropy 33–34, 68–69 with bender elements 64–66 extension extension tests 33–34, 231–235 isotropic consolidation 69 permeability 69, 253–255 principal stress 33 reconsolidation stage 66–68 rock testing methods 58 shear stage 68 simulations 92–93, 232–233 tube sampling 59–61 tunnel boring machines (TBM) 141–143 tunnels 8 , 49 , 141–143, 155 tweaks 185 ultimate limit states (ULS) 163, 164, 168–181 dual factoring 172–175 input factoring factoring 168–171, 168–171, 173–174 output factoring 169, 170, 171–175, 178 rock 180 strength reduction 174–177, 179, 238–239 unconfined groundwater flows 118–119 undrained, definition 109 undrained analysis assumptions 109–113 concepts 109 drained parameter derivation 90 limit states 165–166

methodology 113–117 parameters 27 shaft excavation example 239–241 undrained shear strength 116–117, 178–179, 231–235, 239 validation acquired parameters 91–98 concepts 188 embankment construction example 259–261 output comparisons 188–191 parametric studies 192–196 raft and pile example 220–224 sensitivity sensitivity analysis 167, 192 shaft excavation example 241–243 values, partial factors 175 variable head tests 83 verification 183–184, 188–191 see also validation vertical boundary fixities 15–16 vertical effective stress 19 void ratio changes 61, 85 volume loss control method 142 von Mises failure surface 43 water see groundwater wireline wireline drilling drilling 60 wished in place 23 yield limits 181 yield surfaces 39–42 Young’s modulus 125


Geotechnical Finite Element Analysis

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