Geotechnical Earthquake Engineering-Géotechnique Symposium in Print 2015

Geotechnical Earthquake Engineering Géotechnique Symposium in Print 2015 Edited by

Stuart Haigh

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Géotechnique Advisory Panel Sub-Committee for the Symposium in Print 2015: Dr Stuart Haigh, University of Cambridge, UK Professor Angelo Amorosi, Politecnico di Bari, Italy Professor George Gazetas, National Technical University of Athens, Greece Dr Erdin Ibraim, Bristol University, UK Professor Boris Jeremic, University of California, Davis, USA Dr Jonathan Knappett, University of Dundee, UK Dr Stavroula Kontoe, Imperial College London, UK Dr William Murphy, University of Leeds, UK Professor Ellen Rathje, University of Texas at Austin, USA Professor Ikuo Towhata, University of Tokyo, Japan Liam Wotherspoon, The University of Auckland, New Zealand Related titles from ICE Publishing: Partial Saturation in Compacted Soils (Géotechnique Symposium in Print 2011). D. Gallipoli (ed). ISBN 978-0-7277-4175-2. Bio- and Chemo-mechanical Processes in Geotechnical Engineering (Géotechnique Symposium in Print 2013). L. Laloui (ed). ISBN 978-0-7277-6053-1. Earthquake Design Practice for Buildings, third edition. E.D. Booth. ISBN 978-0-7277-5794-4. Designers’ Guide to Eurocode 8: Design of buildings for earthquake resistance. M.N. Fardis, E. Carvalho, A.S. Elnashai, E. Faccioli, P. Pinto, A. Plumier. ISBN 978-0-7277-3348-1. Earthworks: A guide, second edition. P Nowak, P. Gilbert. ISBN 978-0-7277-5735-7. ICE Manual of Geotechnical Engineering (2 volumes). J. Burland, T. Chapman, H. Skinner, M.J. Brown (eds). ISBN 978-0-7277-3652-9. Core Principles of Soil Mechanics. S.K. Shukla. ISBN 978-0-7277-5847-7. ISBN 978-0-7277-6149-1 © Thomas Telford Limited 2016 Papers extracted from Géotechnique © Authors and Institution of Civil Engineers All rights, including translation, reserved. Except as permitted by the Copyright, Designs and Patents Act 1988, no part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying or otherwise, without the prior written permission of the Publishing Director, ICE Publishing, 1 Great George Street, London SW1P 3AA. This book is published on the understanding that the authors are solely responsible for the statements made and the opinions expressed in it and that its publication does not necessarily imply that such statements and/or opinions are or reflect the views or opinions of the publishers. While every effort has been made to ensure that the statements made and the opinions expressed in this publication provide a safe and accurate guide, no liability or responsibility can be accepted in this respect by the authors or publishers. Commissioning Editor: Laura Marriott Production Editor: Rebecca Taylor Market Development Executive: Elizabeth Hobson Typeset by Manila Typesetting Company Printed and bound in Great Britain by TJ International Ltd, Padstow


Preface Damage to buildings and infrastructure due to geotechnical failures has been observed in all the major earthquakes of the last half century, causing many thousands of deaths and many billions of dollars of economic damage. Since the 1964 Niigata earthquake, this has resulted in a substantial research effort to understand soil behaviour under cyclic loading, to predict the onset of damaging phenomena such as soil liquefaction and to design foundations and geotechnical systems to survive earthquake loading. While this research effort has continued for the last fifty years, great advances have recently been made using state of the art laboratory testing, dynamic centrifuge modelling and high quality field investigations of system performance during earthquakes such as those in Christchurch, New Zealand in 2010 and 2011 and the Tohoku earthquake in Japan in 2011. The ability of infrastructure to continue to perform post-earthquake is of particular importance in allowing disaster mitigation efforts to rapidly relieve suffering in the affected areas. It was also shown to play a major role in resisting multi-hazards during the Tohoku earthquake, after which coastal defences already damaged by earthquake shaking were required to resist tsunami loading. The response to this symposium in print was substantial with abstracts being submitted by authors throughout the world. Following the standard review process of Géotechnique, ten papers have been accepted for publication in this issue of the journal, with several further papers missing the deadline to be ready for publication according to the planned schedule. These remaining papers have been moved to the regular publication process of the journal and will appear in future issues. The papers selected for this symposium cover a wide range of topics, from liquefaction prediction based on site investigation to the interaction between buildings owing to coupling through the soil between them. Despite the wide range of subjects, these

papers all share the similarity of advancing the current state of the art in geotechnical earthquake engineering research. This issue was accompanied by a full-day symposium held on 15th June 2015 at the Institution of Civil Engineers (ICE) in London. The symposium included two sessions on field behaviour of soils during earthquakes and model testing of geotechnical systems during earthquakes. The ten papers were all presented by their authors on the day, enabling a wide-ranging discussion to take place around current issues in geotechnical earthquake engineering. The symposium provided an excellent opportunity to discuss the current state of the art in geotechnical earthquake engineering and future opportunities in both research and practice. This volume also contains three papers published in Géotechnique in 2014 on geotechnical earthquake engineering. These papers, together with those presented during the Symposium in Print are representative of the high quality work on this subject produced by researchers from around the world and published in Géotechnique. It was my great pleasure to act as chair of the Géotechnique advisory panel subcommittee and to have been guest editor of this issue of Géotechnique. I wish to thank the other members of the subcommittee who have assisted me over the last 18 months for their enormous amount of effort in making this issue and event occur. I would also like to thank all those who have reviewed papers for their thorough and timely reviews. Finally, I would like to thank Craig Schaper, Journals Editor at ICE Publishing, for his great efforts in efficiently organising this issue as well as for his ongoing efforts in making Géotechnique a success.


Stuart Haigh University of Cambridge, UK August 2015


Contents

Preface iii Session 1: Field behaviour of soils during earthquakes Assessment of CPT-based methods for liquefaction evaluation in a Liquefaction Potential Index (LPI) framework B. W. Maurer, R. A. Green, M. Cubrinovski and B. A. Bradley

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Correlation between liquefaction resistance and shear wave velocity of granular soils: a micromechanical perspective X. M. Xu, D. S. Ling, Y. P. Cheng and Y. M. Chen

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An interpretation of the seismic behaviour of reinforced-earth retaining structures L. Masini, L. Callisto and S. Rampello

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Cyclic and dynamic behaviour of a soft pyroclastic rock L. Verrucci, G. Lanzo, P. Tommasi and T. Rotonda

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Empirical predictive relationship for seismic lateral displacement of slopes: models for stable continental and active crustal regions J. Lee and R. A. Green

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Session 2: Model testing of geotechnical systems during earthquakes Dynamic response of flexible square tunnels: centrifuge testing and validation of existing design methodologies G. Tsinidis, K. Pitilakis, G. Madabhushi and C. Heron Influence of initial stress distribution on liquefaction-induced settlement of shallow foundations D. Bertalot and A. J. Brennan

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89

Seismic structure–soil–structure interaction between pairs of adjacent building structures J. A. Knappett, P. Madden and K. Caucis

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A new macro-element model encapsulating the dynamic moment–rotation behaviour of raft foundations C. M. Heron, S. K. Haigh and S. P. G. Madabhushi

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Importance of seismic site response and soil–structure interaction in the dynamic behaviour of a tall building founded on piles E. Bilotta, L. De Sanctis, R. Di Laora, A. D’Onofrio, F. Silvestri

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Related content Some remarks on the seismic behaviour of embedded cantilevered retaining walls R. Conti, G. M. B. Viggiani, and F. Burali D’Arezzo

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Revisiting Nigawa landslide of the 1995 Kobe earthquake H. Ling, H. I. Ling and T. Kawabata

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Static and cyclic rocking on sand: centrifuge versus reduced-scale 1g experiments P. Kokkali, I. Anastasopoulos, T. Abdoun and G. Gazetas

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Maurer, B. W. et al. (2015). Geótechnique 65, No. 5, 328–336 [http://dx.doi.org/10.1680/geot.SIP.15.P.007]

Assessment of CPT-based methods for liquefaction evaluation in a Liquefaction Potential Index (LPI) framework B. W. M AU R E R � , R . A . G R E E N �, M . C U B R I N OV S K I † a n d B. A . B R A D L E Y †

In practice, several competing liquefaction evaluation procedures (LEPs) are used to compute factors of safety against soil liquefaction, often for use within a liquefaction potential index (LPI) framework to assess liquefaction hazard. At present, the influence of the selected LEP on the accuracy of LPI hazard assessment is unknown, and the need for LEP-specific calibrations of the LPI hazard scale has never been thoroughly investigated. Therefore, the aim of this study is to assess the efficacy of three CPT-based LEPs from the literature, operating within the LPI framework, for predicting the severity of liquefaction manifestation. Utilising more than 7000 liquefaction case studies from the 2010–2011 Canterbury (NZ) earthquake sequence, this study found that: (a) the relationship between liquefaction manifestation severity and computed LPI values is LEP-specific; (b) using a calibrated, LEP-specific hazard scale, the performance of the LPI models is essentially equivalent; and (c) the existing LPI framework has inherent limitations, resulting in inconsistent severity predictions against field observations for certain soil profiles, regardless of which LEP is used. It is unlikely that revisions of the LEPs will completely resolve these erroneous assessments. Rather, a revised index which more adequately accounts for the mechanics of liquefaction manifestation is needed. KEYWORDS: earthquakes; liquefaction; sands; seismicity

assumed that the severity of liquefaction manifestation is proportional to the thickness of a liquefied layer, the proximity of the layer to the ground surface, and the amount by which FSliq is less than 1 .0. Given this definition, LPI can range from 0 to a maximum of 100 (i.e. where FSliq is zero over the entire 20 m depth). Analysing SPT data from 55 sites in Japan, Iwasaki et al. (1978) proposed that severe liquefaction should be expected for sites where LPI . 15 but not where LPI , 5. This criterion for liquefaction manifestation, defined by two threshold values of LPI, is subsequently referred to as the Iwasaki criterion. However, in using the LPI framework to assess liquefaction hazard in current practice, it is not always appreciated that the Iwasaki criterion is inherently linked to the LEP that was in common use in Japan in 1978, which differs significantly from those commonly used today. Also, it has been shown that the various LEPs used in today’s practice can result in different FSliq values for the same soil profile and earthquake scenario (e.g. Green et al., 2014), and thus different LPI values. These differences have led to confusion as to which LEP is the most accurate, and whether the Iwasaki criterion is equally effective for all LEPs. The 2010–2011 Canterbury earthquake sequence (CES) resulted in a liquefaction dataset of unprecedented size and quality, presenting a unique opportunity to assess the efficacy of liquefaction analytics (e.g. Cubrinovski & Green, 2010; Cubrinovski et al., 2011; Bradley & Cubrinovski, 2011). Towards this end, Maurer et al. (2014) evaluated LPI during the CES at approximately 1200 sites using the R&W98 CPT-based LEP. Although the Iwasaki criterion was found to be effective in a general sense, LPI hazard assessments were erroneous for a portion of the study area. In practice, several competing LEPs are used to assess liquefaction hazard in an LPI framework (e.g. Sonmez, 2003; Toprak & Holzer, 2003; Baise et al., 2006; Holzer et al., 2006a, 2006b; Lenz & Baise, 2007; Cramer et al., 2008; Hayati & Andrus, 2008; Holzer, 2008; Chung & Rogers, 2011; Kang et al., 2014), but the need for LEP-specific calibration of the LPI hazard scale has never been thoroughly investigated.

INTRODUCTION The objective of this study is to assess the efficacy of three common cone penetration test (CPT)-based liquefaction evaluation procedures (LEPs), operating within a liquefaction potential index (LPI) framework, for predicting the severity of surficial liquefaction manifestation, which is commonly used as a proxy for liquefaction damage potential. Utilising data from the 2010–2011 Canterbury earthquakes, this study investigates the influence of the selected LEP on the accuracy of hazard assessments, and assesses the need for LEPspecific calibrations of the LPI hazard scale. Towards this end, the deterministic LEPs of Robertson & Wride (1998) (R&W98), Moss et al. (2006) (MEA06), and Idriss & Boulanger (2008) (I&B08) are evaluated. While the ‘simplified’ LEP (Seed & Idriss, 1971; Whitman, 1971) is central to most liquefaction hazard assessments, the output from an LEP is not a direct quantification of liquefaction damage potential, but rather is the factor of safety against liquefaction triggering (FSliq) in a soil stratum at depth. Iwasaki et al. (1978) proposed the LPI to link liquefaction triggering at depth to damage potential, where LPI is computed as LPI ¼

20ðm

Fw(z)dz

(1)

0

where F ¼ 1 � FSliq for FSliq < 1 and F ¼ 0 for FSliq . 1; w(z) is a depth weighting function given by w(z) ¼ 10 � 0 .5z; and z is depth in metres below the ground surface. Thus, it is Manuscript received 31 March 2014; revised manuscript accepted 9 January 2015. Published online ahead of print 23 February 2015. Discussion on this paper closes on 1 October 2015, for further details see p. ii. � Department of Civil and Environmental Engineering, Virginia Tech, Blacksburg, Virginia, USA. † Department of Civil and Natural Resources Engineering, University of Canterbury, Christchurch, New Zealand.

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MAURER, GREEN, CUBRINOVSKI AND BRADLEY naissance and using high-resolution aerial and satellite imagery (CGD, 2012b) performed in the days immediately following each of the earthquakes. CPT sites were assigned one of six damage classifications, as described in Table 1, where the classifications describe the predominant damage mechanism and manifestation of liquefaction. For example, some ‘severe liquefaction’ sites also had minor lateral spreading, and likewise, many ‘lateral spreading’ sites also had some amount of liquefaction ejecta present. Of the more than 7000 cases compiled, 48% are cases of ‘no manifestation’, and 52% are cases where manifestations were observed and classified in accordance with Table 1.

Therefore, the objective of this study is to assess the efficacy of the R&W98, MEA06 and I&B08 CPT-based LEPs, operating within the LPI framework, for predicting the severity of surficial liquefaction manifestation. Utilising more than 7000 liquefaction case studies from the CES, this study evaluates the influence of the selected LEP on the accuracy of hazard assessment, and assesses the need for LEP-specific calibrations of the LPI hazard scale. This evaluation is performed using receiver operating characteristic (ROC) analyses, which are commonly used to assess the performance of medical diagnostics (e.g. Zou, 2007). In the following, the high-quality liquefaction case history dataset resulting from the CES is briefly summarised. This is followed by a description of how LPI was computed using three common CPT-based LEPs. An overview of ROC analyses is then presented, which is followed by the analysis of the LPI data. The influence of the LEP on the accuracy of LPI hazard assessment is then discussed.

Estimation of amax (peak ground acceleration) To evaluate FSliq using the three LEPs (i.e. R&W98, MEA06 and I&B08), the peak ground accelerations (PGAs) at the ground surface were computed using the robust procedure discussed in detail by Bradley (2013a) and used by Green et al. (2011, 2014) and Maurer et al. (2014). The Bradley (2013a) procedure combines unconditional PGA distributions estimated by the Bradley (2013b) ground motion prediction equation, recorded PGAs from strong motion stations, and the spatial correlation of intra-event residuals to compute the conditional PGA distribution at sites of interest.

DATA AND METHODOLOGY The 2010–2011 CES began with the moment magnitude (Mw) 7 .1, 4 September 2010 Darfield earthquake and includes up to ten events that are known to have induced liquefaction in the affected region (Quigley et al., 2013). However, most notably, widespread liquefaction was induced by the Darfield earthquake and the Mw 6 .2, 22 February 2011 Christchurch earthquake (e.g. Green et al., 2014). Ground motions from these events were recorded by a dense network of strong motion stations (e.g. Bradley & Cubrinovski, 2011), and due to the extent of liquefaction, the New Zealand Earthquake Commission funded an extensive geotechnical reconnaissance and characterisation programme (Murahidy et al., 2012). The combination of densely recorded ground motions, well-documented liquefaction response and detailed subsurface characterisation comprises the high-quality dataset used for this study. To evaluate the influence of the LEP operating in the LPI framework, a large database of CPT soundings performed across Christchurch and its environs (CGD, 2012a) are analysed in conjunction with liquefaction observations made following the Darfield and Christchurch events.

Estimation of ground-water table depth Given the sensitivity of liquefaction hazard and computed LPI values to GWT depth (e.g. Chung & Rogers, 2011; Maurer et al., 2014), accurate measurement of GWT depth is critical. For this study, GWT depths were sourced from the robust, event-specific regional ground-water models of van Ballegooy et al. (2014). These models, which reflect seasonal and localised fluctuations across the region, were derived in part using monitoring data from a network of ,1000 piezometers and provide a best estimate of GWT depths immediately prior to the Darfield and Christchurch earthquakes. Liquefaction evaluation and LPI The value of FSliq was computed using the deterministic CPT-based LEPs of R&W98, MEA06 and I&B08, where the soil behaviour type index, Ic, was used to identify

CPT soundings This study utilises 3616 CPT soundings performed at sites where the severity of liquefaction manifestation was well documented following both the Darfield and Christchurch earthquakes, resulting in more than 7000 liquefaction case studies. In the process of compiling these case studies, CPT soundings were first rejected from the study as follows. First, CPTs were rejected if the depth of ‘pre-drill’ significantly exceeded the estimated depth of the ground-water table (GWT), a condition arising at sites where buried utilities needed to be safely bypassed before testing could begin. Second, to identify soundings prematurely terminating on shallow gravels, termination depths of CPT soundings were geo-spatially analysed using an Anselin local Moran’s I analysis (Anselin, 1995) and soundings with anomalously shallow termination depths were removed from the study. For a complete discussion of CPT soundings and the geospatial analysis used herein, see Maurer et al. (2014).

Table 1. Liquefaction severity classification criteria (after Green et al., 2014) Classification

Criteria

No manifestation

No surficial liquefaction manifestation or lateral spread cracking Small, isolated liquefaction features; streets had traces of ejecta or wet patches less than a vehicle width; ,5% of ground surface covered by ejecta Groups of liquefaction features; streets had ejecta patches greater than a vehicle width but were still passable; 5–40% of ground surface covered by ejecta Large masses of adjoining liquefaction features, streets impassable owing to liquefaction; .40% of ground surface covered by ejecta Lateral spread cracks were predominant manifestation and damage mechanism, but crack displacements ,200 mm Extensive lateral spreading and/or large open cracks extending across the ground surface with .200 mm crack displacement

Marginal manifestation Moderate manifestation Severe manifestation Lateral spreading

Liquefaction severity Observations of liquefaction and the severity of manifestations were made by the authors for each of the CPT sounding locations following both the Darfield and Christchurch earthquakes. This was accomplished by ground recon-

Severe lateral spreading

4


Frequency

ASSESSMENT OF CPT-BASED METHODS FOR LIQUEFACTION EVALUATION true positives (i.e. liquefaction is observed, as predicted) and non-liquefiable strata; soils having Ic . 2 .6 were considered false positives (i.e. liquefaction is predicted, but is not too plastic to liquefy. Soil unit weights were estimated for observed). Setting the threshold too low will result in each procedure using the method of Robertson & Cabal numerous false positives, which is not without consequences, (2010). For the MEA06 procedure, the stress-reduction coefwhile setting the threshold unduly high will result in many ficient, rd, was computed using the Vs-independent equation false negatives (i.e. liquefaction is observed when it is given in Moss et al. (2006); in addition, the probability of predicted not to occur), which comes with a different set of liquefaction (PL) was set to 0 .15, as proposed by Moss et al. consequences. ROC analyses are particularly valuable for (2006) for deterministic assessments of FSliq. For the I&B08 evaluating the relative efficacy of competing diagnostic tests, procedure, fines content (FC) is required to compute normalindependent of the thresholds used, and for selecting an ised tip resistances (in lieu of FC, R&W98 and MEA06 use optimal threshold for a given diagnostic test. Ic and CPT friction ratio, Rf, respectively); as such, FC In this study, the competing diagnostic tests are the LEPs, values were estimated using both the generic Ic–FC correlaand the index test results are the computed LPI values. tion proposed by Robertson & Wride (1998) and a ChristchAccordingly, in analysing the case histories, true and false urch-soil-specific Ic–FC correlation developed by Robinson positives are scenarios where surficial liquefaction manifestaet al. (2013). Henceforth herein, I&B081 and I&B082 refer tions are predicted, but were and were not observed, respecto the use of the generic and Christchurch-specific Ic–FC tively. Fig. 2 illustrates the relationship among the positive correlations, used in conjunction with the I&B08 procedure. and negative distributions, the selected threshold value and The two Ic–FC correlations are shown in Fig. 1; it can be the corresponding ROC curve, where the ROC curve plots the seen that the Christchurch-specific correlation suggests diftrue positive rate (TPR) and false positive rate (FPR) for ferent Ic–FC trends for Ic , 1 .7 and Ic > 1 .7, where FC is varying threshold values. Fig. 3 illustrates how a ROC curve estimated to be 10 for all Ic < 1 .7. While thin layer correcis used to assess the efficiency of LPI hazard assessment, tions (i.e. adjustments to CPT data in thin strata to account for the influence of over- or underlying soils) are applicable to the LEPs used herein, their use requires judgement, and an automated implementation of these corrections does not No surficial liquefaction manifestation yet exist. Given the quantity of case studies analysed, thin Surficial liquefaction manifestation layer corrections were not performed. FSliq was computed at 1- or 2-cm depth intervals (i.e. the measuring rate of CPT B A C D soundings); LPI was then computed with each of the four 8·75 LPI 5 14 18 LEPs as per equation (1). OVERVIEW OF ROC ANALYSES Receiver operating characteristic analyses are used herein to assess: (a) the efficacy of each LEP for predicting the severity of liquefaction manifestation within the LPI framework; and (b) the need for LEP-specific calibrations of the LPI hazard scale. ROC analyses have been extensively used in assessing medical diagnostic tests in clinical studies (e.g. imaging tests for identifying abnormalities), as well as in machine learning and data-mining research (e.g. Swets et al., 2000; Eng, 2005; Fawcett, 2006; Metz, 2006). In any ROC application, the distributions of ‘positives’ (e.g. liquefaction is observed) and ‘negatives’ (e.g. no liquefaction is observed) overlap when the frequency of the distributions are expressed as a function of index test results (e.g. LPI values). In such cases, threshold values for the index test results are selected considering the relative probabilities of

0

True positive rate, TPR

Apparent fines content, FC: %

60 50 40

R&W98 low-plasticity bound

R&W98 high-plasticity bound 1·50

1·75

22·5

A LPI 5

B LPI 8·75

0·7 Increasing LPI threshold

0·6 0·5

C LPI 14

0·4 0·3

0·1

20

0 1·25

20·0

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10

7·5 10·0 12·5 15·0 17·5 Liquefaction potential index, LPI (a)

0·8

Generic correlation (R&W98)

70

ROC curve

0·9

Christchurch-specific correlation (REA13)

80

5·0

1·0

100 90

2·5

2·00 2·25 2·50 2·75 3·00 Soil behaviour type index, Ic

3·25

0

3·50

0

D LPI 18 0·1

0·2

0·3 0·4 0·5 0·6 0·7 False positive rate, FPR (b)

0·8

0·9

1·0

Fig. 2. ROC analyses: (a) frequency distributions of no surficial liquefaction manifestation and surficial liquefaction manifestations as a function of LPI, with four different threshold LPI values shown; (b) corresponding ROC curve

Fig. 1. Correlations between Ic and apparent FC: Christchurchspecific correlation (Robinson et al., 2013) and generic correlation (Robertson & Wride, 1998)

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MAURER, GREEN, CUBRINOVSKI AND BRADLEY 1·0

1·0

Perfect model

0·9

0·6

ROC curve

s

es

0·5

m do

gu

n Ra

0·4 0·3

pe

Im

p rfo rov rm in an g ce

0·2 0·1 0


0·7

0·1

0·2

0·3 0·4 0·5 0·6 0·7 False positive rate, FPR

0·8

0·6

r fo

r pe

e nc

lin

m

o-

Is

0·5

1·0

I&B081

a

I&B082 0·80

0·4

0·75

0·3

0·70

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0·65

6·0 4·5

0·60 0·20

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MEA06

e

0·7

0 0

R&W98

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Perfect model


0·8

See inset figure

0·9

OOP

0

0·1

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4·0 5·5 0·25

0·30

0·3 0·4 0·5 0·6 0·7 False positive rate, FPR

0·35 0·8

0·40 0·9

1·0

Fig. 4. ROC analysis of LPI model performance in predicting the occurrence of surficial liquefaction manifestation. The optimum threshold LPI values (i.e. OOPs) for each LEP are highlighted in the inset figure

Fig. 3. Illustration of how a ROC curve is used to assess the efficiency of a diagnostic test. The optimum operating point (OOP) indicates the threshold value for which the misprediction rate is minimised, as described in the text

ing is discussed later in this paper). It can be seen in Fig. 4 that, while the four LPI models perform similarly, MEA06 and I&B081 are respectively the least and most efficacious, with AUC ranging from 0 .71 (MEA06) to 0 .78 (I&B081). To place this performance in context, AUCs of 0 .5 and 1 .0, respectively, indicate random guessing and a perfect model. Also, as highlighted in Fig. 4, the optimum threshold LPI values for the R&W98, MEA06, I&B081 and I&B082 models are 4 .0, 5 .5, 6 .0 and 4 .5, respectively. Thus, while the lower Iwasaki criterion (i.e. LPI ¼ 5) is generally appropriate for predicting liquefaction manifestation in Christchurch, the optimum threshold is LEP dependent. The presence of different optimum threshold LPI values for each LEP is not surprising given that different LEPs have been shown to commonly compute notably different FSliq values for the same soil profile (e.g. Green et al., 2014). Although not unexpected, these findings may have important implications for liquefaction hazard assessment, as the risks corresponding to particular LPI values depend on the LEP used to compute LPI. Also of interest is the influence of the Ic–FC correlation used within the I&B08 LEP. As shown in Fig. 1, the Christchurch-specific correlation infers a higher FC than does the generic correlation for all values of Ic, resulting in higher computed FSliq values, and thus lower computed LPI. As a result, the LPI hazard scale computed using I&B082 (i.e. using the Christchurch-specific correlation) is shifted towards lower values relative to the hazard scale computed using I&B081 (i.e. using the generic correlation) such that the median LPI values computed using I&B081 and I&B082 are 7 .2 and 4 .1, respectively. In addition to influencing the LPI hazard scale, the Ic–FC correlation affects model efficacy (i.e. efficiency segregating sites with and without liquefaction manifestations), with I&B081 correctly classifying 3% more cases than I&B082 when operating at their respective OOPs. The slightly weaker performance of I&B082 might be due to the fact that the Robinson et al. (2013) Christchurch-specific Ic–FC correlation was developed using data from along the Avon River only, while the database assessed herein consists of sites distributed throughout Christchurch, although further analysis is needed to evaluate this hypothesis. As research continues in Christchurch, refined region-specific Ic–FC correlations, which

where TPR and FPR are synonymous with ‘true positive probability’ and ‘false positive probability’, respectively. In ROC curve space, random guessing is indicated by a 1:1 line through the origin (i.e. equivalent correct and incorrect predictions), while a perfect model plots as a point at (0,1), indicating the existence of a threshold value which perfectly segregates the dataset (e.g. all sites with manifestation have LPI above the selected threshold; all sites without manifestation have LPI below the same selected threshold). While no single parameter can fully characterise model performance, the area under a ROC curve (AUC) is commonly used for this purpose, where AUC is equivalent to the probability that sites with manifestation have higher computed LPI than sites without manifestation (e.g. Fawcett, 2006). As such, increasing AUC indicates better model performance. The optimum operating point (OOP) is defined herein as the threshold LPI value which minimises the rate of misprediction (i.e. FPR + (1 � TPR), where TPR and FPR are the rates of true and false positives, respectively). As such, contours of the quantity [FPR + (1 � TPR)] represent points of equivalent performance in ROC space. Thus, in plotting the LPI data as ROC curves for each LEP, it is possible to assess both the influence of LEPs on the accuracy of hazard assessments, and the need for LEP-specific calibrations of the LPI hazard scale. RESULTS AND DISCUSSION Utilising more than 7000 combined case studies from the Darfield and Christchurch earthquakes, LPI values were computed using the LEPs of R&W98, MEA06, I&B081 and I&B082. Prediction of liquefaction occurrence In Fig. 4, ROC curves are plotted to evaluate the performance of each LPI model in segregating sites with and without liquefaction manifestation; this initial analysis assesses only whether LPI accurately predicts the occurrence of manifestations and does not yet consider manifestation severity. Included in Fig. 4 are data from both the Darfield and Christchurch earthquakes for all investigation sites, except for those where lateral spreading was the predominant manifestation (the separate assessment of lateral spread-

6


ASSESSMENT OF CPT-BASED METHODS FOR LIQUEFACTION EVALUATION result in different risk levels for the same LPI value, particularly with I&B081.

might improve the efficacy of LPI hazard assessment in Christchurch, are likely to be developed. While the preceding ROC analysis showed that optimum threshold LPI values are LEP dependent, the implications for liquefaction hazard assessment are not intuitively clear. For example, it was shown that for the considered dataset the R&W98 and I&B081 LPI models have optimum threshold LPI values of 4 .0 and 6 .0, respectively, but the potential consequence of failing to account for different optimum thresholds is not easily discerned. To elucidate the significance of these differences, the probability of surficial liquefaction manifestation is computed herein using the Wilson (1927) interval for a binomial proportion. This assessment also allows for application to risk-based frameworks, complementing the prior evaluation of deterministic threshold values. The resulting probabilities are plotted in Fig. 5 where each data point represents one-twentieth of the corresponding dataset (,350 case histories) and is plotted as a function of the median-percentile for each data bin (i.e. 2 .5th-percentile, 7 .5th-percentile, and so on); also shown are third-order polynomial regressions for each LPI model. It can be seen from these regressions that, at an LPI value of 5 .0, the probabilities of liquefaction manifestation corresponding to the I&B081, MEA06, R&W98 and I&B082 LPI models are 0 .44, 0 .53, 0 .58 and 0 .58, respectively. Conversely, using the optimum threshold LPI values found previously, the probabilities corresponding to the respective LPI models are 0 .50, 0 .55, 0 .53 and 0 .55. Thus, the optimum thresholds correspond to roughly the same probability of manifestation, whereas failing to account for the influence of the LEP could

Prediction of liquefaction severity While prediction of the occurrence of surficial manifestation is an important component of liquefaction hazard analysis, the severity of manifestation is of greater consequence to the built environment and is thus of added importance for hazard mapping and engineering design. To investigate the capacity of each LPI model for predicting manifestation severity, additional ROC analyses were performed for each classification of severity in Table 1; the results are summarised in Table 2 in the form of AUC and recommended threshold LPI values. Where the prior ROC analysis assessed each model’s capacity for predicting any surficial manifestation (i.e. having at least marginal severity), the additional analyses assess their ability to predict that manifestations will be of a particular severity (e.g. moderate as opposed to marginal). As mentioned previously, lateral spreading is treated separately in this study, and the ‘marginal’, ‘moderate’ and ‘severe’ classifications refer only to sand-blow manifestations. This distinction is made because lateral spreading is a unique manifestation associated with large permanent ground displacements, and because there are separate criteria for assessing its severity (e.g. Youd et al., 2002), including the ground slope and height of the nearest free face (e.g. river bank), among others. Consequently, although site profiles with thin liquefiable layers may have low LPI values, these sites are susceptible to lateral spreading if located on sloping ground or near rivers. Since the factors pertinent to lateral spreading cases are not considered in the formulation of LPI, such cases should not be used to assess its performance. From Table 2, the following observations are made.

1·0 I&B082 (R2 0·99)

Probability of liquefaction manifestation

0·9

R&W98 (R2 0·99)

(a) Relative trends in model performance, as suggested by AUC, are consistent for each classification of manifestation severity. While the LPI models perform similarly, the I&B081 and MEA06 models are consistently the most and least efficacious, respectively. (b) Unsurprisingly, the models are more efficient in predicting the incidence of liquefaction manifestation than in predicting the severity of manifestation (e.g. distinguishing between marginal and moderate manifestations); nonetheless, the expected severity of manifestation increases with increasing LPI. (c) Differences in optimum threshold LPI values extend throughout the LPI hazard scale, indicating that the utility of the Iwasaki criterion varies among LEPs. (d ) Considering the potential for damage to infrastructure, lateral spreading manifestations have relatively low optimum threshold LPI values. For example, lateral

0·8 0·7 0·6 I&B081 (R2 0·99)

0·5

MEA06(R2 0·98)

0·4 0·3

R&W98

0·2

MEA06

1σ

I&B081

0·1

I&B082

0 0

5

10 LPI

15

20

Fig. 5. Probability of liquefaction manifestation

Table 2. Summary of receiver operator characteristic (ROC) analyses LPI Model

R&W98 MEA06 I&B081 I&B082

All manifestations†

Marginal manifestation†

Moderate manifestation†

Severe manifestation†

Lateral spreading

Severe lateral spreading

OOP‡

AUC§

OOP‡

AUC§

OOP‡

AUC§

OOP‡

AUC§

OOP‡

AUC§

OOP‡

AUC§

4 .0 5 .5 6 .0 4 .5

0 .73 0 .71 0 .78 0 .75

3 .0 5 .0 5 .0 3 .0

0 .68 0 .66 0 .72 0 .70

5 .5 7 .5 9 .0 6 .0

0 .62 0 .60 0 .64 0 .63

10 .5 14 .0 16 .0 11 .0

0 .69 0 .68 0 .69 0 .69

4 .5 5 .0 6 .5 5 .0

0 .83 0 .83 0 .79 0 .86

10 .0 12 .0 8 .0 8 .0

0 .66 0 .64 0 .62 0 .63

Where manifestation severity is characterised as described in Table 1. Excludes sites where lateral spreading was the predominant manifestation, as described in text. ‡ Optimum operating point: recommended optimum threshold LPI value found from ROC analysis. § Area under ROC curve: general index of model efficacy, where higher AUC indicates better performance. †

7


MAURER, GREEN, CUBRINOVSKI AND BRADLEY the LEP-specific calibrations in Table 2. The prediction error (E) is computed using the thresholds assigned to each manifestation category, such that E ¼ LPI � (min or max) of the relevant range. For example, using the Iwasaki criterion, if the computed LPI is 14 for a site with no manifestation, E ¼ 14 � 5 ¼ 9 (where 5 is the maximum of the range of LPI values for no manifestation), whereas if the computed LPI is 6 for a site with severe liquefaction, E ¼ 6 � 15 ¼ �9 (where 15 is the minimum of the range of LPI values for severe liquefaction). Thus, positive errors indicate overpredictions of manifestation severity and, conversely, negative errors indicate under-predictions. While there is no precedent for using a ‘moderate manifestation’ threshold with the Iwasaki criterion, an LPI value of 8 .0 is used herein to facilitate comparisons among the models. Also, in light of the separate criteria for assessing lateral spreads, lateral spreading is assigned a wide range of expected LPI values consistent with any manifestation, independent of spreading severity (i.e. lateral spread sites are only expected to have LPI > the threshold for marginal liquefaction). The distributions of LPI prediction errors are shown for each model in Fig. 7 using both the Iwasaki (Fig. 7(a)) and LEP-specific (Fig. 7(b)) hazard scales. It can be seen in Fig. 7(a) that the distributions of errors among LEPs vary using the Iwasaki criterion, as expected. Because the models have different LPI hazard scales, applying the Iwasaki criterion to each results in dissimilar performance. For example, R&W98 and I&B081 under-predict manifestation severity for 38% and 18% of cases, respectively. Conversely, using the

spreading and marginal sand-blow manifestations have similar OOPs for each respective LPI model (i.e. similar LPI distributions), but the potential for damage to infrastructure is generally much greater with lateral spreading. This illustrates that while LPI may be useful for hazard assessment, the influence of local conditions on the manifestation of liquefaction must also be considered. As such, the damage potential of lateral spreading may not be well estimated by LPI. As was done previously, the probability of manifestation is computed to assess the significance of different optimum thresholds, and to allow for application to risk-based frameworks. Because damage to infrastructure (e.g. settlement of structures, failure of lifelines and cracking of pavements) is more likely a consequence of moderate or severe liquefaction, these cases are used to compute the likelihood of damaging liquefaction due to sand blows, where marginal liquefaction is considered non-damaging. Using the methodology previously discussed, the probability of moderate or severe liquefaction is plotted in Fig. 6 along with third-order polynomial regressions for each LPI model. It can be seen from these regressions that, at an LPI value of 15 .0 (i.e. the upper Iwasaki criterion), the probabilities corresponding to the I&B081, MEA06, R&W98 and I&B082 LPI models are 0 .37, 0 .40, 0 .43 and 0 .47, respectively. Conversely, using the threshold LPI values found previously for severe liquefaction (Table 2), the probabilities corresponding to the respective LPI models are 0 .39, 0 .39, 0 .38 and 0 .40. Thus, the optimum thresholds correspond to roughly the same probability of damaging manifestation, whereas failing to account for the influence of the LEP results in different risk levels. Similarly, the optimum thresholds for moderate liquefaction correspond to the same level of risk (,27%).

50

Prediction rate (7232 cases): %

45

Comparative performance in an applied framework The preceding analyses have suggested the four LPI models may be capable of assessing liquefaction hazard, but that LEP-specific correlations relating LPI values and severity of surficial liquefaction manifestations are required. To compare LEP performance in an applied setting, and to determine whether any LEP is superior for practical intents and purposes, deterministic ‘prediction errors’ are computed for each case history using both the Iwasaki criterion and

25 20

35% 0·2

15

0

0·2

10

I&B08

I&B082

40

0·4

0·3

0·2

40%

30

45

Prediction rate (7232 cases): %

Probability of moderate or severe manifestation

0·5

I&B082

15

20

25

50

R&W98 (R2 0·98)

1

45%

I&B081

0 5 25 20 15 10 5 0 10 Prediction error: LPI units (a)

I&B082 (R2 0·99)

MEA06

35

MEA06

5

0·6 R&W98

40

R&W98

2

MEA06 (R 0·97)

35 30

R&W98 MEA06

45%

I&B081 I&B082

40%

25 20 15

35% 0·2

0

0·2

10 5

0·1

I&B081 (R2 0·98)

0 5 25 20 15 10 5 0 10 Prediction error: LPI units (b)

1σ

0 0

5

10 LPI

15

20

15

20

25

Fig. 7. Distribution of LPI prediction errors, computed from the LPI hazard scales defined by: (a) the Iwasaki criterion; (b) LEPspecific calibrations given in Table 2

Fig. 6. Probability of moderate or severe liquefaction manifestation

8


ASSESSMENT OF CPT-BASED METHODS FOR LIQUEFACTION EVALUATION 35

LEP-specific calibrations of the LPI hazard scale (Fig. 7(b)), the distributions of errors among LEPs are more similar. For example, R&W98 and I&B081 under-predict manifestation severity for 24% and 20% of cases, respectively. In addition, the rate of accurate prediction (i.e. zero error) is improved for each LEP; R&W98, MEA06, I&B081 and I&B082 accurately predict 44%, 42%, 46% and 44% of cases, respectively. These performance trends mirror those of the ROC analyses, which indicated that, although the models performed similarly, I&B081 and MEA06 were respectively the most and least efficacious. However, although accurate predictions of manifestation severity are important, so too is limiting the rate of highly erroneous predictions, which are not necessarily mutually inclusive. While I&B081 has the most zero-error predictions (46%), it also has the most predictions with |E| . 15 (5%). Conversely, MEA06 has the least zero-error predictions (42%), but it also has the fewest predictions with |E| . 15 (2 .5%). Given these inconsistencies, and considering the variety of metrics that might be used to gauge performance, it is difficult to argue that any one LEP is superior in this applied framework. Thus, using the LEP-specific hazard scales, and based on the prediction errors computed herein, the performance of the LPI models is, for practical intents and purposes, equivalent. While minor errors are to be expected in any deterministic analysis, each model produced significant errors with consequences for hazard assessment. For example, even with calibration, |E| exceeded 10 at 9% of sites, on average, for each model (e.g. severe manifestation predicted, but no manifestation observed) and |E| exceeded 5 at 22% of sites, on average, for each model (e.g. no manifestation predicted, but moderate manifestation observed). To determine whether certain models perform better in particular locations, prediction errors from the calibrated R&W98, MEA06 and I&B082 models are plotted against one another in Fig. 8. It can be seen that prediction errors are generally equivalent; in all, the difference in prediction error between any two of the models exceeds 5 for only 12% of investigation sites. Thus, locations of under-, over- and accurate prediction are generally consistent between models. In addition, maps showing the spatial distributions of errors to be very similar in both earthquakes are provided in an electronic supplement to this paper. Thus, some site profiles have very poor predictions, irrespective of the LEP used (note that Maurer et al. (2014) found no correlation between prediction errors and either PGA uncertainty, ground water fluctuation or CPT termination depth). This suggests that LPI has inherent limitations in its formulation, such that the variables influencing surficial manifestation are not adequately accounted for. While liquefaction triggering has garnered significant research and is a subject of frequent debate, the mechanics of liquefaction manifestation have received less attention. This study highlights that triggering and manifestation are two distinct phenomena contributing to liquefaction hazard, and that an improved framework providing clear separation and accounting of the two phenomena is needed. Lastly, the 12% of cases with inconsistent prediction errors between models can be shown to correspond to ‘exceptional’ site profiles. Since the LEP-specific calibrations are based on the entire dataset (i.e. predominant behaviour across Christchurch), predictions for site profiles that diverge from typical conditions may be inconsistent among models. As an example, it can be seen in Figs 8(a) and 8(c) that a number of cases exist where the MEA06 prediction error significantly differs from that of R&W98 and I&B082. One common cause of this discrepancy is cases where relatively thick, potentially liquefiable layers are present at depths greater than ,10 m. For such cases the LEPs can yield divergent FSliq and hence divergent LPI values. However, the

MEA06 prediction error: LPI units

|E | 5 25

15

5

5

Christchurch Darfield

15 15

5 5 15 25 MEA06 prediction error: LPI units (a)

35

I&B082 prediction error: LPI units

35

25

15

5

5


15 15

5 5 15 25 MEA06 prediction error: LPI units (b)

35

I&B082 prediction error: LPI units

35

25

15

5

5


15 15

5 5 15 25 MEA06 prediction error: LPI units (c)

35

Fig. 8. Comparison of LPI model prediction errors at each investigation site, as computed by: (a) R&W98 plotted against MEA06; (b) R&W98 plotted against I&B082; (c) MEA06 plotted against I&B082

LEP-specific calibrations do not account for this divergence because the median cumulative thickness of soil strata predicted to liquefy below 10 m depth for all the sites in the dataset is only 0 .35 m, according to I&B082. This emphasises that assessments and/or calibrations of the LPI hazard scale are a function not only of the selected LEP, but also of the chosen dataset, including the geometry and soil characteristics of site profiles, as well as the amplitude and duration of ground shaking. As such, the applicability of findings derived herein to other datasets is unknown.

9


MAURER, GREEN, CUBRINOVSKI AND BRADLEY Land Information New Zealand (LINZ) for providing the earthquake occurrence data and the Canterbury Geotechnical Database and its sponsor EQC for providing the CPT soundings, lateral spread observations and aerial imagery used in this study. However, any opinions, findings, and conclusions or recommendations expressed in this paper are those of the authors and do not necessarily reflect the views of NSF, ERDC, EQC, NHRP or LINZ.

SUMMARY AND CONCLUSIONS Utilising high-quality case histories from the CES, this study evaluated the performance of the R&W98, MEA06, I&B081 and I&B082 CPT-based LEPs, operating within the LPI framework, for assessing liquefaction hazard. The findings are summarised as follows. (a) For deterministic analyses, the optimum threshold LPI values for assessing liquefaction hazard were unique to the LEP used in the LPI framework; suggested optimum thresholds for the CES dataset are summarised in Table 2. The use of LPI for assessing lateral spread potential is not recommended. (b) Taking these LEP-specific threshold values into account, receiver-operating-characteristic analyses indicated that, while the models performed similarly, the I&B081 and MEA06 models were respectively the most and least efficacious. (c) LPI probability curves were computed to assess the significance of different optimum thresholds, and to allow for application in probabilistic frameworks. The optimum thresholds were shown to correspond to roughly the same probability of manifestation, whereas failing to account for the influence of the LEP (i.e. using the Iwasaki criterion) resulted in different risk levels for the same LPI value. (d ) To compare model performance in a practical setting, deterministic ‘prediction errors’ were computed for each case history. Using the Iwasaki criterion, the distributions of errors among LEPs varied. These distributions became more similar using the LEP-specific hazard scales given in Table 2, which also improved the rate of accurate prediction for all LEPs. (e) Even with calibration, each model had significant prediction errors (e.g. severe manifestation predicted, but no manifestation observed). This suggests that LPI has inherent limitations in its formulation, such that the variables influencing surficial liquefaction manifestation are not adequately accounted for. ( f ) The findings presented in this study are based on a dataset from the CES; the applicability of these findings to other datasets is unknown.

SUPPLEMENTAL DATA The following are available in an electronic supplement: (a) aerial images representative of the liquefaction manifestation severity classes described in Table 1; and (b) map figures showing the spatial distribution of LPI prediction errors for each LPI model, for both the Darfield and Christchurch earthquakes. NOTE Some of the data used in this study were extracted from the Canterbury Geotechnical Database (https://canterbury geotechnicaldatabase.projectorbit.com), which was prepared and/or compiled for the Earthquake Commission (EQC) to assist in assessing insurance claims made under the Earthquake Commission Act 1993 and/or for the Canterbury Geotechnical Database on behalf of the Canterbury Earthquake Recovery Authority (CERA). The source maps and data were not intended for any other purpose. EQC, CERA and their data suppliers, and their engineers, Tonkin & Taylor, have no liability for any use of these maps and data or for the consequences of any person relying on them in any way. NOTATION Ic PL Rf rd Vs w(z) z

In conclusion, the following points can be made. (a) The risk levels corresponding to the Iwasaki criterion varied among LEPs. (b) Using a calibrated, LEP-specific hazard scale, the performance of the LPI models was, for practical intents and purposes, equivalent. (c) The existing LPI framework has inherent limitations such that all LEPs have very poor predictions for certain soil profiles. It is unlikely that revisions of the LEPs will completely resolve these erroneous assessments. Rather, a revised index which more adequately accounts for the mechanics of liquefaction manifestation is needed.

soil behaviour type index probability of liquefaction cone penetration test friction ratio stress reduction coefficient shear wave velocity depth weighting function given by w(z) ¼ 10 � 0 .5z depth below ground surface (m) standard deviation

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Xu, X. M. et al. (2015). Geótechnique 65, No. 5, 337–348 [http://dx.doi.org/10.1680/geot.SIP.15.P.022]

Correlation between liquefaction resistance and shear wave velocity of granular soils: a micromechanical perspective X . M . X U � † ‡ , D. S . L I N G � , Y. P. C H E N G † a n d Y. M . C H E N �

The shear wave velocity method has become an increasingly popular means to evaluate the liquefaction potential of granular soils. Understanding the fundamental mechanism underlying existing empirical or semi-empirical relationships is important for better assessing their reliability. This paper presents a particle-scale study of the correlation between cyclic resistance ratio (CRR) and the shear wave velocity corrected for overburden stress (V s1 ). The discrete-element method was used to simulate a series of undrained stress-controlled cyclic triaxial tests together with shear wave velocity (V s ) measurements. Discrete-element method modelling with various relative densities, confining pressures and micro-parameters was performed under various cyclic stress ratios (CSRs), and the onset of liquefaction was illustrated through both macroscopic and microscopic responses, for example, inferred excess pore-water pressure, mechanical coordination number and redundancy index. The inter-particle friction was identified as the key micro-parameter that governs the liquefaction resistance of granular soils. A micro-scale CRR–V s1 correlation considering two independent micro-parameters, inter-particle friction and particle shear modulus, was then obtained and further validated with the outcomes from three dynamic centrifuge model tests performed on silica sand no. 8. This study demonstrates that the CRR–V s1 correlation is particle specific, thus soil specific, and the particle mechanical properties should be included in the V s -based method for future liquefaction evaluation of granular soils. KEYWORDS: centrifuge modelling; discrete-element modelling; particle-scale behaviour; liquefaction

geotechnical engineers a promising alternative to evaluate the liquefaction resistance of granular soils. This is especially true for sites underlain by soils that are difficult to penetrate or sample (Andrus & Stokoe, 2000; Kayen et al., 2013). The V s -based simplified method has attracted an increasing number of studies, and various ‘boundary curves’ have been relatively well established on the basis of either field data (Robertson et al., 1992; Andrus & Stokoe, 2000; Juang et al., 2001; Andrus et al., 2004; Juang et al., 2005; Kayen et al., 2013), or laboratory studies (Dobry et al., 1981; de Alba et al., 1984; Tokimatsu & Uchida, 1990; Chen et al., 2005; Wang et al., 2006; Zhou & Chen, 2007; Baxter et al., 2008; Zhou et al., 2010; Ahmadi & Paydar, 2014). Note that these V s -based correlations were developed from empirical or semi-empirical evaluation of field observations and laboratory test data following the general format of the simplified procedure by Seed & Idriss (1971). Their fundamental mechanisms are still open to question. First, the shear wave velocity measurements are made at small strains, whereas pore-water pressure build-up and liquefaction are medium-to-high-strain phenomena (Roy et al., 1996). Whether there is a natural link between these two different characterising variables remains unknown. Second, the uniqueness of this correlation for all types of granular soils is also questionable. Although both V s and liquefaction resistance were reported to be similarly influenced by many of the same macroscopic factors (Andrus & Stokoe, 2000), their parametric laws may be quite different. Taking the relative density for an example, previous studies showed that it has a very strong effect on the liquefaction resistance (Seed & Idriss, 1971), while it has a weak effect on V s : This implies that liquefaction resistance may not uniquely correlate with V s for multiple soils (Baxter et al., 2008), as indicated by the weakness of the V s -based correlation reported by Liu & Mitchell (2006). More recently, Kayen et

INTRODUCTION The simplified method pioneered by Seed & Idriss (1971), based on standard penetration test (SPT) data, has become the standard practice for evaluating the liquefaction potential of granular soils (Youd & Idriss, 2001; Youd et al., 2001). Earthquake-related soil liquefaction is discussed in relation to the cyclic stress ratio (CSR) induced by ground shaking at some depth in the ground av amax v0 CSR ¼ ¼ 0.65 (1) rd v90 g v90 where av is the average equivalent uniform cyclic shear stress caused by the earthquake and is assumed to be 0 .65 of the maximum induced stress; g is the acceleration of gravity; v0 and v90 are total and effective vertical overburden pressures, respectively; and rd is stress reduction coefficient to adjust for the flexibility of the soil profile. The key issue is to characterise the capacity of soil to resist liquefaction based on various routinely used field or laboratory techniques (e.g. SPT, cone penetration test (CPT), Becker hammer test (BHT) and shear wave velocity (V s ) measurement). Among them, the shear wave velocity corrected for overburden stress (V s1 ) is considered to offer Manuscript received 4 April 2014; revised manuscript accepted 13 January 2015. Published online ahead of print 26 March 2015. Discussion on this paper closes on 1 October 2015, for further details see p. ii. � Key Laboratory of Soft Soils and Geoenvironmental Engineering of Ministry of Education, Department of Civil Engineering, Zhejiang University, Hangzhou, China. † Department of Civil, Environmental and Geomatic Engineering, University College London, London, UK. ‡ Department of Engineering, University of Cambridge, Cambridge, UK.

13


XU, LING, CHENG AND CHEN al. (2013) reported that around 50 non-liquefied sites out of a global catalogue of 422 case histories were misclassified, with their data mainly located in the lower intensity zone (i.e. cyclic stress ratio, CSR < 0.20). One possible explanation for these observations is that V s -based correlations are soil specific; however, establishing individual relations would be rather costly and time consuming (Tokimatsu & Uchida, 1990; Zhou & Chen, 2007; Baxter et al., 2008; Zhou et al., 2010; Ahmadi & Paydar, 2014). The shear wave velocity is well known as a comprehensive representative metric for characterising the current state of granular soils, which is highly dependent on both macroscopic parameters (e.g. stress state, void ratio or relative density) and microscopic parameters (e.g. particle properties, fabric and coordination number) (Hardin & Richart, 1963; Tatsuoka, 1999; Yimsiri & Soga, 2000; Agnolin & Roux, 2007; Clayton, 2011). The liquefaction resistance in a soil deposit, commonly quantified by the cyclic resistance ratio (CRR), also depends on its current state (Seed & Idriss, 1971; Tokimatsu & Uchida, 1990; Chen et al., 2005; Wang et al., 2006; Baxter et al., 2008; Ahmadi & Paydar, 2014). CRR is generally estimated by performing cyclic triaxial tests on reconstituted samples following a given stress path (e.g. a series of sinusoidal cyclic stress cycles), and is defined as the CSR to cause initial liquefaction after a certain number of cycles (e.g. 15 cycles corresponding to an earthquake magnitude, M w ¼ 7.5). Exploring the underlying fundamental mechanisms of CRR and V s at both macro and micro level is essential to establish their correlation for the purposes of liquefaction evaluation of granular soils. However, it is still a great practical challenge for experimenters to examine and accurately quantify the intricate characteristics of internal soil structure at the micro scale. Numerical simulations using the distinct-element method (DEM) pioneered by Cundall & Strack (1979) can offer some micromechanical insights to understand the mechanical properties of granular soils (Thornton, 2000; Cheng et al., 2004; Soga & O’Sullivan, 2010; O’Sullivan, 2011; Zhao & Guo, 2013). Its applicability to the modelling of the cyclically induced liquefaction behaviour of granular soils shearing at constant volume has been demonstrated (Shafipour & Soroush, 2008). Ng & Dobry (1994) were among the first researchers who studied the responses of both two-dimesional (2D) disc and three-dimensional (3D) sphere assemblies in undrained cyclic simple shear loading conditions. Their results qualitatively agreed with physical tests on sand. Recently, although there have been a number of both 2D (Sitharam, 2003) and 3D (O’Sullivan et al., 2008; Sitharam et al., 2009; Soroush & Ferdowsi, 2011) numerical simulations that have investigated liquefaction phenomenon based on strain-controlled loading, the authors are not aware of any DEM studies that have explored the micromechanics both of liquefaction and of CRR in a stress-controlled manner, and further developed a CRR–V s1 correlation by microscopically measuring the V s of granular soils. Following similar procedures to laboratory tests, this paper presents numerical simulations using a particle-scale DEM investigation of the CRR–V s1 correlation, by conducting a series of undrained stress-controlled cyclic triaxial test simulations, together with V s measurement. DEM specimens with various initial relative density, confining pressure and microparameters are tested under various cyclic stress amplitudes. The onset of liquefaction is illustrated through both macroscopic and microscopic responses. The key micro-parameters that govern the magnitude of CRR are identified, and a micro-scale CRR–V s1 correlation is then proposed and further validated through the results from three dynamic centrifuge model tests with silica sand no. 8.

METHODOLOGY AND APPROACH Sample preparation The DEM simulations were performed using PFC3D (particle flow code in three dimensions) (Itasca, 2008). In this study, around 21 000 polydisperse spherical particles with diameters ranging from 0 .15 mm to 0 .20 mm were randomly generated in a cylindrical region (diameter 4 mm 3 height 8 mm) with rigid frictionless walls. The particle size distribution of an assembly is shown in Fig. 1. The radius expansion method was adopted to facilitate the creation of an initially isotropic sample. Each particle was prescribed with properties including a radius, density, contact stiffness and coefficient of contact friction. The Hertz–Mindlin contact model was employed in this study as it is suitable for simulating pressure-dependent behaviour at small strain (Sadd et al., 1993; Itasca, 2008; Wang & Mok, 2008). The gravitational force was neglected in this analysis. Unless otherwise stated, the parameters used in the model are listed in Table 1. Once the DEM assembly has been generated, a numerical servo-control mechanism specially written for the Hertz– Mindlin contact model was implemented to compress the specimen to reach a desired isotropic stress state. The minimum void ratio (emin ) was obtained by setting a low initial value (e.g. e < 0 .333) with the inter-particle friction set to zero at the assembly generation stage, followed by a number of numerical cycles until the isotropic stress reached 90% of the required stress. The inter-particle friction was then switched to the required contact friction value and maintained until the assembly reached an equilibrium state (the ratio of the mean unbalanced force to the mean contact u c force f =f < 10�3 ) at the desired stress. The maximum void ratio (emax ) was obtained by assigning a large initial value (e.g. e > 0 .905) with an inter-particle friction of 0 .50, and by preventing particle spin when the assembly was initially generated. Particle rotation was then permitted when

Percent finer by weight: %

100

80

60

40

20

0

0·01

0·1 Diameter: mm

1

Fig. 1. Particle size distribution selection for DEM simulations; inset diagram is the initially generated sample of spheres

Table 1. Model parameters used in DEM simulations Parameters Particle shear modulus, Gg : GPa Particle friction coefficient, g Particle Poisson ratio, vg Particle density, rg : kg/m3 Particle diameter, d: mm Model height, H: mm Model diameter, D: mm

14


Value 1 .0 0 .50 0 .20 2 630 0 .15,0 .20 8 4

LIQUEFACTION RESISTANCE AND SHEAR WAVE VELOCITY OF GRANULAR SOILS the isotropic stress level reached 90% of the required stress. For the other relatively loose samples, the inter-particle friction coefficients were adjusted by trial and error during the assembly generation stage to facilitate the sample reaching the desired void ratio. Similar procedures for sample preparation can be found in Salot et al. (2009), Thornton & Zhang (2010) and Gong et al. (2012). In this study, the emax and emin at 100 kPa confining pressure are 0 .809 and 0 .585, respectively. These values are very close to that of the traditional random loose packing (RLP) and random close packing (RCP) for monodisperse spheres, where the emax and emin are 0 .818 and 0 .577 (Song et al., 2008; Silbert, 2010). The relative density (Dr ) of the DEM specimens is expressed as emax � e (2) Dr ¼ emax � emin

Sitharam et al., 2009; Yimsiri & Soga, 2010). The excess pore-water pressure ratio is then defined as U ¼ u= m 9 0, where m 9 0 is the initial mean confining pressure. Initial liquefaction is said to have occurred when the effective stress becomes zero (U ¼ 1.0) owing to the build-up of excess pore-water pressure. Shear wave velocity measurement The measurement of shear wave velocity was implemented using a pair of virtual ‘disc elements’ fully embedded in the sample, as shown in Fig. 2(a). In the DEM model, a group of particles (a circular disc with around 100 particles) in a certain region were selected as a transmitter and another one as a receiver. The excitation wave was created by applying a velocity pulse to the transmitter in the x direction, which would immediately be transferred to neighbouring particles and subsequently propagated through the whole sample. The wave propagation could be visualised by way of the velocity field. Once the disturbance reached the receiver, the average velocity in the same direction would be picked up. Fig. 2(b) plots the normalised velocity–time histories of both the excitation and received waves. The latter is scaled up by five times for better visualisation. The Cartesian coordinate system oxyz, as shown in Fig. 2(a), is used in this paper for all cases analysed in 3D space. The longitudinal direction of the numerical sample is chosen as the z-axis, and the x-axis and the y-axis directions are the radial directions. Note that before performing the shear wave propagation, it is essential to ensure that the sample has reached a further u c equilibrium state (f =f < 10�9 ). To do this, the coefficient of local damping was set to a relatively high level (e.g. 0 .9) with the purpose of dissipating energy more efficiently and saving computational time. After more numerical cycles, the velocities of the particles would eventually become low enough compared to the excitation magnitude. The damping factor was then reduced to zero at the end of this stage for the shear wave analysis. The V s is calculated using the wave travel time (t) and the distance of the travel path (LTR ), in exactly the same way as it is in a laboratory test with bender elements (Clayton, 2011). LTR Vs ¼ (3) t

where e is the void ratio after isotropic consolidation. Undrained stress-controlled cyclic triaxial test After isotropic consolidation, a series of undrained cyclic triaxial tests with various CSRs were simulated in a stresscontrolled manner for each numerical specimen, by applying a number of sinusoidal cyclic stress loadings until initial liquefaction occurred. The velocities of the boundaries were adjusted in such a way that the cyclic deviator stress followed a sinusoidal cyclic stress history while the specimen volume remained constant. The input frequency of the cyclic loading was chosen as 1000 Hz in this study. We performed a comparison among the cyclic-induced boundary responses and backbone curves of another numerical sample (with fewer particles) at various frequencies (1 Hz , 5000 Hz). The results indicated that the cyclic behaviours of the DEM specimen are essentially not sensitive to the input frequency when its value is less than 2000 Hz. This threshold is higher than the experimental findings from a cylindrical sand sample 38 mm in diameter and 78 mm high (Bolton & Wilson, 1989) owing to the much smaller size of the DEM specimen in the present study. These simulations were performed using four workstations (each with 3 .2 GHz Intel CPU and 8– 32 GB memory) and one computer cluster (with eight nodes, each with 4 3 2 .26 GHz Intel CPU and 16 GB memory) over a few years. The equivalent excess pore-water pressure (u) is evaluated under an assumption of fully saturated conditions, by taking the difference between the initial effective confining pressure ( r90 ) at the beginning of shearing and the current effective stress ( r9), that is, u ¼ r90 � r9 (e.g. Ng & Dobry, 1994;

Receiver

Transmitter x y

o (a)

Normalised amplitude

1·0

Normalised shear wave velocity, Vs /Vs0

z

The travel distance of the shear wave component is generally taken as the tip to tip distance between the transmitter and receiver. However, the determination of the

Received wave (5) Excitation wave

0·5 0 0·5 1·0 0·5 0 0·5 1·0

0

5

10

15 20 Time, t: μs (b)

25

30

1·3 1·2

f 125 kHz f 200 kHz

1·1 1·0 0·9 0·8

0·7 1 1010 1 108 1 106 1 104 1 102 1 109 1 107 1 105 1 103 1 101 Ratio of the excitation amplitude to wave length, Adisp /λ (c)

Fig. 2. Measurement of shear wave velocity: (a) half of the DEM model for wave propagation; (b) transmitted wave and received wave; (c) choice of velocity amplitude for shear wave excitation

15


XU, LING, CHENG AND CHEN travel time is much less straightforward (Jovicic et al., 1996; Viana da Fonseca et al., 2009). A number of methods have been commonly used, ranging from the simplest approach based on the immediate observation of the wave traces (e.g. first arrival and peak to peak), to more elaborate signal processing techniques (e.g. cross-correlation analysis, wavelet analysis and phase detection analysis). In this study, the cross-correlation analysis was adopted, owing to its superiority in both determining the travel time and identifying the similarities between two signals (Santamarina & Fratta, 1998). The velocity amplitude used in this paper for the shear wave propagation was selected by comparing the measured V s from the above-mentioned wave propagation, to the actual value V s0 interpreted from a boundary p measurement ffiffiffiffiffiffiffiffiffiffiffi of the small strain shear modulus G0 (V s0 ¼ G0 =r). Fig. 2(c) plots the variation of the normalised shear wave velocity V s =V s0 against the ratio of the excitation amplitude to the wave length Adisp =º: When the excitation amplitude is too high, V s is smaller than V s0 : When Adisp =º reduces to 10�5, then shear wave velocity becomes very close to the boundary measurement (V s =V s0 ¼ 1). In the figure, the value indicated by the single star point (Adisp =º ¼ 10�8 ) is used in this paper. To ensure a linear elastic wave propagation without inducing any frictional work, the authors suggest the ratio of the displacement amplitude of the excitation wave to the mean particle diameter Adisp =d 50 , 10�4 (Xu et al., 2012) or Adisp =º , 10�5 , according to Fig. 2(c).

Cyclic-induced liquefaction: from macro to micro The macroscopic liquefaction phenomena are observed for all numerical simulations in this study. A typical example is presented in Fig. 3, which shows the variation of axial strain and excess pore-water pressure ratio U with the number of cycles (N ) for an isotropically consolidated sample at a confining pressure of 100 kPa (Dr ¼ 50.4%, CSR ¼ 0.125). The inset of this figure illustrates the stress-controlled deviator stress during the cyclic loading. As observed from the figure, with the application of constant cyclic stress, both the cyclic axial strain magnitude and excess pore-water pressure ratio only increase slightly after the first 50 cycles; the axial strain magnitude then increases dramatically when the numerical specimen approaches initial liquefaction after 64 cycles, owing to the development of excess pore-water pressure. In a qualitative sense, this simulation captures the realistic behaviour of sand liquefaction phenomenon observed in laboratory experiments (Seed & Lee, 1966; Zhou & Chen, 2005). To explore the underlying micro-mechanism of liquefaction, the mechanical coordination number (C n ) proposed by Thornton (2000) is adopted herein. It is calculated as an average coordination number, but excludes particles with only one or zero contacts that are not contributing to the stable state of stress. It is expressed as Cn ¼

CSR 0·125 Nl 64

Axial strain, εa: %

1·0

20 1·0

10 0

0·8

10 20

0·5

30

0

0

1

2

3 4 60 61 62 Number of cycles, N

63

64

0·6

65

εa

0·4

0·5 0·2 1·0

Excess pore-water pressure ratio, U

1·5

1·2

30

Deviator stress, σd: kPa

Dr 50·4% σm0 100 kPa

(4)

where N b and N c are the total number of particles and contacts respectively; N b1 and N b0 are the number of particles with only one or no contacts, respectively. Fig. 4 plots the evolution of C n with the number of cycles towards initial liquefaction. Note that the data are taken at the end of each cycle (see the inset of Fig. 3), except that more data points are taken in the last incomplete cycle when the specimen was very close to initial liquefaction. After the first cycle, C n decreases from 4 .83 to 4 .79, and then remains almost constant until the 50th cycle. After that, the number decreases sharply to about 4 .0, which is the minimum requirement for a stable three-dimensional deposit of frictional spherical particles (Edwards, 1998).

RESULTS AND OBSERVATIONS Over 108 numerical samples were cyclically sheared from different initial states (i.e. in terms of relative density, initial confining pressure, particle shear modulus, particle Poisson ratio and inter-particle friction) to initial liquefaction under various CSRs, where the excess pore-water pressure ratio U reaches 1 .0. The onset of liquefaction is illustrated through both macroscopic and microscopic criteria. The key microparameters that govern the magnitude of CRR are identified, and a micro-scale CRR–V s1 correlation is then proposed and further verified. 2·0

2N c � N b1 N b � N b0 � N b1

U

1·5

0

Axial strain Excess pore-water pressure ratio

2·0

0

10

20

30 40 Number of cycles, N

50

60

70

0·2

Fig. 3. Variation of axial strain and excess pore-water pressure ratio with number of cycles for a confining pressure of 100 kPa (Dr 50.4%, CSR ¼ 0.125); inset diagram shows the stresscontrolled cyclic loading

16


LIQUEFACTION RESISTANCE AND SHEAR WAVE VELOCITY OF GRANULAR SOILS 30

Deviator anisotropic coefficients, adr , adn, adt

25

4·8

20

Mechanical coordination number

4·6

Sliding fraction

4·4

15

Dr 50·4%; σm0 100 kPa CSR 0·125; Nl 64

10

4·2

4·0

1·0

Sliding fraction, Sc: %

Mechanical coordination number, Cn

5·0

5

0

10

20

30 40 50 Number of cycles, N

0 70

60

Contact normal anisotropy, ard Normal contact force anisotropy, and

0·8

Tangential contact force anistropy, adt 0·6

0·4

Dr 50·4%; σm0 100 kPa

0·2

and 0·104 adt 0·073

0 0

Fig. 4. Evolution of mechanical coordination number and sliding fraction with the number of cycles for a confining pressure of 100 kPa (Dr 50.4%, CSR 0.125)

ard 0·239

CSR 0·125; Nl 64

10

20

30 40 50 Number of cycles, N

60

can be observed from the figure that all three coefficients ard , and and atd increase (from their initial values of 0 .14, 0 .09 and 0 .03, respectively) dramatically (to 0 .239, 0 .104 and 0 .073, respectively) when initial liquefaction occurs. The final magnitude of each coefficient indicates that the geometrical anisotropy (ard ¼ 0.239) at liquefaction dominates the mechanical anisotropy (and ¼ 0.104; atd ¼ 0.073), which is the signature of anisotropy at liquefaction (Guo & Zhao, 2013). Figure 6 plots the 2D visualisation (in the xz plane) of the contact force network changing as initial liquefaction approaches. The thickness of the lines is proportional to the magnitude of the contact forces of each contact. As the number of cycle increases, the contact force network gradually becomes looser and weaker. The value of average normal contact force is 3.49 3 10�3 N at the initial state, and reduces to 3.78 3 10�4 N at the liquefaction state, where the contact force network becomes insufficient to sustain any more shearing. Fig. 7 illustrates the influence of the CSR on the evolution of the mechanical coordination number towards initial liquefaction. The number of cycles is normalised by the number of cycles when initial liquefaction occurs (N l ). Both cases show that C n remains fairly constant until N =N l . 0.6: Liquefaction resistance and shear wave velocity Figure 8 shows the relationship between CSR and N l for various relative densities at a confining pressure of 100 kPa, together with V s measurement at the initial isotropic state.

z

x

(a)

80

Fig. 5. Variation of deviator anisotropic coefficients with number of cycles for a confining pressure of 100 kPa (Dr 50.4%, CSR 0.125)

During the liquefaction process, the number of interparticle contacts changes as a result of contact destruction and contact reorientation. The former can be characterised by the contact sliding fraction (S c ), which is defined as the ratio of sliding contact number to the total number of contacts. Fig. 4 also shows the variation of S c with the number of cycles. At the beginning of the shearing, the magnitude of S c is only 2 .09%, attributed to the isotropic consolidation, and it increases to 5 .31% after the first cycle. It then increases rapidly from 6 .57% at the 50th cycle up to 29 .5% at the end of shearing. At this stage, the majority of the stored energy in the sample is dissipated. Moreover, the redundancy index (I R ) can also be used to illustrate the occurrence of liquefaction. It is defined as the ratio of the number of constraints to the number of degrees of freedom in the system, and expressed as I R ¼ C n (3 � 2S c )=12 by Gong et al. (2012). If I R . 1, this indicates a redundant system, otherwise there is a non-redundant one. In this study, for the sample shown in Fig. 4, a redundant system is identified at the initial state (I R ¼ 1.190), and an unstable state is recognised when initial liquefaction occurs (I R ¼ 0.809). To further assess the evolution of contact reorientation during the cyclic loading, three deviator anisotropy coefficients are used in contact normals (ard ), normal contact forces (and ) and tangential contact forces (atd ), which are based on a second-order fabric tensor, a second-order normal contact force tensor and a second-order tangential contact force tensor, respectively (Chantawarangul, 1993; Sitharam et al., 2009). For an isotropic assembly, all these three coefficients are zero. The system is anisotropic otherwise. Fig. 5 presents their evolution with the number of cycles. It

o

70

(b)

(c)

(d)

(e)

Fig. 6. Contact force network at initial state, 50th, 60th and 64th cycles and initial liquefaction ( f cmax 0.02078N) (Dr 50.4%, CSR ¼ 0.125): (a) N 0; (b) N 50; (c) N 60; (d) N 64; (e) N 64 .5

17


XU, LING, CHENG AND CHEN 5·0

0·5 σm0 50 kPa

CSR 0·150, Nl 20

4·8

Cyclic resistance ratio, CRR

Mechanical coordination number, Cn

CSR 0·125, Nl 64

4·6

4·4

4·2

σm0 100 kPa

0·4

Fitting curve

0·3 Liquefaction 0·2

Non-liquefaction

0·1

Dr 50·4%; σm0 100 kPa 4·0 0

0·2 0·4 0·6 0·8 Normalised number of cycles, N/Nl

0

1·0

Vs 109·6 m/s

Cyclic stress ratio, CSR

0·1

Dr 52·7% Dr 50·4% Dr 50·0%

0·3

0·2

Dr 55·8% Nl 15

Dr 17·4%

Vs 99·1 m/s Vs 96·1 m/s Vs 91·1 m/s Vs 93·0 m/s

where � v9 is the overburden stress and n is the power exponent. The value of n is 0 .386 for the present DEM simulations (Xu et al., 2013), which is determined by performing the shear wave velocity measurement on a number of samples at different initial states, such as void ratio and confining pressure. It is lower than that of real sand (typically n¼0.5), owing to the idealised sphere–sphere Hertzian contact characteristics in this study, rather than the non-spherical contacts in real sand particles (Santamarina & Cascante, 1996).

Dr 39·3%

Vs 104·8 m/s

σm0 100 kPa

Vs 82·4 m/s 0

1

10 100 Number of cycles to liquefaction, Nl

150

spheric pressure (typically 100 kPa); and V s1 is the overburden stress-corrected shear wave velocity, which is expressed as (Robertson et al., 1992) � �n=2 Pa V s1 ¼ V s (6) � v9

Dr 61·2%

0·4

30 60 90 120 Stress-corrected shear wave velocity, Vs1: m/s

Fig. 9. Correlation between liquefaction resistance and shear wave velocity

Fig. 7. Influence of cyclic stress ratio on the evolution of mechanical coordination number for a confining pressure of 100 kPa (Dr 50.4%) 0·5

0

1000

Fig. 8. Curves of CSR against number of cycles to liquefaction for a confining pressure of 100 kPa

Particle-specific CRR–Vs1 correlation In order to further explore the fundamental micro-mechanism of the CRR–V s1 correlation, the influence of the particle mechanical properties on both liquefaction resistance and shear wave velocity were examined. The benchmark specimen was selected as the one isotropically consolidated to a confining pressure of 100 kPa, with relative density of 50 .4%. Particle properties listed in Table 1, such as particle shear modulus, particle Poisson ratio or friction coefficient, were replaced by various values at the end of the isotropic consolidation stage, and then numerically cycled until the new specimen reach its equilibrium state. During this process, no significant changes in the void ratio, mechanical coordination number and contact normal anisotropy were found. After that, these samples were brought to initial liquefaction as before. Figure 10 shows the effects of particle Poisson ratio vg , shear modulus Gg and friction coefficient �g on CSR–Nl curves. It is evident from Fig. 10(a) and Fig. 10(b) that both Gg and vg have negligible influence on CRR, as all these curves almost overlap. For instance, the values of CRR are 0 .159, 0 .167 and 0 .162 when the magnitudes of Gg are set as 1 GPa, 5 GPa and 10 GPa, respectively. However, a significant effect of �g on CRR is found in Fig. 10(c). With the increase of particle friction coefficient from 0 .3 to 0 .9, the magnitude of CRR increases 3 .65 times, from 0 .074 to 0 .270. The above observations imply that the inter-particle friction is the governing micro-parameter for liquefaction resistance. This is understandable as soil liquefaction is a large deformation process with energy dissipation (e.g.

From the figure, it is evident that the CSR curves move downwards significantly as Dr reduces, which means that a much smaller CSR is required to liquefy a looser soil sample. This suggests that the DEM simulations have captured quantitatively the effect of Dr on liquefaction potential observed in laboratory experiments (e.g. Zhou et al., 2010). The magnitude of CRR is taken as the value of CSR when N l equals 15 for each specimen. As expected, CRR also decreases as Dr decreases. It is worth noting that, in Fig. 8, the single point specified by a dotted circle corresponds to the results previously presented in Figs 3–6. To develop the liquefaction evaluation chart, the CRRs obtained from Fig. 8 are plotted against the converted stresscorrected shear wave velocity measured at the initial state of each sample in Fig. 9. The data from tests under a different confining pressure (� m 9 0 ¼ 50 kPa) are also added to the figure. All the data fall into a very narrow band. This demonstrates that there exists a unique CRR–V s1 correlation for a granular soil at the macro scale. The data are then fitted using the following equation !�=2 rg (V s1 )2 CRR ¼ Æ (5) Pa where Æ ¼ 7.218 3 10�8 and � ¼ 5.339 (with the fitting correlation coefficient R2 ¼ 0.98) are material constants related to the particle mechanical properties; Pa is the atmo-

18


Dr 50·4%; σm0 100 kPa Gg 1 GPa; μg 0·50

0·4


Mobilised angle of internal friction, φ: degrees

LIQUEFACTION RESISTANCE AND SHEAR WAVE VELOCITY OF GRANULAR SOILS 0·5

υg 0·10 υg 0·20 υg 0·30

0·3

0·2

0·1 Nl 15 0

1

10 100 Number of cycles to liquefaction, Nl (a)

1000

0·5 Dr 50·4%; σm0 100 kPa υg 0·20; μg 0·50


0·4

Yimsiri & Soga (2010) (loose, φmax)

40

φmax 34·4°

30 20 10 0

Silica sand no. 8 φg 39·9° 0

10

�max ¼ a(�g )b Nl 15 1

0·5

10 100 Number of cycles to liquefaction, Nl (b)

Gg 1 GPa; υg 0·20

0·4

1000

μg 0·30 μg 0·50 μg 0·70

0·3

μg 0·90

0·2

0·1 Nl 15 1

20 30 40 50 Inter-particle friction angle, φg: degrees

60

10 100 Number of cycles to liquefaction, Nl (c)

(7)

where a ¼ 10.121 and b ¼ 0.332 (for �g ranging from 0 .1 to 0 .9). For the shear wave velocity of granular soils in the initial state, although not presented here, the current DEM simulations with particle shear modulus ranging from 1 .0 GPa to 50 GPa indicate that the relationship between V s and Gg follows a power law, with a power index of m ¼ 0.320 (Xu, 2012). The influence of particle Poisson ratio vg on V s is found to be negligible (e.g. V s only increases from 92 .17 m/s to 94 .68 m/s when vg varies from 0 .1 to 0 .4 with Dr ¼ 50.4% and � m 9 0 ¼ 100 kPa). No effect of inter-particle friction �g on V s was discovered, as the V s measurement of the present study is within the elastic range. All these observations are consistent with the previous theoretical research based on various hypotheses (Chang et al., 1991; Santamarina & Cascante, 1996; Yimsiri & Soga, 2000). It can be concluded that two independent micro-parameters, the inter-particle friction and the particle shear modulus, govern the liquefaction resistance and the shear wave velocity of granular soils, respectively. This implies that the previous CRR–V s1 correlation shown in Fig. 9 should also be dependent on particle properties, rather than being a unique relationship for all types of granular soils. Hence, the present authors carried out some extra DEM simulations to demonstrate the existence of particle-specific CRR–V s1 correlations, by varying the particle shear modulus and friction coefficient. Fig. 12 presents the influence of particle shear modulus on the CRR–V s1 correlation. The curves are the predictions from equation (5), except that the values of Æ are reduced based on the V s –Gg power law. Interestingly, this indicates that they are in good agreement with the results from DEM simulations at various Gg levels. For the effect of inter-particle friction, the obtained DEM data are plotted in Fig. 13, and best fitted using equation (5) by keeping Æ constant. The fitting correlation coefficients for both cases (�g ¼ 0.3 and �g ¼ 0.7) are higher than 0 .90, with � equal to 5 .135 and 5 .440, respectively. Therefore, from the present DEM simulations, the CRR–V s1 correlation can be reasonably characterised by equation (5), where the coefficient Æ is dependent on Gg for a given �g , and the

100 kPa Dr 50·4%; σm0


DEM results (φcv)

Yimsiri & Soga (2010) (dense, φmax)

clearly shown that both �max and �cv from the current DEM simulations are in good agreement with those of the previous research, and the �max –�g relationship for the present DEM assemblies with spherical particles can be expressed as

Gg 10 GPa

0·1

0

DEM results (φmax)

Gg 1 GPa

0·2

0

50

Thornton (2000) (φcv) Thornton (2000) (φmax)

Fig. 11. Relationship between mobilised angle of internal friction and inter-particle friction angle Gg 5 GPa

0·3

60

1000

Fig. 10. Influence of particle mechanical properties on liquefaction resistance: (a) particle Poisson ratio; (b) particle shear modulus; (c) inter-particle friction

through contact sliding, as shown in Fig. 4), which is highly related to the contact frictional property. The higher the value of �g , the more difficult it is to induce contact sliding (Thornton, 2000), resulting in a higher CRR. These findings suggest that CRR is highly dependent on the inter-particle mobility (i.e. void ratio, coordination number, confining pressure and inter-particle friction), rather than the deformability of the soil particle itself. Figure 11 presents the correlation between the mobilised angle of internal friction � (both at peak �max and critical state �cv ) and inter-particle friction angle �g obtained from a series of drained DEM simulations at a confining pressure of 100 kPa, together with some other results from the literature (Thornton, 2000; Yimsiri & Soga, 2010). It is

19


XU, LING, CHENG AND CHEN 0·5 Prediction from equation (5) (Gg 1 GPa) DEM results (Gg 2 GPa)

Prediction from equation (5) (Gg 2 GPa) 0·4

DEM results (Gg 5 GPa)

Prediction from equation (5) (Gg 5 GPa)

DEM results (Gg 10 GPa)


Prediction from equation (5) (Gg 10 GPa)

0·3

� 5·339

0·2

α 2·205 108

α 7·218 108

α 1·406 109

α 4·600 109

0·1

Vg 0·20; μg 0·50 0 50

100

150 Stress-corrected shear wave velocity, Vs1: m/s

200

250

Fig. 12. Influence of particle shear modulus on CRR–V s1 correlation


0·5

and the results from three dynamic centrifuge model tests (cases A, B and C) performed on silica sand no. 8 (Zhou et al., 2010). These tests were conducted in a laminar box, together with three pairs of bender elements, two pairs of earth pressure meters, eight miniature accelerometers, six miniature pore pressure transducers and three laser displacement transducers. The silica sand no. 8 is poorly graded, with a mean diameter of 0 .084 mm (emax ¼ 1.381, emin ¼ 0.721). The peak internal friction angle �max is 34 .48. The relative density of the soil used in the dynamic centrifuge model varies from 61 .7% to 89 .8%. From these tests, 84 liquefied data sets and 120 non-liquefied data sets were produced. As stated previously, the use of equation (8) to characterise the CRR–V s1 correlation requires two micro-parameters: inter-particle friction and particle shear modulus. Direct measurement of these particle properties is not easy to achieve. Instead, one can calibrate them through establishing the micro–macro relationship, as shown in Fig. 11. Based on this �max –�g relationship, equation (7), the value of �g suitable for the silica sand no. 8 (�max ¼ 34 .48) is obtained as 39 .98 (�g ¼ 0.836), leading to � ¼ 5.483 by way of equation 8(b). For the particle shear modulus, it is calculated by comparing the V s obtained from the lab testing using bender elements and the DEM V s measurements from the present study at a similar relative density range, and by assuming that the real sand follows the same V s –Gg power law. This is reinforced by the fact that both sets of the V s data were found to be linearly related to Dr with a similar slope. Using this method, the value of Gg was calibrated as 6 .649 GPa for the silica sand no. 8 (Xu, 2012). This measured value is around four times lower than most published DEM work (Thornton, 2000; Thornton & Zhang, 2010; Gong et al., 2012; Huang et al., 2014). This may be due to particle roughness and the imperfect shape of real sand particles (Chang et al., 1991). Figure 14 presents the predicted CRR–V s1 curve from

DEM results (μg 0·3) DEM results (μg 0·5) � 5·440 DEM results (μg 0·7)

0·4

Fitting by equation (5) (μg 0·3) Fitting by equation (5) (μg 0·5)

0·3

Fitting by equation (5) (μg 0·7)

� 5·339 � 5·135

α 7·218 108

0·2

Liquefaction Non-liquefaction

0·1 Gg 1 GPa; υg 0·20 0

0

30 60 90 120 Stress-corrected shear wave velocity, Vs1: m/s

150

Fig. 13. Influence of inter-particle friction on CRR–V s1 correlation

power index � is determined by �g : This can be expressed explicitly as " #�=2 rg (V s1 )2 Æ0 CRR ¼ (8a) Pa (Gg =Gg0 )m� � ¼ B arctan (A�g ) þ C

(8b)

where Gg0 ¼ 1 GPa; Æ0 ¼ 7.218 3 10�8 ; A ¼ 4.890; B ¼ 0.971; and C ¼ 4.191: These two parameters may be dependent on particle shape and particle size distribution. This requires further investigation. VALIDATION AND DISCUSSION The present authors attempted to validate the above observation by comparing the prediction from equation (8)

20


LIQUEFACTION RESISTANCE AND SHEAR WAVE VELOCITY OF GRANULAR SOILS 0·9 (1) Andrus & Stokoe (2000)

(1)

(5)

(4)

(2) Zhou et al. (2010) (3) Ahmadi & Paydar (2014) (Babolsar sand) (2)

(4) Ahmadi & Paydar (2014) (Firoozkooh sand) (5) Present study 0·6

(3)

CSR or CRR

Equation (8) Gg 6·649 GPa; μg 0·836

Case A (Liq) 0·3

Case A (Non-liq) Case B (Liq) Case B (Non-liq) Case C (Liq) Case C (Non-liq)

0 50

100

150 Corrected shear wave velocity, Vs1: m/s

200

250

Fig. 14. Validation on the CRR–V s1 correlation (after Zhou et al., 2010)

CONCLUSIONS This study has explored the fundamental micromechanics of liquefaction resistance and its correlation with the shear wave velocity of granular soils, by performing a series of undrained, stress-controlled, cyclic triaxial tests on DEM specimens with various relative density, confining pressure and particle mechanical properties, together with V s measurement at the initial state. A number of novel findings were obtained, and these are summarised as follows.

equation (8) using both the calibrated parameters described above and the 204 CSR data sets from the dynamic centrifuge tests (Zhou et al., 2010). The latter were converted from a series of irregular time histories recorded in centrifuge tests according to equation (1). Liquefied data are represented by solid symbols in the figure. Non-liquefied data are represented by hollow symbols. As shown in Fig. 14, the predicted curve separates almost all (about 92%) of the liquefied data properly, and even for the non-liquefied data more than 78% of them are correctly classified. This is very similar to the previous evaluation done by Zhou et al. (2010). For the misclassification of the non-liquefied data, Zhou et al. (2010) had clarified that the initial liquefaction criterion is not always applicable to dense sand. They did observe some considerable settlement after dissipation of pore-water pressure, although these data could not be classified as liquefied according to the initial liquefaction criterion. While the correlation proposed by Andrus & Stokoe (2000) considerably underestimates the liquefaction resistance of silica sand no. 8 within the low-velocity range, it overestimates the resistance in the high-velocity range, as shown in Fig. 14. The more recent relationship established by Ahmadi & Paydar (2014) from two types of clean sands, namely the Firoozkooh and the Babolsar sands, are also shown in Fig. 14, and the one based on the Babolsar sand gives much less satisfactory evaluation than that of the Firoozkooh sand. These observations strongly support the fact that the proposed CRR–V s1 , which separates the liquefied and the non-liquefied data, is soil type specific. This DEM study supports the use of equation (8) and suggests that the peak internal friction angle at the dense state can potentially be used to obtain an appropriate � value, where the Æ value can be obtained from the shear wave velocity measurement. A satisfactory evaluation of the liquefaction potential of granular soil can then be obtained.

(a) The CRR–V s1 correlation of granular soils is particle specific, and thus soil specific, and is governed by two independent micro-parameters, inter-particle friction and particle shear modulus. The proposed micro-scale CRR– V s1 correlation, with two independent coefficients Æ and � explicitly relating to the particle shear modulus and inter-particle friction, respectively, was validated by the outcomes from three dynamic centrifuge model test performed on silica sand no. 8 (Zhou et al., 2010). This study provides clear, yet simple, micromechanical evidence that current liquefaction evaluation with Vs alone is not sufficient, and could lead to overestimation of CRR. The effect of inter-particle friction should also be taken into account to consider the large deformation process of liquefaction in granular soils. (b) The key micro-parameters that govern the liquefaction resistance of granular soils were identified. A systematic parametric study on DEM samples with various particle mechanical properties indicated that it is the inter-particle mobility (e.g. inter-particle friction), rather than the deformability of the soil particle itself (e.g. particle shear modulus and Poisson ratio) that determines the magnitude of CRR of granular soils. The higher the value of inter-particle friction, the more difficult it was to induce contact sliding, giving higher CRR. A micro–macro

21


XU, LING, CHENG AND CHEN relationship between the inter-particle friction and the internal friction angle at peak was also developed. (c) The micromechanics of the initial liquefaction process based on stress-controlled loading were investigated. The numerical data showed that all of the relevant micromechanical parameters, such as the mechanical coordination number, the sliding fraction, the deviator anisotropic coefficients and even the force network, remained almost constant until the number of cycles N reached a relatively high portion (. 0 .6) of the total number of cycles at which initial liquefaction occurred N l : The onset of liquefaction is fairly sudden, leading to a dramatic change of the micro-parameters. When initial liquefaction occurs, the mechanical coordination number reduces to 4 .0, redundancy index becomes smaller than 1 .0, and geometrical anisotropy dominates mechanical anisotropy.

number of cycles to cause initial liquefaction power exponent in equation (6) atmospheric pressure stress reduction coefficient contact sliding fraction wave travel time excess pore water pressure ratio excess pore water pressure shear wave velocity; shear wave velocity interpreted from a boundary measurement V s1 overburden stress-corrected shear wave velocity; Æ material constant in equation (5) Æ0 material constant in equation (8a) � material constant in equation (5) �a axial strain º wave length in Fig. 2(c) �g inter-particle friction coefficient ıg particle Poisson’s ratio r sample density rg particle density �m 9 0 mean effective stress � r9 radial effective stress � r90 radial effective stress after isotropic consolidation � v9 overburden stress � v0 total vertical overburden stress � v90 effective vertical overburden stress �av average equivalent uniform cyclic shear stress � internal friction angle �cv internal friction angle at critical state �g inter-particle friction angle �max peak internal friction angle Nl n Pa rd Sc t U u Vs V s0

ACKNOWLEDGEMENTS Much of the work described in this paper was supported by the National Basic Research Program of China (grant no. 2014CB047005), the National Natural Science Foundation of China (grant no. 51278451), the Zhejiang Provincial Natural Science Foundation of China (grant nos. LZ12E09001 and LY12E08011) and Erasmus Mundus External Cooperation Window (EM ECW) scholarship. These financial supports are gratefully acknowledged. The authors thank Professor Kenichi Soga and Professor Malcolm Bolton at University of Cambridge, Professor J. Carlos Santamarina at Georgia Institute of Technology and Professor M A Curt Koenders at Surrey University for their valuable exchanges of ideas and help with this study.

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NOTATION A Adisp a and ard atd amax B b C Cn D Dr d d 50 e emax emin f f c f cmax f u G0 Gg Gg0 g H IR LTR m Mw N Nb N b0 N b1 Nc

coefficient in equation (8b) displacement of the excitation wave coefficient in equation (7) normal contact force anisotropy coefficient contact normal anisotropy coefficient tangential contact force anisotropy coefficient peak horizontal ground surface acceleration coefficient in equation (8b) power index in equation (7) coefficient in equation (8b) mechanical coordination number model diameter relative density particle diameter mean diameter of particle void ratio after isotropic consolidation maximum void ratio minimum void ratio excitation frequency in Fig. 2(c) mean contact force maximum contact force mean unbalanced force small strain shear modulus particle shear modulus material constant in equation (8a) acceleration of gravity model height redundancy index distance between the transmitter and receiver power index in equation (8a) earthquake magnitude number of cycles total number of particles number of particles with no contact number of particles with only one contact total number of contacts

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Masini, L. et al. (2015). Geótechnique 65, No. 5, 349–358 [http://dx.doi.org/10.1680/geot.SIP.15.P.001]

An interpretation of the seismic behaviour of reinforced-earth retaining structures L . M A S I N I , L . C A L L I S TO a n d S . R A M P E L L O

This paper provides an interpretation of the behaviour of retaining structures made of geosyntheticreinforced earth, subjected to a severe seismic loading. It is seen that during strong ground motion the main source of energy dissipation derives from the transient activation of plastic mechanisms within the soil mass: these mechanisms can be global, local, or a combination of the two. Using numerical pseudo-static analyses and limit analysis methods it is shown that three retaining structures having a similar overall seismic resistance, expressed by their critical seismic coefficient, activate different – global, local, or combined – plastic mechanisms. The seismic performance of the different retaining structures is then evaluated through a series of dynamic analyses in which acceleration–time histories are imposed to the bottom boundary of the same numerical models used for the pseudo-static analyses. The results of the dynamic analyses are interpreted in the light of the plastic mechanisms evaluated with the pseudo-static procedure. They show that for the reinforced-earth structures there is always a local contribution to the dissipation of energy during strong motion, evidenced by the attainment of the available strength in different portions of the soil-reinforcement system, and that this energy dissipation has a substantial influence on the seismic performance of the system. These results extend the current understanding on the seismic behaviour of reinforced-earth retaining structures and can be used to provide some guidance for design. KEYWORDS: design; earthquakes; reinforced soils

soil and the reinforcement: in fact, the design is typically carried out using solutions based on either limit equilibrium (Bathurst & Cai, 1995; Cascone et al., 1995; Motta, 1996; Ling et al., 1997; Biondi et al., 2008; Basha & Basudhar, 2010) or limit analysis (Michalowski, 1998; Ausilio et al., 2000; Leshchinsky, 2001). A direct evaluation of the seismic performance of these structures can be obtained through the sliding-block method of Newmark (1965), in which the critical seismic coefficient is calculated with one of the available pseudo-static solutions (Lin & Whitman, 1986; Ambraseys & Menu, 1988; Yegian et al., 1991; Conte & Rizzo, 1996; Rampello et al., 2010; Biondi et al., 2011; Callisto & Rampello, 2013). A review of existing slidingblock methods was presented by Cai & Bathurst (1996). It is important to emphasise that, although the understanding of the fundamental mechanisms controlling the dissipation capabilities and the ductility of these structures is still incomplete, many design procedures implicitly assume an indefinitely ductile behaviour using, for instance, kinematic solutions to evaluate the resultant force in the whole reinforcement system, and then redistributing quite arbitrarily this total force among the different reinforcement layers (e.g. Ling et al., 1997; Michalowski & You, 2000). The present paper contributes to the understanding of the main factors controlling the seismic performance of geosynthetic-reinforced earth retaining structures, by examining the behaviour of three different idealised structures retaining the same backfill and resting on the same foundation soil. Specifically, a reference structure is conceived in such a way that it has a relatively low resistance, under both static and seismic conditions, but is characterised by a potentially large capability of dissipating energy, thanks to a dense and ductile reinforcement layout. The seismic behaviour of this reference structure is compared to that of two additional structures that have the same overall resistance, but provide progressively less opportunity of an internal dissipation of energy. The first two structures are made of geosynthetic-

INTRODUCTION The seismic behaviour of geosynthetic-reinforced earth retaining structures has been the subject of a large number of research activities, including field observations (Kramer & Paulsen, 2001; Wartman et al., 2006; Shinoda et al., 2007; Maugeri & Biondi, 2008; Tatsuoka, 2008; Koseki et al., 2009), shaking-table experiments on small-scale models (Watanabe et al., 2003; El-Eman & Bathurst, 2004, 2005, 2007) and full-scale models (Ling et al., 1997, 2005; Ling & Leshchinsky, 1998), and centrifuge tests (Izawa et al., 2004; Kramer & Paulsen, 2004). Remarkably, a number of field observations have shown a generally good performance of geosynthetic-reinforced earth retaining structures subjected to severe seismic loading (e.g. Koseki et al., 2009), and this finding is consistent with observations resulting from shaking table experiments on model reinforced-earth structures (Ling et al., 2005). Intuitively, this satisfactory behaviour can be ascribed to the possibility that these structures contribute to energy dissipation through the development of internal plastic mechanisms, and possess an overall ductile behaviour deriving from the large deformation that can be accommodated by the soil-reinforcement system. It is clear that such structures respond to a severe seismic loading by mobilising the strength of different portions of the system, including the reinforcing elements, the soil– reinforcement interfaces, the retained soil and possibly the foundation soil. It may then be anticipated that the seismic behaviour of these structures is controlled by the strength properties, rather than by the pre-failure behaviour of the

Manuscript received 26 March 2014; revised manuscript accepted 29 January 2015. Published online ahead of print 23 March 2015. Discussion on this paper closes on 1 October 2015, for further details see p. ii. Department of Structural and Geotechnical Engineering, Sapienza Universita` di Roma, Rome, Italy.

25


MASINI, CALLISTO AND RAMPELLO B

45 m

40 m

55 m

15 m

reinforced earth, but the first one develops large plastic deformations in the reinforced zone, while the second one, having stronger reinforcements, activates a plastic mechanism that includes a significant portion of the backfill. The third structure is a non-dissipative one and can activate only external mechanisms. The seismic behaviour of these idealised structures is examined employing a combination of pseudo-static analyses, based on kinematic solutions and on finite-difference calculations, and dynamic non-linear timedomain analyses. While these are only three specific idealised cases, the conclusions deriving from the interpretation of their seismic response in terms of predominant plastic mechanisms are indeed quite general, and shed some light on the factors determining a good seismic performance for this type of retaining structures. PROBLEM LAYOUT In the reference scheme A of Fig. 1, a fill of height H ¼ 15 m is retained by an earth structure with a batter ¼ 108, reinforced with geo-grids of uniform length B ¼ 11 .25 m ¼ 0 .75H having a spacing, s ¼ 0 .6 m and a tensile strength, TT ¼ 25 kN/m. The fill is made of a coarsegrained cohesionless material with an angle of shearing resistance 9 ¼ 358, while the foundation soil has 9 ¼ 288 and a cohesion c9 ¼ 10 kPa. The resistance at the soilreinforcement contact is purely frictional with an angle of shearing resistance 9s equal to that of the parent soil. A study of the effect of the strength at the soil–reinforcement interface was also carried out, using an interface strength factor of 0 .7; it was found that this reduction in the contact strength has a small influence on the pattern of behaviour found in the reference analyses. All the materials are dry and have a unit weight of 20 kN/m3. Schemes B and C were devised to have the same resistance of case A, but with a smaller capability of internal energy dissipation: in the case B the reinforcements are shorter (B ¼ 7 .9 m, B/H ¼ 0 .53) and stronger (TT ¼ 35 kN/m), while case C is that of a retaining structure with an infinite internal resistance (modelled as an elastic material) with B ¼ 5 .6 m.

101·25 m

Fig. 2. Detail of the finite-difference grid adopted for case A, showing the boundary conditions adopted in the pseudo-static analyses

while the horizontal displacements were inhibited at the lateral boundaries. Also, in these analyses the soil was modelled as an elastic–perfectly plastic material with a Mohr–Coulomb plasticity criterion, zero dilatancy, a Poisson ratio of 0 .3 and a shear modulus equal to 20 to 50% of the small strain shear stiffness (discussed in the next section), depending on the computed average strain level. The reinforcing levels were modelled as Flac strip elements, reacting only to axial tension. The constitutive model for these elements is shown in Fig. 3(a): the relationship between the axial force T and the axial strain in the strip, simulating that of a polyester (PET) geo-grid, is assumed to be elastic–perfectly plastic, with an axial stiffness EA, a yield strain y and an ultimate strain u; for strains larger than u the strip element loses entirely its tensile strength TT,

ANALYSIS OF PLASTIC MECHANISMS The fundamental analysis tool to study the plastic mechanisms for the above schemes is a pseudo-static numerical finite-difference analysis, carried out in plane strain conditions using the computer code Flac v.5 (Itasca, 2005), in which the seismic forces are increased progressively until the strength of the system is fully mobilised. Fig. 2 shows a detail of the calculation grid employed for the case A: it has a total width of about 100 m and extends 40 m below the earth fill; it includes 19 800 quadrilateral soil zones having a size as small as 0 .2 m in the vicinity of the reinforced area. In the static and pseudo-static analyses, both horizontal and vertical displacements were restrained at the base of the grid,

T εpl lim TT

EA εy

εu

ε

(a) Case A

Case B

�

ΔL L

Flexible wall facing

Case C

Geogrid layer

Backfill soil

s

H �

Connected node

Free node (b)

B 0·75H

0·53H

0·38H

Fig. 3. (a) Constitutive behaviour assumed for the reinforcing layers; (b) detail of the connection of a reinforcing layer to the wall facing

Fig. 1. Layouts of the three idealised earth-retaining structures

26


SEISMIC behaviour OF REINFORCED-EARTH RETAINING STRUCTURES Starting from the end of construction, a pseudo-static analysis was carried out, applying a uniform horizontal body force expressed as a fraction kh of gravity. The value of the seismic coefficient kh was increased progressively until convergence, evidenced by a steady reduction of the unbalanced forces, became no longer possible. Under this circumstance, the numerical model exhibited a well-defined mechanism, associated with a plastic flow of the soil. The seismic coefficient kh that activates the mechanism is termed ‘critical’ and is indicated as kc. Parametric studies were carried out to check that the finite-difference grid was sufficiently fine and adequately extended away from the excavation. It was also found that the solution does not depend on the stiffness of the materials and therefore can be assumed to be a result of the strength properties only. Since the pseudo-static analyses were carried out up to critical conditions, the values of the computed displacements are only conventional, since they refer to a system that is accelerating due to the static activation of a plastic mechanism. Therefore, for the pseudostatic analysis it was assumed that the strip elements are infinitely ductile, that is, their ultimate strain �u is infinite. The critical seismic coefficient kc obtained for the three schemes of Fig. 1 are very similar, varying from 0 .060 to 0 .066. Fig. 4 shows the contours of shear strains ª obtained in critical conditions for the three different structures, with an indication of the corresponding critical seismic coefficient kc. It can be seen that, although for the three cases the critical coefficient is almost identical, the deformation pattern is very different. For case A, two concurrent plastic mechanisms seem to emerge: a first one is confined within the reinforced zone, and does not seem completely developed, in the sense that the strain contours show smaller values near the top of the structure; a second one involves both the reinforced area and the upper part of the backfill, evidencing also a large strain gradient at the contact between the backfill and the reinforced structure. The two mechanisms converge towards the toe of the structure, marginally involving the foundation soil. For case B, there seems to be only one, more definite plastic mechanism, going through the bottom part of the reinforced structure, and mobilising the shear strength in most of the backfill; the foundation soil is still involved only marginally, but to a slightly larger extent than for case A. In case C, any internal failure is

displaying a perfectly brittle behaviour. When this occurs, the calculation algorithm sub-divides the strip into two separate elements that cannot interact with each other. The contact at the soil–strip interface was simulated as elastic– perfectly plastic with a very large stiffness Ks and a purely frictional strength �9s ¼ �9 ¼ 358. The mechanical properties of the reinforcing layers are typical of a medium-strength PET geo-grid and reported in Table 1; it was assumed that the axial stiffness of the reinforcements increases proportionally to their strength – that is, that the yield strain is the same for both cases A and B. A wall facing was modelled by assigning a purely elastic behaviour to the couple of soil zones closest to the lateral surface of the reinforced structure. The external end of each strip element was connected rigidly to the wall facing, while the internal end was left unconstrained (Fig. 3(b)). After initialising the effective stresses in the foundation soil, the construction of the reinforced soil structure and the fill was simulated in 25 steps; each step included the activation of a reinforcement element, the corresponding portion of wall facing, and a 0 .6 m-thick soil layer. At the end of construction, a maximum horizontal displacement equal to 1 .1 % and to 1 .0 % of the wall height was computed for cases A and B, respectively, while a value of 0 .5% H was obtained for the retaining wall of infinite internal resistance (case C). Starting from this stage, an evaluation of the safety with respect to a static collapse was carried out iteratively (e.g. Callisto, 2010) reducing progressively the strength parameters of the soil and the soilreinforcement contact; these analyses yielded a strength factor slightly lower than 1 .25. Table 1. Mechanical parameters adopted for the geosynthetic reinforcements Case A

Case B

25 1250 0 .02 10 35 106

35 1750 0 .02 10 35 106

TT: kN/m EA: kN/m �y �u /�y �9s : degrees Ks: kN/m per m

Case A

kc 0·060 Limit analysis

Case B

kc 0·066

Case C

kc 0·060

Limit analysis

Trasl. – kc 0·101


Rot. – kc 0·115

Rot. – kc 0·106

Rot. – kc 0·132 (a)

(b)

(c)

Fig. 4. Results of the pseudo-static analyses carried out up to critical conditions: contours of shear strains (the actual values are not relevant in critical conditions); and interpretation using kinematic limit analysis solutions

27


MASINI, CALLISTO AND RAMPELLO Case A

inhibited, and the mechanism can only develop within the external soil: it involves the development of the active limit conditions in the backfill and the mobilisation of the strength of the foundation soil. The effect of a uniform vertical acceleration was also studied, by assuming either upward or downward inertial forces equal to 50% of the horizontal ones. These vertical forces caused for all the schemes a 10% increase or decrease of kc, respectively, but had no appreciable effect on the plastic mechanisms. The above mechanisms were back-interpreted implementing two different kinematic approaches based on limit analysis, as illustrated schematically in Fig. 5. The first solution was proposed by Michalowski & You (2000) and assumes that a portion of reinforced soil slides along a log-spiral surface that passes through the toe of the structure; the sliding surface can either include only the reinforced zone, or extend out of the reinforced zone to the upper part of the backfill. For a given value of the angle of shearing resistance �9, a log-spiral sliding surface is fully specified by the angles Ł0 and Łh of Fig. 5(a). The second solution is relative to a two-block mechanism, in which the first soil block is triangular and represents the upper part of the backfill, while the second block is a quadrilateral representing the portion of the reinforced soil involved in the mechanism. Following the upper bound theorem of perfect plasticity, the solution for both cases is obtained by equating the rate of work done by the external force to that dissipated along the sliding surfaces and within the reinforcing elements that are intersected by the surfaces. Using these two solutions, an iterative search was carried out to define the mechanism corresponding to the minimum critical seismic coefficient. Figure 6 shows, for cases A and B, the contours of the critical seismic coefficient kc obtained varying the angles Ł0 and Łh. As a general result, the values of kc obtained from limit analysis are somewhat larger than those resulting from the numerical pseudo-static analyses. This may be due both to the upper bound nature of the kinematic approach, and to the effect of the associated flow rule assumed in limit analysis, in contrast with the assumption of zero dilatancy made in the numerical computations. For case A, the minimum kc is equal to 0 .115 and is relative to a value of Ł0 equal to Łh, that is, to a log-spiral degenerating to a planar surface; this planar surface has an inclination on the horizontal of 908 � Ł0 + �9 � 418 and reaches the backfill in the upper 25% of the structure’s height: it is shown with a dashed line in Fig. 4(a), and it is seen to be in a fair agreement with the prevailing plastic mechanism for case A. Inspection of the contour lines of Fig. 6(a) also shows that there is a relative minimum of kc ¼ 0 .132 corresponding to Ł0 ¼ 528 and Łh ¼ 838. The resultant log-spiral is shown with a dash-dotted line in Fig. 4(a) and approximates very well the internal, secondary mechanism evidenced by the pseudo-static numerical analyses. The continuous line in Fig.

v

W2

φ

v2 �

(a)

A

φ

h

θ 0

θ 0·1 5

30 70

80 90 θh: degrees (a)

100

70

80 90 θh: degrees (b)

100

Fig. 6. Contours of angles Ł0 and Łh of Fig. 5(a) corresponding to different critical seismic coefficients for layouts A and B

4(a) derives from the analysis of the two-block mechanism of Fig. 5(b); it describes effectively the critical mechanism obtained from the numerical analysis and corresponds to a seismic critical coefficient kc ¼ 0 .10 that, although larger than the numerical one, is the smallest among those obtained with the kinematic solutions. For case B, the contour plot of Fig. 6 shows that there is a single minimum for kc ¼ 0 .106, corresponding to a logspiral surface extending to the backfill (the irregularities of the contour are due to the mobilisation of the reinforcement strength and indicate that the sliding surface is partly external). This surface is shown in Fig. 4(b) with a dashed line, and is seen to be roughly consistent with the numerical results. The two-block mechanism of Fig. 5(b) yields a very similar mechanism, shown with a continuous line in Fig. 4(b), but a smaller critical seismic coefficient kc ¼ 0 .086. In summary, for the reference case A it appears that there are two concurring mechanisms at stake. The prevailing one mobilises the resistance of about 75% of the reinforcements, extends to the upper part of the backfill and can be interpreted by a two-block scheme. The secondary one is fully internal, mobilising the resistance of all the reinforcing layers; it is fully consistent with a kinematic log-spiral solution, and is associated with similar strains levels; it should be deemed capable of producing a significant amount of energy dissipation during an earthquake. Conversely, in case B there is a single plastic mechanism, which mobilises the resistance of about half the reinforcing elements and extends well behind the reinforced structure; it is interpreted satisfactorily by the two-block kinematic solution, as the relative displacement between the two blocks is quite evident. For this case, a smaller amount of energy dissipation is expected, since there are no different concurring plastic mechanisms. Case C cannot mobilise any internal dissipation mechanism; therefore it can reach a plastic mechanism only activating the strength of the external soil, including the backfill and the foundation. This idealised wall provides no internal energy dissipation and it is expected to show the worst performance under seismic conditions.

D W1

φ v 1

H

θ

θ θ0: degrees

40

0·5

�

zi

C v21

0·4

B

B

rh

0·3

r0

0·106

0·132

0·15

θi

60

50

θ0 ri

15 0· ·2 0

70

ω

θh

0·115

0

80

5 0·1

h

90

0·2

O

Case B

φ

α1

DYNAMIC BEHAVIOUR Analysis method and seismic input Starting from the end-of-construction stage, time-domain dynamic analyses were carried out by applying time histories

α2 (b)

Fig. 5. Kinematic solutions based on upper bound limit analysis

28


SEISMIC behaviour OF REINFORCED-EARTH RETAINING STRUCTURES a(t ) of the horizontal acceleration to the bottom boundary of the same finite-difference grid used for the pseudo-static analyses. For these analyses the horizontal fixities at the lateral sides of the grid were replaced by Flac free-field boundary conditions. The cyclic behaviour of the soil was described through the hysteretic damping model implemented in Flac, coupled with the same Mohr–Coulomb plasticity criterion as was used in the pseudo-static analyses. The hysteretic damping model is essentially an extension to two dimensions of the non-linear soil models that describe the unloading–reloading stress–strain cycles using the Masing (1926) rules. Some details on the hysteretic damping model are provided by Callisto & Soccodato (2010). The model requires the values of the small-strain shear stiffness and a backbone curve. The small-strain shear stiffness was expressed as a function of the mean effective stress p9 G0 ¼ B þ C 3 (p9)D

Table 2. Mechanical properties of the soil

Foundation soil Backfill soil

c9: kPa

�9: degrees

B: kPa

10 1

28 35

12 750 5100

D

1397 .4 5329 .5

0 .790 0 .500

Assisi (discussed by Callisto & Soccodato, 2010) was used as a seismic input; for case A the additional seismic record of Darfield (taken from the Darfield High School (DFHS) station in New Zealand) was employed. Some properties of the records are reported in Table 3, where amax is the peak ground acceleration, IA is the Arias intensity, Ts is the significant duration and Tm is the mean period as defined by Rathje et al. (1998). Fig. 8, showing the acceleration–time histories (Fig. 8(a)) and the 5% damped elastic spectra (Fig. 8(b)), evidences that the amplitudes of the Darfield record are much larger than in the Assisi record, particularly in the large-periods interval of 0 .3–1 s. Fig. 8(b) shows also the elastic spectra of the horizontal acceleration computed at the

(1)

Values for the coefficients B, C and D were selected to reproduce the typical small-strain stiffness for a medium plasticity sandy-silt for the foundation soil and a dense sand for the backfill. The backbone curve was calibrated to reproduce a modulus decay curve typical for coarse-grained materials. Fig. 7 shows a comparison of the modulus decay curve obtained with the hysteretic damping model and those published by Seed & Idriss (1970) and by Vucetic & Dobry (1991) for coarse-grained soils. The figure also shows a comparison between the equivalent damping ratio evaluated from the model hysteresis loops and that provided by the same authors. Although for ª . 0 .02% the equivalent damping ratio predicted by the hysteretic model is larger than the Seed & Idriss (1970) one, it seems still compatible with the Vucetic & Dobry (1991) curve. In the soil model, damping results entirely from the hysteretic unloading–reloading cycle. A small amount of additional viscous damping was used to attenuate the soil response at very small strain and to reduce spurious highfrequency noise (Joyner & Chen, 1975; Callisto et al., 2013). This was obtained by specifying a Rayleigh damping with both mass and stiffness components corresponding to a damping ratio of 1% at a central frequency of 1 .02 Hz, which is the fundamental frequency of the soil deposit, including the backfill. Table 2 lists the values of the strength and stiffness parameters adopted in the analyses. In most of the analyses, the scaled seismic record of

Table 3. Properties of the input seismic records Record Assisi New Zealand (NZ-DFHS)

amax: g

IA: m/s

Ts: s

Tm: s

0 .28 0 .53

0 .75 2 .62

4 .28 20 .70

0 .24 0 .42

Acc.: g

0·5

Assisi

0

0·5 0·5

Acc.: g

NZ-DFHS 0

0·5 0

5

10

15

20

25

30

35

t: s (a)

50

1·0

C

1·6 Assisi – bedrock

40

0·8

Assisi – free field

1·2

Seed & Idriss (1970) 0·4

20

Vucetic & Dobry (1991)

Sa : g

G/G0

30

Flac model

Damping: %

NZ-DFHS – bedrock 0·6

NZ-DFHS – free field

0·8

0·4 10

0·2

0 0 0·0001

0·001

0·01 γ: %

0·1

1

0

0

1

2 T: s (b)

3

4

Fig. 8. Acceleration–time histories and 5%-damped elastic response spectra for the selected seismic records and for the freefield surface motion

Fig. 7. Modulus decay curve and equivalent damping ratio predicted by the hysteretic damping model

29


MASINI, CALLISTO AND RAMPELLO 1·0

soil surface in the free-field, evidencing a significant amplification for periods in the interval of about 0 .5 to 2 s. Essential to the present discussion is that both records proved sufficiently intense to activate the plastic mechanisms found in the previous pseudo-static analyses. It was found, as expected, that the deformation pattern induced by the two records is quite similar, because it is controlled by the diffuse plastic flow in the soil rather than by the soil stiffness; therefore, the following discussion is restricted to the effects of the Assisi record. However, the computed final seismic displacements, discussed in the next section, are quite different, reflecting the largest intensity of the DFHS record.

u: m 0·4

T G

0·2

G

0 0

2

4

6

8 t: s

10

12

14

16

Fig. 10. Time histories of the horizontal displacements computed with the Assisi record for case A, with reinforcements of infinite ductility (dashed lines), and with reinforcements of limited ductility (continuous lines)

internal log-spiral evaluated with both the FLAC pseudostatic analysis and the kinematic approach; the second surface is somewhat lower than the corresponding pseudo-static one, and engages a larger portion of the foundation soil. The corresponding displacements, indicated in Fig. 10 with dashed lines, show a regular increase during the strong motion part of the seismic record, reaching maximum values of about 0 .43 m and 0 .71 m for points G and T, respectively, corresponding to about 2 .9% and 4 .7% of the wall height. It is interesting to note that point T is located to the left of the internal sliding surface, while the position of point G is intermediate between the two surfaces. Therefore, the displacement of point G is produced only by the activation of the two-block plastic mechanism, while the larger displacement of point T results from the activation of both plastic mechanisms. It is interesting to note that the computed seismic displacements have the same order of magnitude as those resulting from empirical relationships developed for the Italian territory: according to the results published by Callisto & Rampello (2013), relative to Italian seismic records, for a class B subsoil as defined by Eurocode 8, part 5 (EN 1998-5, CEN, 2003) values of the critical seismic coefficient of 0 .06–0 .07 correspond, for a maximum bedrock acceleration of 0 .28g, to a permanent seismic displacement of 0 .4 4 0 .6 m. For the case of reinforcements of infinite ductility, Fig. 11 shows the profiles of the mobilised strength T/TT of the reinforcement layers and of the corresponding maximum tensile strain , divided by the yield strain y. The open symbols represent the initial static condition, while the filled symbols indicate the post-seismic values. For case A (Fig. 11(a)) after the construction the reinforcements located in the lower third of the wall height are already mobilising their strength, the corresponding strains being larger than y. After the earthquake, the strength of 90% of the reinforcing layers has been attained. The post-seismic axial strain increases with depth: in the lower half of the structure it varies from 15 to 20 times the yield strain y, but in the lowest two layers it is as large as 30 to 50 y. Since the ultimate tensile strain u assumed for the reinforcements is equal to 10y, it is unsurprising that the dynamic behaviour of the reinforced structure A changes dramatically if the analysis accounts for the limited ductility

2 0·20 0·16 0·12 0·08 0·04 0 εu ∞ (a) Case B

Limit analysis Trasl. – kc 0·101 Rot. – kc 0·115 (b)

T

0·6

Case A

εu 20 %

εu 20%

0·8

Results of the dynamic analysis The dynamic analyses of the reference retaining structure A were carried out under the two hypotheses that the reinforcing elements either have an infinite ductility, or are characterised by a finite ultimate strain u ¼ 20%, beyond which they lose their strength completely (see Fig. 3(a)). Fig. 9 depicts the contours of the shear strains computed at the end of the Assisi seismic record, while Fig. 10 shows the corresponding time histories of the horizontal displacement u of two selected points, located, respectively, within the reinforced zone (centre of gravity G) and at the top (T) of the retaining structure. For the case of ductile reinforcements, the deformation pattern (Fig. 9(a)) is similar to the plastic mechanisms obtained from the pseudo-static analyses (Fig. 4(a)), with the development of two intensely sheared surfaces. The internal surface is very close to the critical

γ

εu ∞

Rot. – kc 0·132

Fig. 9. Contours of final shear strains computed at t 9 s with the Assisi record for case A, with: (a) indefinitely ductile reinforcements; (b) reinforcements of limited ductility

30


SEISMIC behaviour OF REINFORCED-EARTH RETAINING STRUCTURES 0

Case A T/TT 0·5

1·0

0

Case B

Case B T/TT 0·5

γ _ 2

1·0

T

0 End of construction

0·20 0·16

Post-seismic condition

0·2

G

0·12 0·08

0·4

0·04

z/H

εu /εy10

0

εu /εy10

0·6

Limit analysis 0·8

ε/εy


T/TT

Rot. – kc 0·106 (a)

1·0

Case C

T 0

20

ε/εy (a)

40

60

0

20

ε/εy (b)

40

60

Fig. 11. Profiles of the mobilised strength T/TT and of the corresponding maximum strain ratio /y , computed for reinforcements of infinite ductility: (a) case A; (b) case B

G

of the reinforcing elements. As shown in Fig. 9(b), in this case the shear strains concentrate along the internal sliding surface where, starting from the bottom layers, the largest proportion of the reinforcements reach their ultimate strain progressively. Fig. 10 shows that, in this case (continuous lines), the horizontal displacements of point T diverge: this is a progressive failure, triggered by the attainment of the ultimate strain in the lower reinforcement levels. Since point G is located just outside the internal mechanism, its displacements are scarcely affected by the ductility of the reinforcements. Figure 11(b) relates to the idealised structure B analysed assuming reinforcements of infinite ductility. In this case, fewer reinforcements located at the bottom of the wall mobilise their strength after construction, with axial strains larger than y. After the earthquake, the amount of reinforcing layers that reach their strength is lower than for case A, but the extent of the structure in which the post-seismic strains are larger than 10y is still significant. Therefore, the dynamic behaviour of structure B reinforced with elements of limited ductility (not shown here for brevity) is similar to that for case A – that is, a progressive failure occurs with a rupture surface, which for case B starts within the reinforced zone but also reaches the backfill. Moreover, if case B is analysed assuming that the reinforcements have an infinite ductility, then its seismic behaviour is controlled by the activation of the same plastic mechanism obtained from the pseudo-static analysis: Fig. 12(a), depicting the shear strain contours at the end of the earthquake, shows that a two-block-mechanism emerges, which is analogous to that shown in Fig. 4(b) for the pseudo-static critical condition. The dynamic analysis of the non-dissipative structure C shows the activation of a plastic mechanism (Fig. 12(b)), which is totally external to the wall, and very similar to the corresponding pseudo-static mechanism of Fig. 4(c). Therefore, it appears that the search for the relevant plastic mechanisms that the different structures activate when subjected to a uniform acceleration field is sufficiently indicative of the actual seismic response, provided that the reinforcing elements have a plastic ductility sufficient to

(b)

Fig. 12. Contours of final shear strains computed with the Assisi record for: (a) case B with reinforcements of infinite ductility; (b) case C

accommodate the deformation of the structure. A comparison of the three different mechanisms depicted in Fig. 4 indicates that, although the critical seismic coefficient for the three structures is about the same, in progressing from case A to case C the internal plastic deformations become less and less important; therefore, a general deterioration of the seismic performance should be expected as smaller amounts of kinetic energy can be dissipated by the internal plastic strains. This perception is substantiated by the inspection of the seismic performance of the three different structures subjected to the same Assisi seismic input, as shown in Fig. 13(a). With reference to points G and T, the displacements exhibited by the structure B are from 1 .4 to 2 .6 times larger than the corresponding displacements computed for case A; the non-dissipative structure C undergoes displacements that are from 2 .3 to 4 .5 times larger than those computed for structure A. Differently from case A, for the other two structures the displacement at the top (point T) is always lower than that of the centre of gravity G. This happens because for cases B and – obviously – C, the internal log-spiral concurrent mechanism is absent. This finding is consistent with the fundamental assumption of the present paper that most of the seismic displacements are generated by consecutive, transient activations of the plastic mechanisms evidenced by the pseudo-static analyses. Fig. 13(b) shows the deformed profiles of the wall facing. The average rotation for cases A and B is similar (about 18), whereas that for case C is about three times larger, since in this case only a global plastic mechanism is activated, which implies a significant rotational movement.

31


MASINI, CALLISTO AND RAMPELLO 2·0

Case A

T

γ _ 2

Case C

G

T

1·6 0·40 0·32

1·2

u: m

G

0·24

Case B

0·16 0·8

T

0·08 Case A

G

0·4

0 Limit analysis Trasl. – kc 0·101 εu ∞

0 0

4

2

6

8 t: s (a)

Case A

10

12

Case B

NZ-DFHS

Rot. – kc 0·132

16

Fig. 14. Contours of final shear strains computed with the DFHS record for case A with indefinitely ductile reinforcements

Case C 15

u: m 0

14

Rot. – kc 0·115

CONCLUSIONS The behaviour of an earth retaining structure subjected to an intense seismic loading can be conveniently regarded as a succession of transient activations of plastic mechanisms, each producing an incremental permanent displacement (Richards & Elms, 1979). For a geosynthetic-reinforced structure retaining a coarse-grained backfill, typical wavelengths are generally much larger than the wall height, and therefore the analysis of the plastic mechanism can be carried out with a sufficient accuracy using a pseudo-static method and assuming a uniform distribution of the inertial forces. Limit analysis and limit equilibrium methodologies can be advantageously extended to pseudo-static conditions and used iteratively to analyse a specific mechanism. However, in some cases the critical plastic mechanism for a given structure may not be obvious, and there may be more than one active mechanism. The actual deformation pattern can emerge more accurately from a pseudo-static analysis performed with a numerical discretisation of the entire soil domain, carried out up to the seismic coefficient that mobilises critical conditions (e.g. Callisto, 2014). Implementing this approach, it was shown that the reference structure A undergoes a composite deformation pattern that includes two concurrent plastic mechanisms; conversely, it was found that the seismic behaviour of both structures B and C is controlled by a single, well-defined mechanism, which is partly internal for structure B and totally external for structure C. It was shown that the dynamic behaviour of these retaining structures, resulting from the time-domain dynamic analyses, can be profitably interpreted on the basis of the corresponding pseudo-static plastic mechanisms; different earthquakes may induce different displacements, which, however, derive essentially from the same plastic mechanisms. The three different structures examined in this paper are characterised by about the same critical seismic coefficient, but under the actual dynamic loading they show a very different behaviour. The reference structure A is conceived to mobilise large plastic internal strains; although during construction it undergoes the largest displacements, it shows the best seismic performance with maximum horizontal displacements of 0 .03–0 .05H, close to the range computed with the empirical Callisto & Rampello (2013) relationship. The alternative structure B has shorter and stronger reinforcements and, under the same seismic input, undergoes larger displacements (0 .075H). This happens because the larger reinforcement strength inhibits the development of the internal plastic mechanism, subtracting an important source

3 12

6

y: m

9

3

0 (b)

Fig. 13. (a) Time histories of the horizontal displacements computed with the Assisi record for cases A and B with reinforcements of infinite ductility, and for case C; (b) deformed profiles of the wall façades

Since the seismic displacements derive from the activation of plastic mechanisms that are characterised by a given critical seismic coefficient, it is logical that a different seismic input would induce a different seismic performance resulting from the activation of similar plastic mechanisms. In fact, a dynamic analysis of the retaining structure A with reinforcements of infinite ductility, carried out using the large-amplitudes DFHS seismic record of Fig. 8, produces the larger displacements reported in Table 4, but nearly the same deformation pattern, as shown by a comparison of the contours of shear strains plotted in Fig. 14, relative to the DFHS record, with those of Fig. 9(a), referring to the Assisi record.

Table 4. Dynamic analyses. Final horizontal displacements induced by the input seismic records (m) Case

B/H

Record

A A B C

0 .75 0 .75 0 .53 0 .38

Assisi NZ-DFHS Assisi Assisi

Point G

Point T

0 .43 3 .70 1 .12 1 .92

0 .71 5 .60 1 .00 1 .61

32


SEISMIC behaviour OF REINFORCED-EARTH RETAINING STRUCTURES of energy dissipation from the structure. In this respect, it appears that an important criterion for an effective seismic design is to prefer long reinforcements with a relatively low strength. When analysing the seismic behaviour of a reinforcedearth structure it is recommendable that limit equilibrium and limit analysis methods be used iteratively to study the critical conditions and to inspect the corresponding plastic mechanism, rather than to evaluate the safety coefficient corresponding to a specific value of the seismic coefficient (Callisto, 2014). Contour lines similar to those of Fig. 6 can be used to see whether there is a single, well-defined mechanism, or to detect eventual local minima indicating concurring mechanisms. Structure layouts entailing more than one mechanism should be preferred for design. Structure C is an extreme case in that, having the same overall seismic strength of structure A, it can mobilise only external mechanisms. As there cannot be any internal energy dissipation, its seismic displacements are comparatively large with respect to those shown by the structures made of reinforced earth (0 .13H). This finding is consistent with field observations, indicating a generally superior performance of reinforced earth structures if compared to the behaviour of more traditional retaining structures. Typically, a reinforcing layer intersecting a sheared surface yields because its structural resistance is attained, rather than because of slippage at the soil–reinforcement interface. This is why the seismic behaviour of both retaining structures A and B, relying to a different extent on the strength of the reinforced areas, was found to be critically dependent on the ductility capacity of the reinforcing layers. While the actual mechanical behaviour of a geosynthetic depends on multiple factors, including ageing and environmental effects, it is clear that the choice of the most appropriate reinforcement should be made principally on the basis of the maximum elongation that it can maintain without important strength reduction. For the cases analysed in the present work, it appears that this maximum strain can be evaluated as about ten times the ratio umax /H of the maximum horizontal displacement to the wall height.

t u v z � ª � �u �y Łh Łi Ł0 �9

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ACKNOWLEDGEMENT The research work presented in this paper was partly funded by the Italian Department of Civil Protection under the ReLUIS 2009–2012 research project. NOTATION a amax B B, C, D c9 G G0 H IA Ks kc kh L p9 rh ri r0 Sa T Tm Ts TT

time horizontal displacement velocity vector depth inclination of the wall facing shear strain axial strain ultimate strain yield strain angle of log-spiral radius at the wall toe angle of log-spiral radius angle of log-spiral radius at the top of the wall angle of shearing resistance

acceleration peak ground acceleration length of geo-grid coefficients representing small-strain stiffness cohesion of foundation soil operative shear stiffness small-strain shear stiffness fill height Arias intensity stiffness critical seismic coefficient seismic coefficient reinforcement length mean effective stress log-spiral radius at the wall toe log-spiral radius log-spiral radius at the top of the wall spectral acceleration axial force mean period significant duration tensile strength

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MASINI, CALLISTO AND RAMPELLO sismico delle opere in terra rinforzata: esperienze da recenti terremoti, modellazione fisica e numerica. In Geosintetici in rilevati ed opere di sostegno: atti del XX convegno nazionale geosintetici, Bologna, Italy (ed. D. Cazzuffi), pp. 63–78. Bologna, Italy: Patron Editore (in Italian). Michalowski, R. L. (1998). Soil reinforcement for seismic design of geotechnical structures. Comput. Geotech. 23, No. 1–2, 1–17. Michalowski, R. L. & You, L. (2000). Displacements of reinforced slopes subjected to seismic loads. J. Geotech. Geoenviron. Engng, ASCE 126, No. 8, 685–694. Motta, E. (1996). Earth pressure on reinforced earth walls under general loading. Soils Found. 36, No. 4, 113–117. Newmark, N. M. (1965). Effects of earthquakes on dams and embankments. Fifth Rankine lecture. Geótechnique 15, No. 2, 139–193, http://dx.doi.org/10.1680/geot.1965.15.2.139. Rampello, S., Callisto, L. & Fargnoli, P. (2010). Evaluation of slope performance under earthquake loading conditions. Riv. Ital. Geotec. 44, No. 4, 29–41. Rathje, E. M., Abrahamson, N. A. & Bray, J. D. (1998). Simplified frequency content estimates of earthquake ground motions. J. Geotech. Geoenviron. Engng, ASCE 124, No. 2, 150–159. Richards, R. & Elms, D. G. (1979). Seismic behavior of gravity retaining walls. J. Geotech. Engng Div., ASCE 105, No. GT4, 449–464. Seed, H. B. & Idriss, I. M. (1970). Soil moduli and damping factors for dynamic response analysis, Report No. EERC 70-10. Berkeley, CA, USA: University of California. Shinoda, M., Watanabe, K., Kojima, K., Tateyama, M. & Horii, K. (2007). Seismic stability of reinforced soil structure constructed after the mid Niigata prefecture earthquake. In New Horizons in Earth Reinforcement, pp. 783–787. London, UK: Taylor and Francis. Tatsuoka, F. (2008). Geosynthetics engineering, combining two engineering disciplines. Proceedings of the 4th geosynthetics Asia, Shanghai, China, Special Lecture, pp. 1–35. Vucetic, M. & Dobry, R. (1991). Effect of soil plasticity on cyclic response. J. Geotech. Geoenviron. Engng, ASCE 117, No. 1, 89– 107. Wartman, J., Rondinel-Orviedo, E. A. & Rodriguez-Marek, A. (2006). Performance and analysis of mechanically stabilized earth walls in the Tecoman, Mexico Earthquake. J. Perf. Constr. Facilities, ASCE 20, No. 3, 287–299. Watanabe, K., Munaf, Y., Koseki, J., Tateyama, M. & Kojima, K. (2003). Behaviour of several types of model retaining walls subjected to irregular excitation. Soils Found. 43, No. 5, 13–27. Yegian, M. K., Marciano, E. A. & Ghahraman, V. G. (1991). Earthquake induced permanent deformations probabilistic approach. J. Geotech. Engng, ASCE 117, No. 1, 1158–67.

parameters on the seismic response of reduced-scale reinforced soil retaining walls. Geotextiles Geomembranes 25, No. 1, 33–49. Itasca (2005). FLAC fast Lagrangian analysis of Continua v. 5. 0. User’s manual. Minneapolis, MN, USA: Itasca Consulting Group. Izawa, J., Kuwano, J. & Ishihara, Y. (2004). Centrifuge tilting and shaking table tests on the RSW with different soils. Proceedings of the 3rd Asian regional conference on geosynthetics, Seoul (eds J. B. Shim, C. Yoo and H. Y. Jeon), pp. 803–910. Seoul, Korea: Korean Geosynthetic Society. Joyner, W. B. & Chen, A. T. F. (1975). Calculation of nonlinear ground response in earthquakes. Bull. Seismol. Soc. Am. 65, No. 5, 1315–1336. Koseki, J., Nakajima, S., Tateyama, M., Watanabe, K. & Shinoda, M. (2009). Seismic performance of geosynthetic reinforced soil retaining walls and their performance-based design in Japan. In Proceedings of performance-based design in earthquake geotechnical engineering, Tokyo (eds T. Kokusho, Y. Tsukamoto and M. Yoshimine), pp. 149–161. London, UK: Taylor and Francis Group. Kramer, S. L. & Paulsen, S. B. (2001). Seismic performance of MSE structures in Washington State. Proceedings of the international geosynthetic engineering forum, Taipei, Taiwan, pp. 145– 173. Taiwan: International Geosynthetics Society. Kramer, S. L. & Paulsen, S. B. (2004). Seismic performance evaluation of reinforced slopes. Geosynth. Int. 11, No. 6, 429– 438. Leshchinsky, D. (2001). Design dilemma: use peak ore residual strength of soil. Geotextiles Geomembranes 19, No. 2, 111–125. Lin, J. S. & Whitman, R. V. (1986). Earthquake induced displacements of sliding blocks. J. Geotech. Engng, ASCE 112, No. 1, 44–59. Ling, H. I. & Leshchinsky, D. (1998). Effect of vertical acceleration on seismic design of geosynthetic-reinforced soil structures. Geótechnique 48, No. 3, 347–373, http://dx.doi.org/10.1680/ geot.1998.48.3.347. Ling, H. I., Leshchinsky, D. & Perry, E. B. (1997). Seismic design and performance of geosynthetic-reinforced structures. Geótechnique 47, No. 5, 933–952, http://dx.doi.org/10.1680/geot.1997. 47.5.933. Ling, H. I., Mohri, Y., Leshchinsky, D., Burke, C., Matsushima, K. & Liu, H. (2005). Large-scale shaking table tests on modular – block reinforced soil retaining walls. J. Geotech. Geoenviron. Engng, ASCE 131, No. 4, 465–476. Masing, G. (1926). Eigenspannungen und Verfertigung bim Messing. Proceedings of the 2nd international congress on applied mechanics, pp. 322–355. Zurich, Switzerland and Leipzig, Germany: Fu¨ssli (in German). Maugeri, M. & Biondi, G. (2008). Analisi del comportamento

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Verrucci, L. et al. (2015). Geótechnique 65, No. 5, 359–373 [http://dx.doi.org/10.1680/geot.SIP.15.P.012]

Cyclic and dynamic behaviour of a soft pyroclastic rock L . V E R RU C C I , G . L A N Z O , P. TO M M A S I † a n d T. ROTO N DA

The mechanical behaviour under cyclic and dynamic conditions of a weakly cemented pozzolana, taken from the subsoil of a historical town in a seismically active area, was investigated through laboratory tests and in situ measurements. The geotechnical characterisation at laboratory scale is of particular importance as the small- and large-scale behaviour of these deposits is comparable, owing to the large discontinuity spacing. These soft pyroclastic rocks, which are found in many volcanic regions worldwide, are particularly interesting as they are often alternated with stronger volcanic rocks in thick sequences, constituting a significant impedance contrast that could increase the groundshaking hazard. A cross-hole test and a spectral analysis of surface waves test, carried out on the ground surface and at the floor of an underground cavity, respectively, provided in situ vertical profiles of the shear wave velocity. Cyclic and dynamic properties were investigated in the laboratory through velocity measurements of ultrasonic pulses, cyclic simple shear, torsional shear and resonant column tests. Most testing procedures and devices, which were originally conceived for soils, posed some challenges. The collected data were used to analyse the influence of mean confining stress, strain amplitude and number of cycles on both shear stiffness and material damping. KEYWORDS: dynamics; in situ testing; laboratory tests; particle crushing/crushability; soft rocks

grain surfaces, which in turn affects the micro-mechanisms dominating at the grain contacts (local breakage). The behaviour of volcanic soils for increasing strain amplitude appears more linear than those of other granular materials with similar grain size distribution (e.g. Marks et al., 1998) and the linear elastic threshold ªl (Vucetic, 1994) is higher. Senetakis et al. (2013) suggest that these features of the cyclic behaviour are possibly due to crushing of the asperities at the particle contact, which would also have an impact on energy dissipation mechanisms. Soft tuffs and weakly cemented pozzolanas form a particular class of pyroclastic materials that present a transitional behaviour between rocks and non-cohesive pyroclastites. Laboratory dynamic testing on undisturbed specimens of these materials is not common, owing to difficulties in sampling and in adapting the traditional cyclic and dynamic testing devices. Guadagno et al. (1988) and Papa et al. (1988) tested some cemented pozzolanas together with other incoherent specimens. Similar to other non-cohesive volcanic soils, the stiffness and dissipative behaviour of cemented pozzolanas over the medium strain range is more linear than that of quartz sands. Furthermore, a significant and simultaneous reduction of both stiffness and damping ratio with the number of cycles was attributed to the progressive breakage of particle bonds which occurs without a significant modification of spatial arrangement and size distribution (Papa et al., 1988). This study reports the dynamic and cyclic characterisation of a weakly cemented pozzolana through laboratory and in situ tests. Correlation of stiffness and damping with basic physical properties and their dependence on the stress level, applied strain amplitude and cycle number have been investigated. Furthermore, the different results obtained from various types of test apparatus have been highlighted. Analogies with the mechanical behaviour in static or monotonic loading conditions and possible relationships with the microstructure of the material have also been explored. Certain particular behaviours of the investigated material can be extended to many other slightly cemented pyroclastites in peninsular Italy with similar lythological features and physical properties (Pellegrino, 1969; Cecconi et al., 2010),

INTRODUCTION Pyroclastic deposits cover wide areas, with considerable local heterogeneity that derives from the variable modes of deposition and post-depositional alteration processes. Alternation with other volcanic products, such as effusive rocks, further increases the geotechnical variability and can result in high impedance contrasts and other conditions that influence the local seismic response. The rock-like pyroclastic materials (e.g. welded tuffs and many ignimbrites) are usually dynamically characterised, as with other rocks, only by measuring the small-strain shear wave velocity VS and damping ratio D0. These materials are so stiff that their stress–strain behaviour is linear even during strong earthquakes. Nevertheless comprehensive dynamic characterisations of ash-flow tuffs and ignimbrites from Nevada and New Mexico (US) can be found in Choi (2008) and Jeon (2008), extending up to the mildly non-linear range (up to strains of about 0 .02%). The dynamic properties of incoherent medium- to coarsegrained pyroclastic soils, mainly composed of pumice grains of low density and high porosity, have been investigated through bender element measurements (Liu & Yang, 2014), cyclic triaxial tests (Marks et al., 1998; Miura et al., 2003; Sahaphol & Miura, 2005), resonant column tests (Senetakis et al., 2012; Senetakis et al., 2013) and cyclic torsional shear tests (Orense et al., 2012). Comparison with reference quartz sands has shown that volcanic soils exhibit significantly lower small-strain stiffness and slightly lower smallstrain damping. Differences in the observed response are attributed to the higher void ratio and to the lower dry density of volcanic soils, as well as to the morphology of Manuscript received 1 April 2014; revised manuscript accepted 4 February 2015. Discussion on this paper closes on 1 October 2015, for further details see p. ii. Department of Structural and Geotechnical Engineering, Sapienza University, Rome, Italy. † Institute for Environmental Geology and Geo-Engineering, National Research Council, c/o Faculty of Engineering, Sapienza University, Rome, Italy.

35

Downloaded by [ University of Liverpool] on [12/09/16]. Copyright © ICE Publishing, all rights reserved.

VERRUCCI, LANZO, TOMMASI AND ROTONDA of the experimental investigation presented. Tuff and pozzolana facies were probably deposited by pyroclastic flows with different water vapour contents (Faraone & Stoppa, 1988). More specifically, tuff is believed to be the result of a vapour phase alteration given the widespread growth of zeolite minerals, which are absent in the pozzolana facies. Although the spatial distribution of the two main facies within the slab has not been fully ascertained, a homogeneous stratigraphy is recognisable over a wide sector of the southern side of the hill (site SF in Fig. 1). Here the cliff (35–40 m high) is almost entirely formed by pozzolana, overlying a layer of tuff that is a few metres thick (Fig. 2). At the SF site the pozzolana can be subdivided into an upper weaker lithotype (SFW) and a lower, richer in scoriae, stronger (SFH) lithotype. Pozzolana samples were also collected 200 m to the east (site SB in Fig. 1). Here, too, the pozzolana, similar to the SFW lithotype, overlies the tuff. Although from a macroscopic point of view only the scoria content can differentiate the pozzolana lithotypes, laboratory tests and petrographic analyses highlighted some differences. Focus was on the SFW lithotype, the only one to be investigated through all the test methods. Nevertheless, results of tests conducted on the other lithotypes (SFH and SB), especially in situ tests and laboratory measurements of ultrasonic pulse velocity, offer other insights into the cyclic and dynamic behaviour of the pozzolana overall. A physical and mechanical characterisation under static conditions of the Orvieto pozzolana is reported in Rotonda et al. (2002) and Tommasi et al. (2015), covering all the lithotypes as well as providing some comparisons with the tuff facies. At a macroscopic observation pozzolana appears to be

although the wide variation in pyroclastic materials should not be misestimated. PHYSICAL AND STATIC PROPERTIES The products of the extinct volcanic apparata of central western Italy, located north of Rome, form a wide plateau of layered pyroclastic materials and effusive rocks, which overlies a thick Plio-Pleistocene sedimentary substratum of clayey and sandy units. The hydrographic network has eroded the plateau, especially at its distal areas, exposing the sedimentary substratum and producing a landscape characterised by gorges, canyons and mesas. Many old towns are perched on top of these mesas, often dating back to the Middle Ages. Owing to its inestimable artistic heritage, Orvieto is the most famous of these. It extends over the entire top (about 0 .8 km2) of a pyroclastic slab (Fig. 1), delimited by vertical scarps up to 60 m high, which caps a clay base with gentle slopes (Tommasi et al., 2013). Between the slab and the clay there is a thick layer of fluviolacustrine sediment which hosts a perched groundwater, so that fully saturated conditions do not occur within the pyroclastic slab. The whole slab consists of pyroclastic products belonging to the Tufo di Orvieto e Bagnoregio formation, deposited about 330 000 years BP (before present), in the final period of activity of the Bolsena caldera (Santi, 1991), during a major eruption covering a 200 km2 area (Nappi et al., 1994). From the geotechnical point of view two materials are observed: a rock-like, red–yellow-coloured facies, and a slightly cohesive, grey-coloured facies, hereafter termed ‘tuff’ and ‘pozzolana’, respectively. The latter is the object

SF site

Pyroclastic slab

SB site Fig. 2

Clayey slope

Fig. 1. View of the Orvieto Hill from the south with indication of sampling sites

Duomo

San Francesco

Cross-hole Cavity 536 (SASW)

Cavity 508

SFW pozzolana SFH pozzolana

Tuff

Fig. 2. View of the southern cliff underneath the S. Francesco church (SF zone) with location of the investigation and sampling sites

36


CYCLIC AND DYNAMIC BEHAVIOUR OF A SOFT PYROCLASTIC ROCK constituted of clasts immersed in a grey aphanitic ground mass (Fig. 3(a)). The size of pumices and scoriae, the most numerous clasts, varies from a few millimetres to several centimetres, whereas that of lithic fragments does not exceed 10 mm. The material does not disaggregate when completely soaked, but under an appreciable finger pressure it is slightly friable at any water content. The ground mass is responsible for the disaggregation under loads, pumices and scoriae being markedly tougher. Scanning electron microscopy (SEM) analyses of SF specimens (Tommasi et al., 2015) showed that the ground mass is formed by extremely small glass lumps that are welded to each other over very small areas (Fig. 3(b)). These act as struts of a weak but strongly three-dimensional frame. The size of interparticle pores is similar to that of the glass lumps. At high magnification each lump appears to be an aggregate of closely welded minuscule glassy fragments, in turn containing extremely small voids. The groundmass of the pozzolana sampled at the SB site is more porous and is formed by less continuous and smaller glass particles (Fig. 3(c)). Owing to the high porosity n of most of its constituents (ground mass, pumices, scoriae), the Orvieto pozzolana is highly porous (Table 1), much more than other similar materials, such as the pozzolanas from the Rome area (Cecconi et al., 2011). According to the grain size distribution (Fig. 4), all of the lithotypes are classified as sandy gravel. Based on the main static strength and deformability index parameters (Table 1), the material can be considered to be a rock of extremely low uniaxial strength f and medium to high modulus ratio (Et50 /f). Under oedometric and isotropic compression, permanent strains are observed from the lowest stress values; hence it is not possible to define a linear elastic phase. Of greater note is the variation of the secant Young’s modulus measured during uniaxial monotonic compression in the medium- to

Silt

Sand

100 SFH

Percentage passing: %

80

SFW 60

SB

40

20

0 0·01

0·1

1 d: mm

(a)

10

Fig. 4. Grain size distribution curves of the pozzolana lithotypes

high-strain range (0 .01–0 .1%, Fig. 5). All specimens exhibit, presumably after the coupling effects between testing machine and specimen are extinguished, a decrease of the Young’s modulus up to an axial strain of 0 .02–0 .05%. A further gradual increase of the Young’s modulus, up to large axial strains (about 0 .2%), is then observed. LABORATORY AND IN SITU INVESTIGATIONS A couple of vertical boreholes were drilled to perform cross-hole (CH) measurements in the SFW and SFH pozzolana. A spectral analysis of surface waves (SASW) test was carried out in an ancient sub-horizontal underground pozzolana quarry excavated in the upper portion of the cliff (cavity 536 in the census of Regione Umbria, Fig. 2). The laboratory tests were conducted on specimens retrieved at the SF site from the CH borehole cores and from two block

80 μm

80 μm

10 mm

Gravel

(c)

(b)

Fig. 3. (a) Picture of the pozzolana; SEM microphotographs of ground mass of the (b) SFW and (c) SB lithotypes Table 1. Physical and static mechanical properties of the three pozzolana lithotypes SFW, SFH and SB (< x >: mean value; STD: standard deviation; Nd: number of data) SFW

rs: Mg/m3 rd: Mg/m3 n e0 p9y : MPa Et50: MPa f: MPa t: MPa

SFH

SB

,x.

STD

Nd

,x.

STD

Nd

,x.

STD

Nd

2 .59 1 .07 0 .59 1 .47 4 .4 321 1 .18 0 .56

– 0 .04 0 .02 0 .09 – 132 0 .32 0 .11

4 57 57 57 3 4 4 7

2 .63 1 .16 0 .56 1 .27 – 1106 2 .28 0 .90

– 0 .05 0 .02 0 .09 – 391 0 .51 –

2 37 37 37 – 17 17 3

2 .54 1 .11 0 .57 1 .30 3 .7 394 1 .49 0 .14

– 0 .09 0 .03 0 .15 – 187 0 .71 0 .03

1 48 48 48 4 15 15 10

Note: rs ¼ density of the solid particles; rd ¼ dry bulk density; n ¼ porosity; e0 ¼ void ratio; p9y ¼ isotropic yielding stress; Et50 ¼ Young’s modulus at 50% of failure stress; f ¼ uniaxial strength; t ¼ tensile strength.

37


VERRUCCI, LANZO, TOMMASI AND ROTONDA 1500

the specimen preparation are reported in Tommasi et al. (2015). All tests were carried out on dry specimens; the specimen sizes are shown in Table 2. 50% of failure stress

Measurements of the ultrasonic pulse velocity. Ultrasonic Pand S-wave velocities were measured in the laboratory on specimens prepared for static and cyclic/dynamic tests. The apparatus consisted of a Pundit ultrasonic pulse generator, a pair of piezoelectric contact transducers (diameter 12 .7 mm and natural frequency 1 MHz), an amplifier with gain up to 60 dB, and a digital oscilloscope. A square wave was used as input pulse. To improve the specimen–transducer coupling, a pressure of 320 kPa at contact was applied and a thin film of cane molasses was interposed between the transducers and the specimen ends, which filled the surface voids of the material.

Esec: MPa

1000

SB 500

0

0

0·05

0·10

εa: % (a)

0·15

0·20

Double-specimen direct simple shear apparatus. Tests were conducted through a double-specimen direct simple shear apparatus (DSDSS), an evolution of the single-specimen device developed at the Norwegian Geotechnical Institute (NGI-DSS) (Bjerrum & Landva, 1966). The apparatus, which is schematically illustrated in Fig. 6, is designed for testing soil specimens under simple shear strain conditions. A

0·25

1500

SFH

Table 2. Sizes of specimens for cyclic/dynamic mechanical tests

1000

Esec: MPa

Test type

UPV

Diameter, D: mm Height, H: mm

DSDSS

35 .7–66 .6 20 .0–100 .0

TS–RC

66 .6 20 .0

35 .7 70 .0

Note: UPV ¼ ultrasonic pulse velocity measurement; DSDSS ¼ double specimen direct simple shear test; TS ¼ torsional shear test; RC ¼ resonant column test.

500 SFw

P 0

0

0·05

0·10

εa: % (b)

0·15

0·20

0·25

Top cap D

Fig. 5. Secant Young’s modulus Esec plotted against axial strain a from uniaxial monotonic tests conducted on the (a) SB and (b) SFW and SFH lithotypes

Proximity transducer

Specimen

h δ F

samples that were manually extracted from the cavity 508 wall (Fig. 2). Other core specimens were obtained from a horizontal borehole at the SB site.

Middle cap Load cell O-ring seal

Laboratory test equipment and testing programme The experimental investigation comprised a series of dynamic (ultrasonic pulse velocity (UPV) and resonant column (RC)) as well as cyclic (simple shear and torsional shear) tests. The ultrasonic pulse velocity tests were performed on the three pozzolana lithotypes, but only the SFW lithotype was used for the other cyclic and dynamic tests. Owing to the high drillability contrast between the ground mass and the clasts, the usual coring and grinding procedures could not be used. Therefore, specimens were retrieved from partially saturated frozen samples. On a similar pozzolana (Cecconi, 1998) the decrease of the dynamic Young’s modulus induced by such a preparation method did not exceed 12% of that of the original material. Details about

Specimen

Reinforced rubber membrane

Bottom cap Frame

Fig. 6. Section of the DSDSS apparatus (Doroudian & Vucetic (1995), modified). Adapted, with permission, from Geotechnical Testing Journal, vol. 18, no. 1, March 1995, copyright ASTM International, 100 Barr Harbor Drive, West Conshohocken, PA 19428

38


CYCLIC AND DYNAMIC BEHAVIOUR OF A SOFT PYROCLASTIC ROCK mean effective confining stress p9 between 0 and 500 kPa. In the cyclic TS test a series of five torsional cycles (frequency f ¼ 0 .1 Hz) was applied to the specimen, with ªc increasing from about 0 .0001 to 0 .005%. For the dynamic RC tests, some thousands of high-frequency cycles of the torque (variable frequency f ¼ 30–90 Hz) were applied, yielding a ªc from about 0 .0005 to 0 .05%. In these cases the stiffness was calculated from the resonance conditions and the damping from the frequency response curve, using the half-power bandwidth method. As TS and RC tests were performed consecutively on the same specimen, the sequence of the applied static and dynamic loads was chosen so that the specimen would undergo an increasing maximum strain amplitude. Therefore, the TS tests at all times preceded the RC tests at the same torsional load.

description and details of the DSDSS device are given elsewhere (Doroudian & Vucetic, 1995; D’Elia et al., 2003). To summarise, the vertical load P is applied to the top cap, whereas the horizontal cyclic load F is applied by a micrometer connected to the horizontal piston. A proximity transducer measures the horizontal displacement � of the middle with respect to the bottom cap. Minor false deformations can be induced only by the shear deformability of the caps and possible slips at the specimen–cap interface. These are minimised by using stainless steel caps and by sticking the specimens’ ends to all the caps with a twocomponent adhesive. When used on deformable soils the lateral confinement of the specimens is achieved through wire-reinforced rubber membranes that ensure the absence of lateral strains. Owing to the stiffness of the pozzolana and to the unavoidable, although extremely thin, annular gap between the lateral surface of the cored specimens and the membrane, the lateral constraint cannot be completely achieved by such a method. Nevertheless, a significant horizontal confinement is applied at the end faces, which are firmly fixed to the caps, especially for very short specimens. In some check tests no noticeable differences arose between stress–strain curves obtained with or without membranes. Therefore, an unknown horizontal stress �h acts during the tests, which does not exceed �h,oed ¼ Koed�v, where Koed is the coefficient of earth pressure at rest in oedometric conditions.

In situ measurements The CH test was performed between two 35 m deep vertical boreholes, spaced at 3 .6 m and 15 m from the cliff edge. To measure the P-wave transit time a spark generator and hydrophone were utilised as pulse source and receiver, respectively. Regarding the S-wave, an electro-dynamic shockwave generator and a geophone (natural frequency 14 Hz) were used. The measures were spaced 1 m along the borehole axes. The underground quarry where the SASW test was performed (Rotonda et al., 2002) extends perpendicular to the cliff wall at about 12 m below the top surface of the pyroclastic slab and 30 m east of the CH test site. A 40 m survey line on the floor of the quarry was investigated; the impact on the ground was obtained with a 170 N weight falling from a height of 1 .5 m. In the instrumentation scheme, two piezoelectric transducers, with a 50 kHz resonant frequency, positioned along the same alignment with the impacting weight, were utilised. The common receiver mid-point scheme (Nazarian, 1984) was used. The transmission velocity for each frequency, related to different depths, was calculated using Fourier analyses. The inversion of the dispersion curves provided the vertical profile of the Rayleigh wave velocity. Shear velocity was determined assuming a dynamic Poisson coefficient �dyn ¼ 0 .3.

Torsional shear and resonant column apparatus. The apparatus used for the torsional shear (TS) and the resonant column (RC) tests is an evolution of the first free-fixed type machine designed at the University of Texas at Austin (Isenhower, 1979); its use is widespread in dynamic research on soils (e.g. Allen & Stokoe, 1982; Ni, 1987; Silvestri, 1991). The test system consists of a cylindrical specimen with the bottom end (passive end) fixed on a base pedestal, while the top active end is attached to an excitation device capable of applying a torquing excitation. Transducers are used to measure the rotational vibration amplitudes of the driving system. The specimen ends for the TS–RC tests were also stuck to the platens, thus ensuring full compliance. The vibrating system was positioned in an air pressure chamber where an isotropic pressure p was applied.

EXPERIMENTAL RESULTS Laboratory tests UPV tests. Elastic wave velocities are reported in Fig. 7 for specimens longer than 25 mm for the three lithotypes. Measures for the shortest specimens were rejected because of the difficulty in establishing the arrival time, owing to the near-field source. Moreover, very short ray paths have a high probability of developing almost entirely through the stiffest components, thus overestimating the material global elastic properties. The mean elastic properties of the three lithotypes obtained from the UPV measurements are reported in Table 4. Despite the differences in wave velocity, the Poisson ratio �dyn of the pozzolana does not vary much from 0 .3. The glass forming the scoriae has a very high dynamic stiffness, although the density is similar to that of the overall material. The main physical character influencing the dynamic elastic properties is the abundance of voids, which is well represented by porosity n. Fig. 8 shows that the scattering of a single lithotype is noticeable, but the difference in porosity from one lithotype to another is associated with a significant change in dynamic stiffness. Taking into account all the measures, a correlation between wave velocities and porosity is observed, which is more marked for the longitudinal wave

Testing programme. Table 3 summarises the testing programme applied entirely on dry specimens. VP and VS values measured with the UPV test are listed, together with the derived dynamic elastic moduli Gdyn, Edyn and Poisson ratio �dyn. To assess the Young’s modulus, laterally constrained conditions for the P-wave propagation were assumed. In the DSDSS tests, each series of tests on a pair of specimens consisted of several loading stages, with a � v9 varying between 100 and 800 kPa. Assuming K0 ¼ 0 .43 (coefficient of pressure at rest for an elastic material with Poisson coefficient � ¼ 0 .3 deformed in laterally constrained conditions) the estimated mean effective stress p9 varied between about 60 and 500 kPa. For each loading stage, the specimens were subjected to several steps of strain-controlled cyclic shearing, gradually increasing the amplitude of shear strain ªc from 0 .0004% to 0 .1%. In one test only, a maximum value of ªc equal to 0 .3% was reached. The number of cycles (N) applied in each step, at constant ªc values, varied between 5 and 10. The shape of cyclic loading was approximately sinusoidal, with a frequency ranging between 0 .1 and 0 .3 Hz. The low-amplitude TS and RC tests were performed at a

39


40


0 .59

1 .11

SF-BL2r

1 .42

1 .43

1 .51

1 .42

1 .57

1 .40

1 .52

e0

970

852

916

1060

1255

1271

1218

VP: m/s

468

394

439

556

574

633

618

VS: m/s

0 .35

0 .36

0 .35

0 .31

0 .37

0 .34

0 .32

�dyn

UPV test

244

173

207

345

347

456

408

Gdyn: MPa

658

471

560

904

948

1215

1080

Edyn: MPa

TS–RC

DSDSS

0 .0001–0 .01 (TS) 0003–0 .1 (RC)

0 .0004–0 .3

ªc: %

0 .1 (TS) 30–80 (RC)

0 .1–0 .3

f: Hz

5 (TS) . 1500 (RC)

10

N

DSDSS–TS–RC tests

0 60 120 250 500 0 60 120 250 500 0 60 120 250 500 0 60 120 250 500

223 446 124 248 495 248 495

p9: kPa

150 241 223 288 359 128 211 201 272 321 111 187 210 277 337 162 223 242 290 348

226 235 180 198 284 194 237

G0: MPa

2 .6 1 .8 1 .6 2 .1 1 .1 2 .1 1 .7 0 .8 1 .2 1 .4 1 .9 1 .7 0 .6 1 .5 0 .7 1 .6 2 .3 1 .3 1 .8 1 .8

1 .2 1 .3 1 .5 1 .3 1 .6 1 .8 1 .4

D0: %

Note: VP, VS ¼ elastic wave velocity from UPV tests; �dyn, Gdyn and Edyn ¼ elastic properties from UPV tests; ªc, f, N ¼ strain amplitude range, test frequency and maximum number of cycles of the DSDSS, TS and RC tests; G0, D0 ¼ shear modulus and damping ratio at minimum strain amplitude from DSDSS and TS tests.

0 .59

1 .11

0 .59

1 .11

SF-BL2n

SF-BL2q

0 .61

1 .05

SF-BL2de

0 .60

0 .58

1 .12

SF-BL2bc

1 .07

0 .60

1 .07

SF-BL2ag

SF-BL2o

n

rd: Mg/m3

Code names of specimens

Table 3. Ultrasonic pulse velocity (UPV), double-specimen direct simple shear (DSDSS) and torsional shear and resonant column (TS–RC) testing programme

VERRUCCI, LANZO, TOMMASI AND ROTONDA

CYCLIC AND DYNAMIC BEHAVIOUR OF A SOFT PYROCLASTIC ROCK 2000

VS: m/s

SB

VP: m/s

ν0

SFH

1600

Scoriae

2000

SFW

0·1

1500

1000 1200

Scoriae

0·3

0·50

0·4

800

0·55

n (a)

0·60

0·65

1500

VS: m/s

Scoriae

400

800

1200

1600

1000

500

2000

Vp: m/s

0 0·50

Fig. 7. Longitudinal (VP) plotted against shear (VS) elastic wave velocities from UPV tests for all the pozzolana lithotypes

0·55

n (b)

0·60

0·65

3000 Scoria

Kdyn: MPa

velocity VP. The data trend is consistent with that predicted by models of poroelastic materials with isometric pores (Rotonda & Ribacchi, 1995). DSDSS tests. The results of a typical DSDSS test are shown in Fig. 9 in terms of stress–strain loops for cyclic shear strain amplitude ªc, ranging from 0 .0004% to 0 .3%. The figure highlights that, at strains as small as 0 .0004%, both shear modulus and damping have been accurately determined. Owing to ambient vibrations, measurements of damping ratio at very small strains were rejected in some tests. Plots in Fig. 9 show a typical decrease in shear modulus and an increased damping ratio for increasing shear strain amplitude, up to ªc , 0 .1%. At the highest strain amplitudes (ªc ¼ 0 .1% and 0 .3%) the shape of the loop tips changes. This is more evident in the enlarged picture referring to ªc ¼ 0 .3%, where the stress–strain curve shows an upwards curvature, indicating a stiffening of the material close to the tips of the loops. This strain-hardening behaviour has also been observed in static tests at comparable strain levels (Fig. 5). Average values of the shear modulus G and damping ratio D for the three pairs of specimens tested in the DSDSS apparatus are plotted against ªc in Fig. 10 for different values of p9. G and D data points are connected with solid lines. The maximum shear modulus G0 calculated by extrapolating the stiffness decay curves to ªc ¼ 0 .0001% (Table 3) varies between 180 and 280 MPa in the range of p9,100–500 kPa. At the maximum ªc values (ªc ¼ 0 .1%) one specimen (SF-BL2ag) experienced an approximately 50% reduction in shear modulus with respect to G0. Reduc-

2000

1000

0 0·50

0·55

n (c)

0·60

0·65

Scoria 1500

Gdyn: MPa

SFW 1000

SFH SB

500

0 0·50

0·55

n (d)

0·60

0·65

Fig. 8. Wave velocities ((a) longitudinal Vp and (b) shear VS) and elastic moduli ((c) bulk Kdyn and (d) shear Gdyn) from UPV tests plotted against porosity n of the specimens

tion for the other two specimens (SF-BL2bc, SF-BL2de) is smaller, especially at lower values of confining stress (p9 ¼ 60 kPa and p9 ¼ 124 kPa). Damping ratio is under 2% at small-strain amplitudes and

Table 4. Elastic properties of pozzolana lithotypes from UPV tests (< x >: mean value; STD: standard deviation; Nd: number of data). VP: m/s

SFW SFH SB Scoriae

VS: m/s

,x.

STD

Nd

,x.

STD

Nd

1000 1516 1283 2216

148 166 154 –

35 30 9 2

502 751 766 1312

80 100 73 –

33 29 9 2

41


rd: Mg/m3

Gdyn: MPa

Edyn: MPa

�dyn

1 .07 1 .16 1 .05 1 .00

275 652 618 1715

734 1743 1510 4220

0 .33 0 .34 0 .22 0 .23


γc 0·0004%

0

2 0·001

0 γ: %

0

4 0·001

0·001

20

γc 0·004% N2

τ: kPa

N2

τ: kPa

τ: kPa

N1

0 γ: %

0·001

80

0

10 0·005

γc 0·04%

γc 0·1%

N2

N2

N2

0

20 0·010

0 γ: %

0·010

0 γ: %

0·005

0 γ: %

0·100

200

γc 0·01%

τ: kPa

τ: kPa

10

γc 0·001%

τ: kPa

2

0

80 0·040

0 γ: %

0·040

0

200 0·100

600 γc 0·3% N2

τ: kPa

τ: kPa

100

0

0

600 0·300

0 γ: %

0·300

0

γ: %

0·200

Fig. 9. Stress–strain cycles (�–ª) of a typical series of DSDSS tests (specimen SB-BL2bc) at increasing shear strain amplitude ªc

attains maximum values of about 6% at ªc � 0 .1%. The decrease in damping ratio with strain amplitude, observed at high strains in some tests (e.g. specimen SF-BL2bc), could be associated with the change in shape of the stress–strain loop. Further, a general decrease of D as the mean confining stress increases can be noted. The evolution of stiffness and damping parameters with the number of cycles N, at a given strain amplitude ªc, is shown in Fig. 11 for the highest stress levels (p9 ¼ 240 kPa and p9 ¼ 495 kPa). A maximum of ten cycles was applied during each strain-controlled test. The reduction in damping ratio with the number of cycles is significant in all tests, particularly in the earliest cycles and at higher strain amplitudes. A small reduction of the G modulus with cycles is noticeable only for ªc ¼ 0 .1%. Of particular interest are the results of a test on the SB pozzolana lithotype conducted with the specific aim of investigating the effect of a large number (130) of cycles. The adopted vertical stress corresponds to that acting in situ

at the sampling site (corresponding to a mean stress value of 440 kPa). In Fig. 12, degradation indexes �G ¼ GN/G1 and �D ¼ DN/D1 are plotted against N for ªc ¼ 0 .1%, G1 and GN being the shear moduli at the first and at the Nth cycle, and D1 and DN the analogous damping ratio values. A progressive reduction in stiffness (,15%) and damping (. 20%) can be observed. Although these results refer to a lithotype (SB) characterised by a more delicate structure (Tommasi et al., 2015), they indicate the noticeable effect of a large number of cycles on the stiffness and damping degradation of the pozzolana, similar to the behaviour observed by Papa et al. (1988) in cemented pozzolanas. TS/RC tests. Results from both TS and RC tests are illustrated in Fig. 13 in terms of variation of G and D with the shear strain amplitude at different values of the mean confining stress p9. The strain ranges investigated by TS and RC tests span from 0 .0001% to about 0 .01% and from

42



8

SF-BL2ag

300 200

6 p 446 kPa

D: %

G: MPa

400

SF-BL2ag

4

p 223 kPa 2

100

p 223 kPa p 446 kPa

0

0·0001

0·001

γc: %

0·01

500

0

0·1

100 0

p 248 kPa p 124 kPa p 62 kPa

0·0001

0·001

γc: %

0·01

0

0·1

p 124 kPa p 248 kPa 0·0001

p 495 kPa 0·001

γc: %

0·01

0·1

SF-BL2de

6 p 495 kPa

D: %

G: MPa

0

SF-BL2bc

8

SF-BL2de

p 248 kPa p 124 kPa

100

0·1

4

2

400

200

0·01

6

500

300

γc: %

p 495 kPa

D: %

G: MPa

200

0·001

8

SF-BL2bc

400 300

0·0001

2

p 62 kPa 0·0001

4

0·001

γ c: %

0·01

0

0·1

p 248 kPa p 495 kPa 0·0001

0·001

γc: %

0·01

0·1

Fig. 10. Shear modulus (G) and damping ratio (D) plotted against strain amplitude ªc from DSDSS tests

0 .0004% to about 0 .1%, respectively. The stiffness values obtained from the TS tests are in satisfactory accordance with those from dynamic RC tests. Generally in the medium strain range, where experimental data from the two tests overlap, G values from RC tests are slightly smaller than those from TS tests for the same mean stress. In fact such a weak material suffers significant damage during RC tests, which apply a very high number of cycles. Therefore G from RC tests is underestimated with respect to that obtained from the other test types. At small-strain amplitudes the shear modulus G0 varies in the range of 150–350 MPa for all the tested specimens. The increase of the mean stress results in a general increase in shear stiffness. In some tests, at high values of p9 the slope of the decay curves decreases, similarly to that observed in the DSDSS tests. Damping ratios from TS and DSDSS tests show a similar trend. Conversely, damping ratios from RC tests are smaller than those obtained from TS tests at a comparable mean confining stress and often show a significant decrease at the highest strain amplitudes.

increases from about 400 to 570 m/s, with a low, steady gradient from the surface down to 29 m depth, where the boundary between the two different pozzolana lithotypes is apparent, as already observed on the cliff wall. The values measured in the underlying SFH lithotype vary from 650 to 730 m/s. P-wave measurements were rejected because a strong attenuation of the signal, typical for this kind of materials, caused uncertainty regarding arrival time. The VS profile obtained with the SASW test (Fig. 14) reaches a depth of about 8 m below the floor of the underground quarry, that is, 20 m below the ground surface. A 2 .5 m thick layer below the floor of the underground cavity shows very low dynamic characteristics, due to both loosening and excavationinduced damage around the cavity. In the 15–20 m depth range, VS varies between 460and 500 m/s and this is in close agreement with that obtained from the CH test. Thus the SASW technique is reliable for these materials and can be used when borehole tests cannot be performed. In Fig. 14 UPV measurements on specimens recovered from the CH boreholes are also reported. Within the SFW lithotype, VS exhibits the same gradient that was described in CH measurements. Laboratory UPV values are only slightly larger than field values, because of the lower bulk density of dry specimens. This evidence suggests that the widely spaced discontinuities crossing the pozzolana scarcely have an influence on the propagation of elastic waves, at least at the scale of the borehole distance.

In situ tests In Fig. 14, S-wave velocities from CH and SASW tests are plotted against depth, together with the borehole stratigraphy. In the SFW pozzolana layer, VS from the CH test

43



250

p 240 kPa

p 240 kPa

200

γc 0·01%

150

6

D: %

G: MPa

0·04% 0·1%

4

γc 0·1% 0·04%

100

0·01%

2 50

0

0

4

8 Number of cycles

250

0

12

0

4

8 Number of cycles

8

γc

12

p 495 kPa

0·01% 200

0·04%

6

0·1%

D: %

G: MPa

150

4

γc

100

0·1% 0·04% 2

50

0·01% p 495 kPa

0

0

4

8 Number of cycles

0

12

0

4

8 Number of cycles

12

Fig. 11. Variation of shear modulus (G) and damping ratio (D) plotted against number of cycles in a typical series of DSDSS tests (specimens SF-BL2de) at different strain amplitudes ªc and for the two highest mean stresses

1·0

DISCUSSION The values of the small-strain shear modulus G0 from the cyclic tests (DSDSS and TS) on the SFW pozzolana are reported in Fig. 15 plotted against the mean confining stress p9. The G0 values from TS tests are generally higher than those from DSDSS tests, which are measured at a slightly higher strain. Owing to the early development of permanent strains and the consequent decay of the shear modulus, the most precise measure of G0 is provided by TS tests, and determined even at a strain amplitude as small as 0 .0001%. Variation of G0 with the mean stress p9 can be interpolated by a linear or a power law function with stress exponent equal to 0 .5. A mean shear modulus G0 ¼ 127 MPa might be assumed for a null confinement. A general decrease in the small-strain modulus as the void ratio increases is observed for both TS and DSDSS data sets (Fig. 15). This behaviour is confirmed by UPV tests (Fig. 16), which indicate an apparent inverse correlation between Gdyn and porosity n. Data show a moderate dispersion due to the heterogeneity of the material. The values of Gdyn obtained from the UPV tests generally agree with the G0 values for a mean pressure p9 ranging from 60 to 250 kPa. Plots of normalised shear modulus G/G0 and damping ratio D against ªc for all the cyclic and dynamic tests

0·9

GN /G1, DN /D1

GN /G1 0·8

0·7 DN /D1 0·6

0·5

20

40

60 80 Number of cycles

100

120

140

Fig. 12. Stiffness (�G) and damping (�D) degradation indexes plotted against number of cycles in a DSDSS test at ªc 0 .1% (specimen SB10-5)

44



300

SF-BL2n TS

p 500 kPa UPV 250

RC

4

D: %

G: MPa

400

6

SF-BL2n

120 60 200 p 0 100

250

p 0 2

60 120 p 500 kPa

0

0·0001

0·001

γc: %

0·01

0

0·1

0·0001

0·001 γc: %

0·01

0·1

6

500 SF-BL2o

SF-BL2o

400

200 100 0

4

250 UPV 120 60

0·0001

0·001

γc: %

0·01

500 kPa

0

200 100 0

SF-BL2q

120

60 250

p 0

500 kPa 0·0001

0·001

γc: %

0·01

0

0·1

0·0001

0·001 γc: %

0·01

6

SF-BL2r

0·1

SF-BL2r

500 kPa 4

250 120 UPV 60

250

p 0

2

p 0

0·0001

60

D: %

G: MPa

300

0·1

p 0 2

500 400

0·01

4

250 120 UPV 60

100

0·001 γc: %

6

SF-BL2q

120 250 500 kPa

500

0·0001

D: %

G: MPa

0

0·1

60

120 60 250

p 0

400

200

p 0 2

500

300

p from RC

D: %

G: MPa

500 kPa 300

120

0·001

γc: %

0·01

0·1

0

500 kPa 0·0001

0·001 γ c: %

0·01

0·1

Fig. 13. Plots of shear modulus (G) and damping ratio (D) against strain amplitude ªc for different specimens subjected to TS (black thin lines) and RC (grey thick lines) tests. Values of the mean stress p9 are indicated. Dynamic shear moduli obtained through the UPV tests are shown by arrows on the vertical axes

their band. Consequently the linear strain threshold ªl, from DSDSS tests (0 .001–0 .01%), too, is higher than that from TS–RC tests (0 .0003–0 .003%). The reduced values of G/G0 for the same stress level, obtained from the RC–TS data, can be explained by the fact that the delicate microstructure can be irreversibly damaged, even at the lowest strain levels, when a high number of cycles is applied. In particular, damage under repeated strains could concentrate within the very weak ground mass, interposed between the stiffer clasts, which are not in contact each other. DSDSS results also indicate that the rate of decay of shear stiffness diminishes at the highest investigated strain amplitudes (ªc > 0 .1%). This strain-hardening behaviour, which

carried out on the SFW pozzolana, grouped according to the applied mean stress, are reported in Fig. 17. The G/G0–ªc curves from TS and RC tests, merged into a single curve for each specimen, extend up to a strain of 0 .06%, whereas the DSDSS curves reach a maximum ªc of 0 .1–0 .3%. In the same figure, the G/G0–ªc and D–ªc curves proposed by Rollins et al. (1998) for granular soils merged into a single band, and those obtained by Papa et al. (1988) for cemented pozzolanas, are plotted as reference. Curves by Rollins et al. mostly refer to tests on poorly graded clean gravels and gravelly sands, with few well-graded gravels. G/G0–ªc curves from DSDSS tests lie to the right of those by Rollins et al., while curves from TS–RC tests lie within

45


VERRUCCI, LANZO, TOMMASI AND ROTONDA 200 0

VS: m/s 600

400

500 800

1000

500 kPa

G0

250

250 kPa

400

p: kPa 500

120 60

G0, Gdyn: MPa

5

10

Depth: m

120 kPa 60 kPa 0

20

Gdyn 0·58

0·59

n

0·60

0·61

0·62

Fig. 16. Small-strain shear modulus G0 from cyclic (DSDSS and TS) tests and dynamic shear modulus Gdyn from UPV tests plotted against porosity n. Regression lines (dashed) of G0 from data obtained at different mean stresses p9 are also reported

25

has already been observed in the upwards curvature of the tips of the stress–strain loops in Fig. 9, might also affect the damping ratio. Material damping has been determined in the strain range from 0 .0001% to about 0 .001% from TS tests, whereas reliable damping ratio measurements have been obtained between 0 .0004% and 0 .3% in the DSDSS tests. Values of damping ratio from TS and DSDSS tests compare very favourably at the same mean confining stress p9. Conversely, damping ratios from the RC test are considerably smaller. Possible explanations for this are currently being explored. Values of damping ratio from DSDSS and TS plot within the range proposed by Rollins et al. (1998) up to about 0 .002–0 .003%. At higher shear strain amplitudes (up to 0 .1%), values of damping ratio plot well below the literature curves, especially for higher stress levels. Again, a similar trend has been previously reported in several studies on pyroclastic materials (e.g. Guadagno et al., 1988; Papa et al., 1988; Senetakis et al., 2013). Finally, the dependence on the number of cycles observed in cyclic tests at high strain levels is different for stiffness and damping properties (Fig. 11). The stiffness decrease is moderate and gradual, while the damping ratio sharply decreases, especially in the first cycles at the lowest mean stresses. More specifically the damping ratio quickly diminishes down to 20% of the initial value and then decreases at a progressively lower rate. Tests with over 100 cycles show a non-negligible residual damping decrease (Fig. 12). This behaviour is correlated to the different results yielded by DSDSS and TS–RC tests.

SFH

30

SASW

35

CH test UPV test (linear interpolation) 40

Fig. 14. Profiles of shear wave velocity VS from cross-hole measurements (CH), SASW investigation and UPV tests on specimens from cores recovered in the CH boreholes. The UPV data of the SFW lithotype are interpolated with a linear regression (dashed line)

400 G 126·8 7·8p0·5

300

1·42 1·42 1·43 1·51

1·42 1·51

1·4

1·42

G0: MPa

200

0

1·51

200

1·51

1·4

1·51 1·57 1·43

100

1·57

1·57

1·4 1·42 1·42

1·57

1·52

1·42

CONCLUSIONS The cyclic/dynamic behaviour of the Orvieto pozzolana shows a number of peculiarities that deserve special consideration in geotechnical earthquake engineering studies. The small-strain stiffness from geophysical measurements (cross-hole and surface wave techniques) compares quite well with that obtained from pulse velocity measurements on laboratory specimens. This result, a consequence of the very wide spacing of discontinuity, is of undoubted practical importance. Small-strain shear moduli obtained from laboratory simple shear and cyclic torsional tests also compare satisfactorily with the dynamic moduli from UPV tests. Overall, the

1·57

DSDSS TS 0

0

100

SFW

15

300

0

100

200

300 p: kPa

400

500

600

Fig. 15. Small-strain shear modulus G0 plotted against mean stress p9 for cyclic (DSDSS and TS) tests. Void ratio values of the specimens are reported

46


CYCLIC AND DYNAMIC BEHAVIOUR OF A SOFT PYROCLASTIC ROCK 1·0

10

p 0

8

Papa et al. (1988) Rollins et al. (1998)

G/G0

D: %

0·8

0·6

DSDSS

0·0001

0·001

4 2

TS–RC 0·4

6

γc: %

0·01

0

0·1

p 0 0·0001

0·001

γc: %

0·01

0·1

0·01

0·1

0·01

0·1

0·01

0·1

0·01

0·1

10 p 60 kPa

1·0

8

G/G0

D: %

0·8

0·6

0·0001

4 2

p 60 kPa 0·4

6

0·001

γc: %

0·01

0

0·1

0·0001

10 1·0

D: %

G/G0 0·6

p 125 kPa

0·0001

0·001

6 4 2

p 125 kPa

γc: %

0·01

0

0·1

0·0001

10 1·0

D: %

G/G0 0·6

γc: %

p 215–250 kPa

0·0001

0·001

6 4 2

p 215–500 kPa

γc: %

0·01

0

0·1

10 1·0

0·0001

0·001

γc: %

p 425–450 kPa

8

G/G0

D: %

0·8

0·6

0·0001

0·001

γc: %

6 4 2

p 425–500 kPa 0·4

0·001

8

0·8

0·4

γc: %

8

0·8

0·4

0·001

0·01

0

0·1

0·0001

0·001

γc: %

Fig. 17. Variation of normalised shear modulus G/G0 and damping ratio D with the shear strain amplitude ªc from all the cyclic and dynamic tests (DSDSS, TS, RC), grouped for different stress level

pronounced, especially at high strain level and low confinement stress. Values of damping ratio from cyclic tests (TS and DSDSS) are in satisfactory agreement with those exhibited by other pyroclastic materials (both uncemented and weakly cemented). With respect to other granular materials, a more linear dissipative behaviour is therefore observed, with a

influence of mean confining stress and void ratio on G0 is similar to that observed in other granular materials. The high linearity threshold and the decay rate of G/G0 with ªc obtained from DSDSS tests are comparable with those observed for other pyroclastic materials (both uncemented and weakly cemented). The decay of G/G0 observed in TS–RC tests starts at a lower strain level and is more

47


VERRUCCI, LANZO, TOMMASI AND ROTONDA rs �f �t �v, �h �

slight dependence of the damping ratio on shear strain, even at ªc of about 0 .1%. According to Papa et al. (1988) and Senetakis et al. (2013), it is possible that micro-crushing at the particle contacts is a major mechanism during cyclic loading, thus preventing significant particle rearrangement and increase in damping. The Orvieto pozzolana particles are to be considered not only as single pumices or scoriae, but also as aggregates of weakly bonded minuscule glassy masses. Finally, a dependence of stiffness and damping on the number of cycles has been observed. Both the cyclic degradation indexes and the normalised stiffness provided by dynamic RC tests suggest that the microstructure of the pozzolana has a marked influence on the cyclic behaviour. In fact, the delicate three-dimensional lattice structure of the glassy mass, separating the stiffer and stronger clasts, seems to be easily damaged when subjected to a high number of cycles. Further studies of the cyclic/dynamic behaviour of these pyroclastic materials are in progress to confirm the findings and to be more conclusive regarding the issues at hand.

REFERENCES Allen, J. C. & Stokoe, K. H. (1982). Development of a resonant column apparatus with anisotropic loading, Geotechnical Engineering Report GR82-28. Austin, TX, USA: The University of Texas at Austin. Bjerrum, L. & Landva, A. (1966). Direct simple-shear test on a Norwegian quick clay. Geótechnique 16, No. 1, 1–20, http:// dx.doi.org/10.1680/geot.1966.16.1.1. Cecconi, M. (1998). Sample preparation of a problematic pyroclastic rock. Proceedings of international symposium on problematic soils, Sendai, Japan, pp. 165–168. Rotterdam, the Netherlands: Balkema. Cecconi, M., Scarapazzi, M. & Viggiani, G. MB. (2010). On the geology and the geotechnical properties of pyroclastic flow deposits of the Colli Albani. Bull. Engng Geol. Environ. 69, No. 2, 185–206. Cecconi, M., Rotonda, T., Tommasi, P. & Viggiani, G. M. B. (2011). Microstructural features and compressibility of volcanic deposits from Central Italy. Proceedings of the international symposium on deformation characteristics of geomaterials, Seoul, Korea, pp. 884–891. Amsterdam, the Netherlands: IOS Press. Choi, W. K. (2008). Dynamic properties of ash-flow tuffs. PhD thesis, The University of Texas at Austin, Austin, TX, USA. D’Elia, B., Lanzo, G. & Pagliaroli, A. (2003). Small-strain stiffness and damping of soils in a direct simple shear device. Proceedings of the 2003 Pacific conference on earthquake engineering, Christchurch, New Zealand, paper no. 111. Wellington, New Zealand: New Zealand National Society for Earthquake Engineering. Doroudian, M. & Vucetic, M. (1995). A direct simple shear device for measuring small strain behaviour. ASTM Geotech. Testing J. 18, No. 1, 69–85. Faraone, D. & Stoppa, F. (1988). Il tufo di Orvieto nel quadro dell’evoluzione vulcano-tettonica della caldera di Bolsena, Monti Vulsini. Bollettino della Societa` Geologica Italiana 107, 383– 397 (in Italian). Guadagno, M., Rapolla, A., Ni, S. & Stokoe, K. H. (1988). Dynamic properties of pyroclastic soils of the Phlegraean Fields. Proceedings of the 9th world conference on earthquake engineering, Tokyo-Kyoto, vol. 3, pp. 35–40. Tokyo, Japan: Maruzen. Isenhower, W. M. (1979). Torsional simple shear/resonant column properties of San Francisco Bay mud. MSc thesis, The University of Texas at Austin, TX, USA. Jeon, S. Y. (2008). Dynamic and cyclic properties in shear of tuff specimens from Yucca mountain, Nevada. PhD thesis, The University of Texas at Austin, TX, USA. Liu, X. & Yang, J. (2014). Laboratory measurements of small-strain shear modulus of volcanic soils. In Geo-congress 2014 technical papers: geo-characterization and modeling for sustainability (eds M. Abu-Farsakh and L. R. Hoyos), Geotechnical Special Publication no. 234, pp. 113–122. Reston, VA, USA: ASCE. Marks, S., Larkin, T. J. & Pender, M. J. (1998). The dynamic properties of a pomiceous sand. Bull. New Zealand Nat. Soc. Earthquake Engng 31, No. 2, 86–102. Miura, S., Yagi, K. & Asonuma, T. (2003). Deformation–strength evaluation of crushable volcanic soils by laboratory and in-situ testing. Soils Found. 43, No. 4, 47–57. Nazarian, S. (1984). In situ determination of elastic moduli of soil deposits and pavement systems by spectral-analysis-of-surfacewaves method. PhD thesis, The University of Texas at Austin, TX, USA. Nappi, G., Capaccioni, B., Renzulli, A., Santi, P. & Valentini, L. (1994). Stratigraphy of the Orvieto-Bagnoregio Ignimbrite eruption (Eastern Vulsini district, central Italy). Memorie Descrittive della Carta Geologica d’Italia, vol. XLIX, pp. 241–254. Rome, Italy: Servizio Geologico d’Italia..

ACKNOWLEDGEMENTS The authors would like to give special thanks to Luigi Callisto (DISG – Sapienza Universita` di Roma) for suggestions and helpful insights into various aspects of the TS– RC testing. The support of Alessandro Pagliaroli, who conducted part of the DSDSS testing, is also gratefully acknowledged. The help of Silvano Silvani, Maurizio Di Biase and Francesco Coni in carrying out mechanical tests is greatly appreciated. The cross-hole test was carried out by Solgeo Srl, Seriate. Underground sampling was carried out with the support of Speleotecnica Srl, Orvieto.

NOTATION A D D0 D1, DN d Esec Et50 e F f G Gdyn, Kdyn, Edyn G0 G1, GN h Koed K0 N Nd n P p9 p9y VS, V P ª ªc ªl � �G, �D �a �dyn rd

density of the solid particles uniaxial strength tensile strength vertical and horizontal total stress shear stress

specimen transversal area damping ratio small-strain damping ratio from cyclic tests damping ratio at the first, Nth cycle diameter of the particles Young’s secant modulus Young’s tangent modulus at 50% of failure void ratio horizontal force in DSDSS test cycle frequency in cyclic tests shear modulus shear, volumic bulk, Young’s modulus from UPV tests small-strain shear modulus from cyclic tests shear modulus at the first and the Nth cycle specimen height earth pressure coefficient in oedometric conditions earth pressure coefficient number of cycles number of data in computations of statistical parameters porosity vertical load in DSDSS test effective mean stress isotropic yielding stress shear and longitudinal wave velocity shear strain cyclic shear strain amplitude strain linear threshold horizontal displacement in DSDSS tests degradation indexes of the shear modulus and damping ratio with number of cycles axial strain Poisson coefficient from UPV tests dry bulk density

48


CYCLIC AND DYNAMIC BEHAVIOUR OF A SOFT PYROCLASTIC ROCK volcanic rocks, Madeira, vol. 2, pp. 137–146. Lisbon, Portugal: Sociedade Portuguesa de Geotecnia. Sahaphol, T. & Miura, S. (2005). Shear moduli of volcanic soils. Soil Dynamics Earthquake Engng 25, No. 2, 157–165. Santi, P. (1991). New geochronological data of the Vulsini volcanic district (central Italy). Plinius 4, 91–92. Senetakis, K., Anastasiadis, A. & Pitilakis, K. (2012). The smallstrain shear modulus and damping ratio of quartz and volcanic sands. Geotech. Testing J. 35, No. 6, 964–980, http://dx.doi.org/ 10.1520/GTJ20120073. Senetakis, K., Anastasiadis, A., Pitilakis, K. & Coop, M. R. (2013). The dynamics of a pumice granular soil in dry state under isotropic resonant column testing. Soil Dynam. Earthquake Engng 45, 70–79. Silvestri, F. (1991). Analisi del comportamento dei terreni naturali in prove cicliche e dinamiche di taglio torsionale. PhD thesis, Federico II University of Naples, Italy (in Italian). Tommasi, P., Sciotti, M., Rotonda, T., Verrucci, L. & Boldini, D. (2013). The role of geotechnical conditions in the foundation expansion and preservation of the ancient town of Orvieto (Italy). In Geotechnics and heritage (eds E. Bilotta, A. Flora, S. Lirer and C. Viggiani), pp. 49–73. London, UK: CRC Press, Taylor and Francis Ltd. Tommasi, P., Verrucci, L. & Rotonda, T. (2015). Mechanical properties of a weak pyroclastic rock and their relationship with microstructure. Can. Geotech. J. 52, No. 2, 211–223, http:// dx.doi.org/10.1139/cgj-2014-0149. Vucetic, M. (1994). Cyclic threshold shear strains in soils. J. Geotech. Engng 120, No. 12, 2208–2228.

Ni, S. H. (1987). Dynamic properties of sand under true triaxial stress state from resonant column/torsional shear test. PhD thesis, The University of Texas at Austin, TX, USA. Orense, R. P., Hyodo, M. & Kaneko, T. (2012). Dynamic deformation characteristics of pumice sand. Proceedings of the New Zealand Society for Earthquake Engineering conference (2012 NZSEE), Christchurch, New Zealand, paper no. 6. Wellington, New Zealand: New Zealand National Society for Earthquake Engineering. Papa, V., Silvestri, F. & Vinale, F. (1988). Analisi delle proprieta` di un tipico terreno piroclastico mediante prove dinamiche di taglio semplice. Convegno ‘Deformazioni dei terreni ed interazione terreno-struttura in condizioni di esercizio’, Monselice, Padova, Italy, vol. 1, pp. 265–285. Rome, Italy: CNR – Gruppo Nazionale di Coordinamento per gli Studi di Ingegneria Geotecnica (in Italian). Pellegrino, A. (1969). Proprieta` fisico-meccaniche dei terreni vulcanici del Napoletano. Proceedings of the VIII national congress of geotechnics, Cagliari, Italy, vol. 3, pp. 113–145. Napoli, Italy: Edizioni Scientifiche Italiane (in Italian). Rollins, K. M., Evans, M., Diehl, N. & Daily, W. (1998). Shear modulus and damping relationships for gravels. J. Geotech. Geoenviron. Engng 124, No. 5, 396–405. Rotonda, R. & Ribacchi, R. (1995). Caratteristiche dinamiche di rocce porose e fessurate. Rivista Italiana di Geotecnica 1, No. 1995, 17–36 (in Italian). Rotonda, R., Tommasi, P. & Ribacchi, R. (2002). Physical and mechanical characterization of the soft pyroclastic rocks forming the Orvieto cliff. Proceedings of Eurock 2002, workshop on

49


Lee, J. & Green, R. A. (2015). Geótechnique 65, No. 5, 374–390 [http://dx.doi.org/10.1680/geot.SIP.15.P.011]

Empirical predictive relationship for seismic lateral displacement of slopes: models for stable continental and active crustal regions J. L E E a n d R . A . G R E E N †

The objective of this study is to develop an empirical predictive relationship for permanent lateral displacements for use in assessing the seismic stability of slopes, earthen dams and/or embankments subject to stable continental earthquake motions. The empirical relationship is developed from 620 horizontal motions for stable continental regions, consisting of 28 recorded motions and 592 scaled motions. For each motion, the permanent relative displacements are computed using the Newmark sliding block procedure for a suite of yield accelerations. The proposed predictive relationship is derived by performing separate regression analyses for each yield acceleration. This allows the relationship to be simply formulated in terms of ground motion characteristic parameters, independent of yield acceleration, and results in lower standard deviations than those for relations developed by regressing all the data in a single analysis. The non-linear mixed-effects technique is used to regress the data as functions of maximum ground accelerations and velocities. To account for the zero displacement data, logistic regression was conducted to model the probability of zero-displacement occurrences. Then, the probability models were applied as weighting functions to the non-linear mixed-effects regression results. Also, a similar relationship for active shallow crustal motions is developed and compared with the stable continental region relationship. Lastly, the predicted displacements from the proposed model are shown to be in good agreement with those computed using motions recorded during the Mineral, Virginia earthquake of 23 August 2011. KEYWORDS: dams; deformation; dynamics; earthquakes; embankments; retaining walls; slopes

lated permanent relative displacement to the ratio of ky and the maximum peak horizontal ground acceleration (Amax), in addition to other ground motion parameters. Furthermore, this study differs from the previous studies in that the relationships proposed herein were developed by performing separate regressions for each ky using the non-linear mixedeffects (NLME) technique. Performing separate regressions for each ky allowed the relationships to have relatively low standard deviations and to have simple functional forms that are independent of ky. Using NLME regression analyses results in unbiased fits of the data, irrespective of the varying amount of data from different earthquakes. The authors present a similarly developed relationship for active shallow crustal regions (e.g. western North America: WNA). This WNA relationship was developed so that consistent comparisons in permanent relative displacement for the two regions could be made. Although there are existing relationships for permanent relative displacement for WNA, the authors felt that in order to avoid issues related to differences in predicted displacement due to disparities in database size, analysis techniques and so on, consistently developed relationships for the two regions were needed to make valid comparisons. The correlations were developed by performing NLME regression analyses on data derived from horizontal motion recorded in active shallow crustal regions (e.g. WNA), and horizontal motions for stable continental regions (e.g. CENA) that were composed of both recorded motions and scaled motions; this is discussed in more detail later in the paper. (Note that the acronyms CENA and WNA are used in this paper for convenience to refer to ‘stable continental’ and ‘active shallow crustal’ regions, respectively. However, the authors contend that the use of these respective relationships are not solely limited to North America, but rather are applicable for use in stable continental and active shallow tectonic regions in other parts of the world too.) The paper is organised as follows: first, the strong ground

INTRODUCTION The objective of the study presented herein is to develop an empirical predictive relationship for permanent relative displacements for use in assessing the seismic stability of slopes, earthen dams and/or embankments subjected to stable continental earthquake motions (e.g. central eastern North America (CENA)) for different site conditions (rock and soil). The Newmark sliding block method was used to compute the permanent relative displacements. This method was proposed by Newmark (1965) for evaluating the seismic stability of slopes, wherein the sliding mass is modelled as a block on an inclined plane. Displacement of the block relative to the plane initiates when the yield acceleration (ky) is exceeded and continues until the velocities of the block and ground coincide. The permanent relative displacement is defined as the cumulative relative displacement at the end of ground shaking, as illustrated in Fig. 1. Numerous empirical relationships for estimating permanent relative displacements have been developed over the past 30 years (e.g. Franklin & Chang, 1977; Richards & Elms, 1979; Ambraseys & Menu, 1988; Ambraseys & Srbulov, 1994; Bray & Travasarou, 2007; Jibson, 2007; Saygili & Rathje, 2008; Rathje & Saygili, 2009; Rathje & Antonakos, 2011). However, these relationships are for active shallow crustal tectonic regimes, whereas the focus of the study presented herein is stable continental regimes. Also, most of these previous studies used fixed-effects regression techniques (e.g. least-squares method) and correManuscript received 1 April 2014; revised manuscript accepted 5 February 2015. Discussion on this paper closes on 1 October 2015, for further details see p. ii. Arup, San Francisco, CA, USA. † Department of Civil and Environmental Engineering, Virginia Tech, Blacksburg, VA, USA.

51


LEE AND GREEN Acceleration: g

0·1

Block

Yield acceleration

Ground

0

0·1

Velocity: cm/s

20

0

Relative displacement: cm

20 15 Permanent lateral displacement 10

5

0

0

1

2

3

4

5 Time: s

6

7

8

9

10

Fig. 1. Example of Newmark sliding block analyses for ky 0 .03g and a ground acceleration–time history (BES090: M 6 .9; R 49 .9 km) from the 1989 Loma Prieta earthquake

0 .1 km to 199 .1 km, where site-to-source distance is defined as the closest distance to the fault rupture plane. Because there are few recorded strong ground motions in stable continental regions, only 28 of the motions in the CENA dataset are recorded motions, with the remaining 592 motions being ‘scaled’ WNA motions for CENA conditions. A brief summary of the scaling procedure is provided below, with a more detailed description of the scaling procedure given in the Appendix. The moment magnitudes (Mw) for these motions range from 4 .5 to 7 .6, and the site-to-source distances range from 0 .1 km to 199 .1 km. The recorded motions include the 1988 Saguenay (Mw5 .9 mainshock and Mw4 .5 aftershock), the 1985 Nahanni (Mw6 .8), and the 1989 New Madrid, MO (Mw4 .7) earthquakes. Fig. 2 shows the magnitude and site-to-source distance distributions for both regions. McGuire et al. (2001) scaled the WNA motions for CENA conditions using response spectral transfer functions generated from the single-corner frequency point source model in conjunction with random vibration theory (RVT) (e.g. Brune, 1970, 1971; Boore, 1983; Silva & Lee, 1987; McGuire et al., 2001). The transfer functions account for the differences in seismic source, wave propagation path properties, and site effects between the WNA and CENA regions. Many seismological publications have shown successful results of the RVT point source model for generating strong ground motions for both WNA and CENA (Hanks & McGuire, 1981; Boore, 1983, 1986; McGuire et al., 1984; Schneider et al., 1993; Silva, 1993). In generating the scaled CENA motions, recorded WNA motions were used as ‘seed’ motions in the spectral scaling process, resulting in scaled motions that have realistic characteristics. In this context, the stochastic point source model is a reliable and reasonable approach for estimating spectral characteristics of strong ground motions for engineering analyses. The scaling method, however, should be validated as additional recordings of stable continental motions become available. The ground motions were classified as either ‘rock’ or

motion databases used in this study are described. Then, basic concepts of the NLME regression method are reviewed, and the proposed functional form of the predictive model is introduced. Next, the results of the regression analyses and a comparison of permanent relative displacements predicted by this study’s empirical relationships for stable continental and active shallow crustal regions are presented. A comparison of this study’s relationship to existing relationships is then presented. Additionally, in an effort to validate the proposed relationship, the displacements computed using motions recorded during the recent 2011 Mineral, Virginia earthquake are compared with those predicted by the CENA model proposed herein. Finally, the procedure used to scale the scaled CENA motions is summarised in the Appendix. STRONG GROUND MOTION DATA The datasets used in this study consist of 324 twocomponent sets of horizontal strong ground motion time histories from WNA and 310 sets for CENA. Thus, totals of 648 and 620 horizontal time histories were used to develop the empirical relationships for WNA and CENA, respectively. This study adopted the ground motion dataset assembled by McGuire et al. (2001) from the strong ground motion database processed by Dr Walter Silva of Pacific Engineering and Analysis. Primarily, this dataset was intended to provide a library of strong ground motion time histories suitable for engineering analyses. Hence, the data selection criteria (e.g. earthquake magnitude range) were established based on engineering interest. The criteria were categorised by: site condition, earthquake magnitude, site-tosource distance and strong ground motion duration (McGuire et al., 2001; Lee, 2009). The strong motion data for WNA were from 49 mainshock events, with the 1999 Chi-Chi earthquake being the most recent event included in the database. The moment magnitudes of these events range from 5 .0 to 7 .6, and the site-to-source distances range from

52


PREDICTIVE RELATIONSHIP FOR SEISMIC LATERAL DISPLACEMENT OF SLOPES 8·0

dual data points in the regression analyses. The permanent relative displacements were computed for a suite of yield accelerations. The displacement data were then correlated to the maximum ground acceleration (Amax) and velocity (Vmax). Note that the Amax and Vmax are defined as a peak acceleration and velocity in the opposite direction as the displacement of a sliding block. They may differ from the peak ground acceleration (PGA) and the peak ground velocity (PGV), which are the maximum ‘absolute’ acceleration and velocity.

7·5

Magnitude, M

7·0 6·5 6·0 5·5 5·0

REGRESSION ANALYSES The empirical predictive relationship developed in this study consists of two models: a model for conditional expected non-zero displacement and a model for the probability of non-zero displacement occurrence. This can be expressed using the total probability theorem

Rock Soil

4·5 1 10

100 101 102 Site-to-source distance, R: km

103

E[D] ¼ E[DjD . 0] 3 p(D . 0) þ E[DjD ¼ 0] 3 p(D ¼ 0) (1)

8·0 7·5

where E[ .] and p( .) represent expected value and probability, respectively. The second term on the right side of Equation 1 is zero since E[D|D ¼ 0] is zero. The expected value for D, therefore, becomes equal to the quantity of the expected value for a given non-zero displacement (i.e., E[D|D . 0]) multiplied by the probability of non-zero displacement occurrence p(D . 0). The NLME regression was performed for modeling the expected value of displacement for a given non-zero displacement. A logistic regression method was employed to model the probability of non-zero displacement occurrence. The proposed models were derived by performing separate regression analyses for each ky. This approach allows the predictive relationship to be formulated only in terms of ground motion characteristic parameters, independent of ky, and allows the standard deviations to be estimated for each ky value. This inherently allows the uncertainty in ky to be treated as epistemic uncertainty, which can be reduced by way of field investigations, as opposed to being treated as aleatory variability in the predicted displacements. Also, this approach is in contrast to previous studies where permanent relative displacements were correlated to the ratio ky /Amax, which results in complex functional forms and relatively large total standard deviations. This is attributable primarily to the large variations in displacements for a given ky /Amax, as shown in Fig. 3. A suite of discrete ky values were considered for the separate regressions, which were distributed with an approximately equal interval in a log scale (listed in Tables 2–5). A more detailed discussion of the merits/disadvantages of performing separate regression analyses for each ky against ky /Amax is presented subsequently. As mentioned previously, the regression analyses were performed on the permanent relative displacement data of the individual horizontal components of the dataset using the

Magnitude, M

7·0 6·5 6·0 5·5 5·0 4·5 1 10

Rock Soil 100 101 102 Site-to-source distance, R: km

103

Fig. 2. Earthquake magnitude and site-to-source distance distributions (recorded motions for CENA are shown in bold): (a) active shallow crustal; (b) stable continental

‘soil’, based on the site conditions at the respective seismograph stations. The site classification scheme used by McGuire et al. (2001) is based on the third letter of the Geomatrix three-letter site classification system shown in Table 1. Site categories A and B were considered to represent rock sites, and site categories C, D and E were considered to represent soil sites (note that there were only a few motions in the McGuire et al. database that were recorded in category E sites). As may be surmised from Fig. 1, the permanent relative displacements may vary with the orientation (or sign) of the ground motion. Accordingly, permanent relative displacements were computed for both directions (i.e. +/�) of a ground motion. These displacements were treated as indivi-

Table 1. Third letter: geotechnical subsurface characteristics of Geomatrix three-letter site classification Third letter

Site description

Comments

A B C

Rock Shallow (stiff) soil Deep narrow soil

D E

Deep broad soil Soft deep soil

Instrument on rock (VS . 600 m/s) or , 5 m of soil over rock Instrument on/in soil profile up to 20 m thick overlying rock Instrument on/in soil profile at least 20 m thick overlying rock, in a narrow canyon or valley no more than several kilometres wide Instrument on/in soil profile at least 20 m thick overlying rock, in a broad valley Instrument on/in deep soil profile with average VS , 150 m/s

53


LEE AND GREEN CENA – rock

103

102

ky 0·01 ky 0·05

101

ky 0·10

D: cm

ky 0·20 100

Davg.: ky 0·01 Davg.: ky 0·05 Davg.: ky 0·10

101

Davg.: ky 0·20

102 2 10 103

100

101 ky/Amax ky/Amax 0·05

103

ky/Amax 0·10

102

ky/Amax 0·20

101

ky/Amax 0·50

102

D: cm

D: cm

D: cm

101

101

100 10

100 0·045

100

101

D: cm

10

2

0·050 ky/Amax

0·055

101

0

101

0·09

0·10 ky/Amax

101 0·16 0·18 0·20 0·22 ky/Amax

0·11

102 0·4

0·5 ky/Amax

0·6

Fig. 3. Displacement data and averages for ky /Amax 0 .05, 0 .10, 0 .20 and 0 .50 for each ky value considered in this study; plots for each ky /Amax in semi-log scale (bottom) and a plot for all the ky /Amax in log–log scale (top)

inter-event error is defined as the difference between the median for the ith event and the median of the entire database (i.e. model median) and has a mean of zero and a variance of 2. The intra-event error is designated by ij, where the subscripts ij indicate the jth record of the ith event. The intra-event error is defined as the difference between the data value of the jth record and the median for the ith event and has a mean of zero and a variance of 2. The total error for the jth record of the ith event is defined as the sum of the corresponding inter- and intra-event errors (i.e. i + ij). The standard deviation of the total error is given by

NLME regression technique. The displacements less than 1 cm were considered as zero displacements, owing to engineering insignificance, and were not included in the NLME regression analyses. The NLME modelling is a maximum likelihood method based on normal (Gaussian) distribution and is primarily used for analysing grouped data (i.e. databases comprised of subsets). The NLME regression method allows regression models to account for both random effects that vary from subset to subset and fixed-effects that do not. In this study, a subset consists of motions recorded during a given earthquake. In comparison to applying a fixed-effects regression technique (e.g., the least squares method) to the entire dataset, a mixed-effects regression method allows both interand intra-earthquake uncertainty to be quantified. This regression method produces unbiased fittings for each subset having different numbers of ground motion recordings. This is important because of the number of motions from each earthquake can widely vary. The NLME modelling estimates the variation in the mean values among earthquakes (i.e. inter-event variability) and the variation in the data for a single earthquake (i.e. intraevent variability) by way of the variances of inter-event errors and intra-event errors, respectively. The inter-event error is designated by i, where the subscript i represents the ith event (i.e. set of motions from a given earthquake). The

total ¼

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 þ 2

(2)

where total is the standard deviation of total error, also called the total standard deviation. The NLME method assumes the normal distribution for intra-event errors and random-effects (Pinheiro & Bates, 2000), which underlies the theoretical formulation of the NLME regression analyses. The inherent distributional assumptions are checked by the normal quantile–quantile (Q–Q) plots, wherein the data points will plot approximately as a straight line if the data are normally distributed. The statistical analysis program-R (version 3.0.2 (R Foundation, 2013)) was used to perform

54


1 .272 0 .992 0 .690 0 .177 0 .459 0 .465 0 .473 0 .465 0 .678 1 .412 2 .320 2 .356 2 .582 1 .528 1 .535 0 .751

C1

0 .564 0 .618 0 .702 0 .645 0 .685 0 .818 0 .967 1 .119 1 .177 1 .197 1 .183 1 .231 1 .297 1 .411 1 .618 1 .801

C2

1 .020 1 .049 1 .058 1 .215 1 .205 1 .145 1 .101 0 .995 0 .965 1 .023 1 .163 1 .124 1 .021 0 .614 0 .511 0 .165

C3 0 .355 0 .299 0 .267 0 .194 0 .240 0 .221 0 .187 0 .262 0 .304 0 .352 0 .333 0 .408 0 .338 0 .385 0 .403 0 .437

ln 0 .476 0 .493 0 .509 0 .508 0 .459 0 .486 0 .514 0 .530 0 .525 0 .555 0 .597 0 .559 0 .531 0 .514 0 .524 0 .484

ln

55


0 .01 0 .014 0 .02 0 .03 0 .04 0 .05 0 .06 0 .08 0 .1 0 .14 0 .18 0 .2 0 .28 0 .36 0 .4

ky: g

1 .673 1 .556 1 .044 0 .557 0 .210 0 .145 0 .044 0 .731 0 .980 0 .927 1 .337 1 .479 1 .967 2 .361 1 .673

C1

0 .516 0 .671 0 .711 0 .837 0 .974 1 .133 1 .237 1 .277 1 .450 1 .775 1 .870 1 .845 1 .946 2 .406 0 .516

C2

1 .055 1 .062 1 .134 1 .189 1 .236 1 .209 1 .228 1 .322 1 .327 1 .186 1 .178 1 .159 1 .088 1 .072 1 .055

C3 0 .489 0 .484 0 .448 0 .398 0 .350 0 .343 0 .288 0 .168 0 .073 0 .201 0 .210 0 .247 0 .305 0 .263 0 .489

ln 0 .361 0 .400 0 .411 0 .418 0 .439 0 .452 0 .469 0 .471 0 .521 0 .533 0 .560 0 .547 0 .585 0 .550 0 .361

ln 0 .608 0 .628 0 .608 0 .577 0 .562 0 .568 0 .551 0 .500 0 .526 0 .570 0 .599 0 .600 0 .660 0 .609 0 .608

(ln)total

Regression results

0 .593 0 .577 0 .575 0 .543 0 .519 0 .534 0 .547 0 .591 0 .607 0 .657 0 .684 0 .692 0 .630 0 .642 0 .661 0 .652

(ln)total

Regression results

Table 3. Regression results and data statistics: CENA – soil

0 .01 0 .014 0 .02 0 .03 0 .04 0 .05 0 .06 0 .08 0 .1 0 .14 0 .18 0 .2 0 .28 0 .36 0 .4 0 .5

ky: g

Table 2. Regression results and data statistics: CENA – rock

12 .85 28 .26 10 .32 14 .68 13 .16 13 .10 10 .73 9 .62 11 .95 12 .27 11 .36 10 .82 13 .93 12 .86 16 .02

0

11 .51 8 .53 11 .08 9 .70 8 .30 10 .52 9 .52 10 .26 11 .86 9 .91 11 .08 9 .37 9 .51 12 .88 12 .76 9 .72

0

75 .46 158 .84 99 .16 122 .67 66 .78 61 .74 40 .93 32 .66 33 .04 28 .20 24 .70 20 .24 21 .37 16 .74 20 .28

1

132 .26 78 .88 84 .48 62 .92 34 .42 35 .12 23 .88 26 .40 27 .46 17 .30 15 .42 11 .65 6 .64 7 .70 7 .86 5 .66

1

3 .496 6 .939 0 .897 0 .775 0 .790 0 .578 0 .408 0 .201 0 .220 0 .146 0 .076 0 .069 0 .058 0 .044 0 .043

2

1 .448 0 .912 0 .718 0 .307 0 .279 0 .362 0 .355 0 .148 0 .113 0 .074 0 .080 0 .070 0 .092 0 .104 0 .078 0 .028

2

606 603 581 550 497 456 409 332 280 207 165 142 96 67 54

No. data

532 506 451 375 294 259 234 202 178 142 120 113 84 69 64 47

No. data 0 .002 0 .003 0 .004 0 .006 0 .008 0 .011 0 .013 0 .017 0 .021 0 .030 0 .038 0 .042 0 .059 0 .076 0 .085 0 .106

(ky /Amax)min

37 37 37 37 35 35 29 29 23 21 19 15 14 10 10

0 .006 0 .009 0 .013 0 .019 0 .026 0 .032 0 .039 0 .052 0 .064 0 .090 0 .116 0 .129 0 .180 0 .232 0 .258

(ky / (ky /Amax)max Amax)min

31 29 28 26 26 25 25 22 21 19 17 17 14 13 12 12

No. event 0 .046 0 .046 0 .064 0 .076 0 .098 0 .150 0 .169 0 .204 0 .204 0 .351 0 .384 0 .384 0 .559 0 .689 0 .935 0 .935

4 .72 4 .72 4 .72 4 .72 4 .72 4 .72 4 .72 4 .72 4 .72 4 .72 4 .72 4 .72 4 .72 4 .72 4 .72 4 .72

(Amax)min: g (Amax)max: g

0 .31 0 .44 0 .49 0 .50 0 .55 0 .54 0 .59 0 .62 0 .66 0 .75 0 .67 0 .74 0 .74 0 .73 0 .74

(Amax)min: g

0 .032 0 .032 0 .041 0 .060 0 .073 0 .092 0 .103 0 .130 0 .152 0 .188 0 .267 0 .269 0 .378 0 .494 0 .540

(Amax)max: g

1 .55 1 .55 1 .55 1 .55 1 .55 1 .55 1 .55 1 .55 1 .55 1 .55 1 .55 1 .55 1 .55 1 .55 1 .55

(Vmax)min: cm/s

Statistics of used data for D > 1 cm

0 .218 0 .305 0 .311 0 .394 0 .408 0 .334 0 .355 0 .393 0 .491 0 .399 0 .468 0 .520 0 .501 0 .523 0 .428 0 .535

(ky /Amax)max


1 .78 2 .28 2 .52 2 .62 3 .87 4 .37 4 .37 6 .75 8 .82 9 .41 9 .41 13 .28 17 .28 30 .95 30 .95

(Vmax)max: cm/s

0 .92 1 .04 1 .51 2 .00 3 .42 3 .42 4 .56 4 .56 6 .76 7 .80 10 .40 10 .80 19 .88 28 .04 28 .04 28 .04

(Vmax)min: cm/s

146 .42 146 .42 146 .42 146 .42 146 .42 146 .42 146 .42 146 .42 146 .42 146 .42 146 .42 146 .42 146 .42 146 .42 146 .42

127 .31 127 .31 127 .31 127 .31 127 .31 127 .31 127 .31 127 .31 127 .31 127 .31 127 .31 127 .31 127 .31 127 .31 127 .31 127 .31

(Vmax)max: cm/s

PREDICTIVE RELATIONSHIP FOR SEISMIC LATERAL DISPLACEMENT OF SLOPES

1 .084 0 .901 0 .897 0 .362 0 .444 0 .536 0 .499 0 .0017 0 .077 0 .810 1 .812 2 .021 1 .195 0 .959

C1

0 .571 0 .652 0 .788 0 .842 0 .985 1 .125 1 .170 1 .226 1 .377 1 .622 1 .465 1 .410 1 .893 2 .473

C2

1 .208 1 .210 1 .158 1 .221 1 .139 1 .072 1 .021 1 .072 1 .031 1 .115 1 .252 1 .242 0 .884 0 .732

C3 0 .240 0 .230 0 .239 0 .213 0 .243 0 .265 0 .313 0 .335 0 .365 0 .350 0 .420 0 .428 0 .484 0 .408

ln 0 .491 0 .493 0 .492 0 .486 0 .485 0 .503 0 .480 0 .470 0 .487 0 .572 0 .509 0 .497 0 .458 0 .491

ln

56


0 .01 0 .014 0 .02 0 .03 0 .04 0 .05 0 .06 0 .08 0 .1 0 .14 0 .18 0 .2 0 .28 0 .36

ky: g

1 .687 1 .325 0 .922 0 .493 0 .892 0 .681 0 .519 0 .107 0 .032 0 .245 0 .609 0 .725 0 .722 1 .607

C1

0 .633 0 .715 0 .809 0 .966 1 .244 1 .374 1 .441 1 .668 1 .967 2 .236 2 .288 2 .354 2 .241 1 .661

C2

1 .116 1 .177 1 .234 1 .299 1 .172 1 .204 1 .204 1 .245 1 .244 1 .161 1 .121 1 .105 0 .866 0 .866

C3

0 .446 0 .391 0 .352 0 .286 0 .295 0 .229 0 .140 0 .070 1 .85 3 105 0 .188 0 .219 0 .199 0 .267 0 .174

ln 0 .424 0 .444 0 .436 0 .441 0 .437 0 .481 0 .494 0 .519 0 .561 0 .577 0 .578 0 .566 0 .618 0 .501

ln 0 .615 0 .591 0 .560 0 .526 0 .527 0 .532 0 .513 0 .524 0 .561 0 .607 0 .618 0 .599 0 .673 0 .531

(ln)total

Regression results

0 .547 0 .544 0 .547 0 .531 0 .543 0 .569 0 .573 0 .578 0 .609 0 .671 0 .660 0 .657 0 .667 0 .638

(ln)total

Regression results

Table 5. Regression results and data statistics: WNA – soil

0 .01 0 .014 0 .02 0 .03 0 .04 0 .05 0 .06 0 .08 0 .1 0 .14 0 .18 0 .2 0 .28 0 .36

ky: g

Table 4. Regression results and data statistics: WNA – rock

1

2

2

531 454 378 305 267 239 214 182 155 126 96 86 67 37

No. data

1 .820 1 .092 0 .733 0 .760 0 .628 0 .335 0 .231 0 .139 0 .116 0 .120 0 .025 0 .026 0 .014 0 .010

1 .473 0 .844 0 .695 0 .499 0 .460 0 .551 0 .201 0 .135 0 .152 0 .100 0 .084 0 .092 0 .073 0 .032

229 .51 126 .82 104 .83 75 .03 64 .47 53 .58 46 .58 58 .69 54 .25 58 .78 43 .40 36 .57 31 .51 30 .09

1

86 .94 101 .86 81 .98 83 .75 71 .79 75 .61 52 .90 58 .18 52 .87 36 .84 25 .95 27 .93 13 .11 9 .02

11 .96 9 .36 9 .24 11 .07 12 .00 10 .21 10 .10 11 .89 13 .08 19 .45 14 .75 13 .63 14 .33 17 .02

0

7 .14 8 .35 8 .97 10 .79 11 .64 14 .98 10 .46 12 .61 14 .31 12 .26 11 .33 13 .21 9 .12 7 .85

0

583 541 495 410 341 298 263 240 205 142 110 98 63 35

No. data

30 29 27 25 24 24 23 19 18 18 14 14 12 10

36 36 32 29 27 23 20 20 18 16 15 13 10 8

0 .019 0 .027 0 .033 0 .058 0 .070 0 .091 0 .096 0 .120 0 .139 0 .187 0 .259 0 .290 0 .353 0 .485

1 .58 1 .58 1 .58 1 .58 1 .58 1 .58 1 .58 1 .58 1 .58 1 .58 1 .58 1 .58 1 .58 1 .58

(Amax)min: g (Amax)max: g


0 .54 0 .51 0 .61 0 .52 0 .57 0 .55 0 .62 0 .67 0 .72 0 .75 0 .70 0 .69 0 .79 0 .74

(ky /Amax)max

1 .32 2 .02 3 .03 3 .33 4 .50 4 .66 6 .16 6 .16 8 .22 9 .82 16 .38 16 .38 24 .48 26 .24

(Vmax)min: cm/s

0 .011 0 .015 0 .021 0 .032 0 .043 0 .053 0 .064 0 .085 0 .106 0 .149 0 .192 0 .213 0 .298 0 .383

0 .52 0 .56 0 .57 0 .65 0 .68 0 .73 0 .71 0 .69 0 .72 0 .70 0 .73 0 .80 0 .80 0 .74

0 .019 0 .025 0 .035 0 .046 0 .059 0 .068 0 .085 0 .116 0 .139 0 .201 0 .247 0 .251 0 .348 0 .484

0 .94 0 .94 0 .94 0 .94 0 .94 0 .94 0 .94 0 .94 0 .94 0 .94 0 .94 0 .94 0 .94 0 .94

1 .73 2 .24 3 .16 4 .12 4 .35 5 .61 6 .76 6 .76 8 .91 12 .22 15 .59 15 .59 22 .23 36 .33

(ky /Amax)min (ky /Amax)max (Amax)min: g (Amax)max: g (Vmax)min: cm/s

0 .006 0 .009 0 .013 0 .019 0 .025 0 .032 0 .038 0 .050 0 .063 0 .088 0 .114 0 .126 0 .177 0 .227

(ky /Amax)min

No. event

No. event


263 .21 263 .21 263 .21 263 .21 263 .21 263 .21 263 .21 263 .21 263 .21 263 .21 263 .21 263 .21 176 .65 127 .18

(Vmax)max: cm/s

125 .13 125 .13 125 .13 125 .13 125 .13 125 .13 125 .13 125 .13 125 .13 125 .13 125 .13 125 .13 125 .13 125 .13

(Vmax)max: cm/s

LEE AND GREEN

PREDICTIVE RELATIONSHIP FOR SEISMIC LATERAL DISPLACEMENT OF SLOPES was, however, consistent with many other functional forms attempted in the regression analyses. Thus, the functional form of equation (3) was kept because of its simplicity and associated lower standard deviation. As described above, the distributional assumptions for intra-event errors and random effects were assessed by the normal Q–Q plots. Fig. 5 shows the Q–Q plots for CENA and ky ¼ 0 .05. In these figures, the theoretical quantiles of the standard normal distribution against the standardised intra-event errors (i.e. intra-event errors divided by their standard deviation) and random effects are plotted. As shown in Fig. 5, both intra-event errors and random effects plot approximately as straight lines, indicating that the data follow normal distributions, consistent with the assumptions inherent to NLME modelling. The Q–Q plots for the other N values, and so on, were similar to Fig. 5 and thus are not shown herein. To estimate the probability of non-zero displacement occurrence, logistic regressions were implemented separately for each ky, tectonic regime and site condition as a function of Amax and Vmax. Using the logistic function, the probability of non-zero displacement for a given ky, tectonic regime and site condition is expressed as follows

the NLME and logistic regression analyses (Pinheiro & Bates, 2000; Lee, 2009). PROPOSED MODEL AND REGRESSION RESULTS After observing the trends from the data and considering numerous functional forms (Lee, 2009), the following model was selected for the predictive relationship for all the ky values: E[DjD . 0] ¼ exp[C 1 þ C 2 ln (Amax ) þ C 3 ln (V max )]

(3)

where D is the permanent relative displacement (cm); Amax is the maximum ground acceleration (g); Vmax is the maximum ground velocity (cm/s); and C1, C2, C3 are regression coefficients. The selected functional form (equation (3)) produced a smaller standard deviation than the others, and the resulting residuals showed no bias and had a normal distribution (presented later this section). The regression coefficients and standard deviations determined from NLME regression analyses are listed in Tables 2–5 for CENA – rock, CENA – soil, WNA – rock and WNA – Soil, respectively. Also, the Amax and Vmax ranges associated with the displacement data used in the regression analyses for each ky are listed in these tables. It is recommended that equation (2) be used only for Amax and Vmax values that are within the ranges listed in Tables 2–5. Figure 4 shows the comparison of the non-zero displacements computed directly from the motions (i.e. Dobserved) data and the predicted displacements (i.e. E[D|D . 0]) from the NLME regression results for CENA and three ky values: 0 .01, 0 .1 and 0 .2. Also Fig. 4 shows a straight line with a y intercept of zero and a slope of 1 representing the predicted values equal to the observed data. Overall the NLME results with the functional form of equation (3) represent the observed data well without a significant bias except for small displacements. The over-estimation for small displacements CENA – rock: ky 0·01

7

CENA – rock: ky 0·20 4

4 3 2

E[D|D > 0] in ln unit


5

4 3 2

3 2 1

1 0

0

0 0

1

2

3 4 5 ln Dobserved

6

7

0

CENA – soil: ky 0·01

1

2 3 ln Dobserved

4

5

0


7

5

3 2 1



4

4 3 2 1 0

1

2

3 4 5 ln Dobserved

6

7

2 3 ln Dobserved

4

4 3 2 1 0

0 0

1


5

6 5

0

1

(4)

The regression coefficients (1 through 3) determined from logistic regression analyses are listed in Tables 2–5 for CENA – rock, CENA – soil, WNA – rock and WNA – soil, respectively. Also, the resulting weighting functions (probability of non-zero displacement) for different tectonic regions and site conditions are shown in Fig. 6. As shown in equation (1), the proposed model for permanent relative displacements is defined as the multiplication

5

1


1 1 þ exp [1 þ 2 Amax þ 3 V max ]

CENA – rock: ky 0·10

6


p(D . 0jAmax , V max ) ¼

2 3 4 ln Dobserved

5

6

0

1

2 3 ln Dobserved

4

5

Fig. 4. Predicted displacements from NLME regression compared with observed data in a natural log unit for CENA

57


LEE AND GREEN CENA – rock (ky 0·05 g)

CENA – soil (ky 0·05 g) 3

2

Quantiles of standard normal


3

1 0 1 2

1 0 1 2 3

3 4

2 0 Standardised residuals

2

3

2

2 1 0 1 Standardised residuals

2

2



2

1

0

1

1

0

1

2

2 0·4

0·2

0 Random effects (a)

0·2

0·4

0·5

0 Random effects (b)

0·5

Fig. 5. Normal Q–Q plots of intra-event errors (top) and random-effects (bottom) for (a) CENA – rock and (b) CENA – soil; ky 0 .05g

given that the combination of Amax and Vmax has been used to quantify the characteristic period of the ground motion, and Vmax has been shown to be better correlated to the energy of the motions (e.g. Green & Cameron, 2003). However, as noted in Green et al. (2011), quantifying the frequency content of earthquake ground motions by a single characteristic period is very approximate. As a result, using a functional form for the regression equation that includes both Amax and Vmax is not able to account completely for the influence in the variation of frequencies in WNA as opposed to CENA motions or rock as opposed to soil motions. For ky values other than those listed in Tables 2–5, logarithmic interpolation of D computed using the bounding ky values listed in Tables 2–5 is recommended.

of equations (3) and (4). The permanent relative displacements predicted for CENA and WNA motions using equations (3) and (4), in conjunction with the coefficients listed in Tables 2–5, are shown in Fig. 7. As may be observed from this figure and as expected, for a given site condition and tectonic setting the permanent relative displacements decrease with increasing ky /Amax (i.e. increasing ky for a given Amax) but increase with increasing Vmax. In comparing the permanent relative displacements for CENA and WNA for a given site condition, WNA motions have greater D than CENA motions, especially for rock sites. In comparison between rock and soil sites, larger D values are estimated for soil sites than for rock sites. Finally, the curves in Fig. 7 show the general trend that longer period motions produce larger permanent relative displacements (D) than shorter period motions (e.g. WNA as opposed to CENA motions and soil as opposed to rock motions). This trend makes sense in that for the same peak acceleration, long-period accelerations result in a larger displacement than shortperiod accelerations. To better capture this trend in their predictive relation, Saygili & Rathje (2008) correlated the D with Amax and Tm (mean period of the ground motion). However, as shown in Saygili & Rathje (2008) the inclusion of Tm resulted in larger standard deviations than similar models that correlated D with Amax and Vmax (this is independent of the additional uncertainty associated with estimating Tm versus Vmax). This is not altogether surprising

COMPARISON WITH EXISTING RELATIONSHIPS One of the earliest ground motion predictive relations for permanent relative displacements for evaluating seismic slope stability was proposed by Newmark (1965). Subsequent relations have been proposed by Franklin & Chang (1977), Richards & Elms (1979), Jibson (2007), Saygili & Rathje (2008) and NCHRP (2009), to name a few, where all of these relations were developed using displacements computed using Newmark’s sliding block analysis. However, as noted above, the distinguishing characteristics of the relations proposed herein are that they are for stable continental

58


PREDICTIVE RELATIONSHIP FOR SEISMIC LATERAL DISPLACEMENT OF SLOPES 1·0

Rock: Vmax 25 cm/s

(14%). However, using screening criteria, Jibson (2007) compiled a subset of 875 motions that were evenly distributed across regressed seismic demand parameters. Jibson (2007) computed the permanent relative displacements for this subset of motions using the Newmark sliding block analysis for ky equal to 0 .05, 0 .1, 0 .2, 0 .3 and 0 .4g. However, no information is given with regard to whether the polarity of the motions was considered (i.e. whether or not Amax was scaled by factors of 1 and �1 prior to computing D) or what regression technique was used, although it is assumed that the linear least-squares technique was used. Jibson (2007) used several different functional forms for his regression equation having varying numbers and combinations of parameters (i.e. Mw, ky and Ia, where Ia is Arias intensity), with three of the five proposed models having terms that are functions of ky /Amax and neither of the remaining two models having terms with ky and Amax considered independently. Of the relationships proposed by Jibson (2007), only one did not include Mw or Ia; this model is only a function of ky /Amax. It is this relation that is used for comparison in Figs 8 and 9. As may be observed in these figures, the Jibson (2007) relation tends to predict larger displacements at lower values of ky /Amax and smaller displacements at higher values of ky /Amax, relative to the relations proposed herein for WNA – rock (Fig. 8) and WNA – soil (Fig. 9). However, the ky /Amax values corresponding to the crossover points for larger and smaller predicted displacements varied with Amax, Vmax and site classification. Also, the largest differences in the predicted displacements tended to be at lower values of ky /Amax.

P[D > 0|Amax, Vmax, ky

0·8

0·6

0·4

0·2

0 102

1·0

101 Amax: g

100

Soil: Vmax 25 cm/s

P[D > 0|Amax, Vmax, ky

0·8

0·6

0·4

0·2

0 102

101 Amax: g

100

Saygili & Rathje (2008) Saygili & Rathje (2008) developed their predictive relation using next generation attenuation (NGA) ground motion database for active shallow crustal tectonic regimes. The magnitude and distance ranges for the motions in the NGA database are Mw5 .0 to Mw7 .9 and R ¼ 0 .1 to 100 km, respectively. Saygili & Rathje (2008) scaled the Amax of the motions by factors of 1, 2 and 3, but excluded scaled motions having Amax . 1g (no explicit scaling was simultaneously applied to Vmax, as was done by Newmark (1965) and Franklin & Chang (1977)). They computed the permanent relative displacements for scaled motions using the Newmark sliding block analysis for ky equal to 0 .05, 0 .1, 0 .2 and 0 .3g and performed a linear least-squares regression analysis on the displacements. They used several different functional forms for their regression equation having varying numbers and combinations of ground motion parameters (i.e. Amax, Vmax, Tm and Ia, where Ia is Arias intensity), but all of the forms included several terms that are functions of ky /Amax. Of the relationships having two ground motion parameters, the one expressing displacements as a function of Amax and Vmax had the lowest standard deviation. Accordingly, it is this relation by Saygili & Rathje (2008) that is used for comparison in Figs 8 and 9. As may be observed in Figs 8 and 9, the Saygili & Rathje (2008) relation varies similarly with Amax, Vmax, and ky /Amax to the relations proposed herein for WNA – rock (Fig. 8) and WNA – soil (Fig. 9). However, from Fig. 8 it can be seen that there is a general tendency for the Saygili & Rathje (2008) relation to predict larger displacements at intermediate values of ky /Amax (i.e. ,0 .08 to ,0 .4) and to predict smaller displacements at lower and higher values of ky /Amax for rock sites. For soil sites in WNA (Fig. 9), the predicted displacements by Saygili & Rathje (2008) compare well to the relation proposed herein for Amax ¼ 0 .25g and Vmax ¼ 25 cm/s over the entire range of ky /Amax considered.

WNA: ky 0·05 WNA: ky 0·10 WNA: ky 0·20 CENA: ky 0·05 CENA: ky 0·10 CENA: ky 0·20

Fig. 6. Probability of permanent relative displacement (D) greater than zero as a weighting function for WNA and CENA for rock and soil sites; Vmax 25 cm/s

regimes, NLME regression analysis was used to account for biases in the ground motion databases, and that the relationships were derived by performing separate regression analyses for each value of ky. This latter characteristic is based on the observation that the influence of N cannot be completely normalised by Amax as assumed by Newmark (1965) and subsequent researchers, as shown in Fig. 3. However, given the prevalence of existing relationships, Figs 8–10 compare the relations proposed herein only with selected relations from the literature, namely Jibson (2007), Saygili & Rathje (2008) and NCHRP (2009). Jibson (2007) Jibson (2007) developed his predictive relation starting with 2270 single-component horizontal motions recorded during 30 earthquakes, having a range in magnitude from Mw5 .3 to Mw7 .6. The latest motions in the database were from the 2004, Mw6 .6 Niigata-Ken-Chuetsu, Japan earthquake. The site classes of the recording stations were hard rock (10%), soft rock (27%), stiff soil (49%) and soft soil

59


LEE AND GREEN 103

Vmax 25 cm/s

103

Vmax 50 cm/s

Amax 0·25g: WNA – rock Amax 0·25g: CENA – rock Amax 0·25g: WNA – soil Amax 0·25g: CENA – soil 102

D: cm

D: cm

10

2

101

100 102

101

100 102

100

101 ky/Amax

101 ky/Amax

100

(a) 10

3

Vmax 25 cm/s

103

Vmax 50 cm/s

Amax 0·5g: WNA – rock Amax 0·5g: CENA – rock Amax 0·5g: WNA – soil Amax 0·5g: CENA – soil 102

D: cm

D: cm

10

2

101

101

100 2 10

101 ky/Amax

100 2 10

100

101 ky/Amax

100

(b)

Fig. 7. Permanent relative displacements (D) predicted for WNA and CENA for rock and soil sites, with Vmax 50 cm/s (right): (a) Amax 0 .25g; (b) Amax 0 .50g

However, for Amax ¼ 0 .25g and Vmax ¼ 50 cm/s and for Amax ¼ 0 .5g and Vmax ¼ 25 cm/s, the Saygili & Rathje (2008) relation tends to predict smaller and larger displacements, respectively, than the relation proposed herein. Finally, for Amax ¼ 0 .5g and Vmax ¼ 50 cm/s, the Saygili & Rathje (2008) relation matches very closely to the displacements predicted by the relation proposed herein for intermediate ranges of ky /Amax (i.e. ,0 .13 to ,0 .4), but tends to predict smaller displacements for smaller and larger values of ky /Amax.

25 cm/s (left) and

squares technique was used). Also, the form of the NCHRP (2009) proposed relations have multiple terms that are functions of ky /Amax. As may be observed in Figs 8–10, the NCHRP (2009) relations vary similarly with Amax, Vmax and ky /Amax to the relations proposed herein for WNA – rock (Fig. 8), WNA – soil (Fig. 9), CENA – rock (Fig. 10), and CENA – soil (Fig. 10). From Fig. 8 it can be seen that the NCHRP (2009) for WNA rock sites and the relation proposed herein match very closely for Amax ¼ 0 .25g and Vmax ¼ 25 cm/s and for Amax ¼ 0 .5g and Vmax ¼ 50 cm/s over the entire range of ky /Amax considered. However, for Amax ¼ 0 .5g and Vmax ¼ 25 cm/s and for Amax ¼ 0 .25g and Vmax ¼ 50 cm/s, the NCHRP (2009) WNA – rock relation tends to predict smaller and larger displacements, respectively, than the relation proposed herein. As shown in Fig. 9, the trends identified for WNA rock sites between the displacements predicted by NCHRP (2009) and the relations proposed herein hold for WNA soil sites, except the match between the predicted displacements for Amax ¼ 0 .5g and Vmax ¼ 50 cm/s for WNA soil sites are not as good as they are for WNA rock sites. A comparison of the predicted permanent relative displace-

NCHRP (2009) The NCHRP (2009) relations for permanent relative displacement used the same ground motion database as was used in the present study (i.e. McGuire et al., 2001). As a result, they developed relations for WNA – rock, WNA – soil, CENA – rock and CENA – soil similar to those proposed herein. However, as with Jibson (2007), no information is given with regard to whether the polarity of the motions was considered (i.e. whether or not Amax was scaled by factors of 1 and �1 prior to computing D) and what regression technique was used (although it is assumed that the linear least-

60


PREDICTIVE RELATIONSHIP FOR SEISMIC LATERAL DISPLACEMENT OF SLOPES 103

Vmax 25 cm/s; Amax 0·25g

103

102

Vmax 50 cm/s; Amax 0·25g This study NCHRP (2009) Jibson (2007) Saygili & Rathje (2008)

D: cm

D: cm

102

101

100 102

101

101 ky/Amax

100 102

100

100

101 ky/Amax

(a) 103


103


102

D: cm

D: cm

102

101

100 102

101

101 ky/Amax

100 102

100

101 ky/Amax

100

(b)

Fig. 8. Comparison of this study’s model and the existing relationships (Jibson, 2007; Saygili & Rathje, 2008; NCHRP, 2009) for WNA – rock: (a) Amax 0 .25g; (b) Amax 0 .50g

COMPARISON WITH THE 2011 MINERAL, VIRGINIA EARTHQUAKE MOTIONS The Mw5 .8 Mineral, Virginia earthquake of 23 August 2011 occurred in the Piedmont region of Virginia (central Virginia seismic zone). The epicentre was located about 61 km northwest of Richmond and 8 km south-southwest of the town of Mineral. Its fault mechanism was a reverse slip fault on a north to northeast striking plane dipping to the southeast, and the estimated focal depth was 8 .0 km (Chapman, 2013). The fault rupture comprised three slip events that occurred over about 1 .57 s, with the second slip episode releasing approximately 60% of the total moment (Chapman, 2013). The permanent relative displacements (D) computed from the recorded motions from the mainshock of the 2011 Mineral earthquake are compared with those predicted by the proposed CENA model. The ground motion records were obtained from the Center for Engineering Strong Motion Data (www.strongmotioncenter.org), except for those recorded at North Anna nuclear power plant, which were

ments by the NCHRP (2009) and the relations proposed herein for CENA rock and soil sites are shown in Fig. 10. (Note that of the existing predictive relations presented herein, only the NCHRP (2009) is shown in Fig. 10 because it is the only one that applies to CENA.) As shown in Fig. 10, for Amax ¼ 0 .25g and 0 .5g and for Vmax ¼ 25 cm/s and 50 cm/s, the NCHRP (2009) relations tend to predict similar displacements to those predicted by the relations proposed herein at smaller values of ky /Amax (i.e. ky /Amax , ,0 .05) and to predict smaller displacements than the relationships proposed herein at larger values of ky /Amax for CENA rock sites. On the contrary, for Amax ¼ 0 .25g and for Vmax ¼ 25 cm/s and 50 cm/s, the NCHRP (2009) relation for CENA soil sites tends to predict larger displacements than the relation proposed herein for ky /Amax values less than about 0 .4, while the predicted displacements are similar at larger ky /Amax values. Finally, for Amax ¼ 0 .5g and for Vmax ¼ 25 cm/s and 50 cm/s, the NCHRP (2009) for CENA soil sites matches very closely the displacements predicted by the relation proposed herein for the range of ky /Amax considered.

61


LEE AND GREEN 103


103

102


D: cm

D: cm

102

101

100 102

101

101 ky/Amax

100 102

100

100

101 ky/Amax

(a) 10

3


103

102


D: cm

D: cm

102

101

100 102

101

101 ky/Amax

100 102

100

101 ky/Amax

100

(b)

Fig. 9. Comparison of this study’s model and the existing relationships (Jibson, 2007; Saygili & Rathje, 2008; NCHRP, 2009) for WNA – soil: (a) Amax 0 .25g; (b) Amax 0 .50g

data from the Mineral earthquake shows that the displacements generally increase with decreasing distance and increasing Amax and Vmax, as expected. In comparison with the proposed CENA – rock model shown in Fig. 12, the permanent relative displacements computed from the Mineral, Virginia earthquake motions are in overall good agreement with those predicted by the proposed model, except for those at CW026 station. For this station, the proposed model predicts greater displacements than those computed from the recorded earthquake motions. There are many factors that may have resulted in the CW026 motions being outliers (e.g. the recorded motions at this site showed pronounced nearfault directivity velocity pulses) and further study is required to examine this in more detail.

obtained directly from Dominion Power. Table 6 lists the 2011 Mineral earthquake motions used herein and their recording stations. Also, Fig. 11 shows the locations of the epicentre and the recording stations within 200 km from the epicentre, where 200 km is approximately the maximum applicable distance of the proposed relationships. No information about the site conditions at the recording stations was yet available at the time of writing of this paper; thus, for comparison purposes, it was assumed that the sites were ‘rock’ (i.e. site class A or B in Table 1). Fig. 12 shows the permanent relative displacement comparisons of both horizontal components of motion recorded at each station during the Mineral earthquake with those predicted by the proposed CENA – rock model, along with the range of +/ one standard deviation of the proposed model. Note that only the recorded motions with D greater than 1 .0 cm are considered in this comparison. It is also noted that the 2011 Mineral earthquake motions were not used to compute displacements that were included in the regression analyses to develop the proposed model herein. The overall trend of the permanent relative displacement

CONCLUSIONS Empirical predictive relationships for permanent relative displacement for use in assessing the seismic stability of slopes, earthen dams and/or embankments subjected to stable continental motions have been developed. The predictive

62


PREDICTIVE RELATIONSHIP FOR SEISMIC LATERAL DISPLACEMENT OF SLOPES CENA – rock: Amax 0·25g

103

CENA – soil: Amax 0·25g

103

Vmax 25 cm/s; Present study




Vmax 25 cm/s; NCHRP (2009)




102

D: cm

D: cm

102

101

101

100 102

100 102

100

101 ky/Amax

100

101 ky/Amax

(a) CENA – rock: Amax 0·5g

103

CENA soil: Amax 0·5g

103









102

D: cm

D: cm

102

101

101

100 102

100 102

100

101 ky/Amax

100

101 ky/Amax

(b)

Fig. 10. Comparison of this study’s model and the NCHRP (2009) model for CENA – rock (left) and soil (right) sites: (a) Amax (b) Amax 0 .50g

0 .25g;

Table 6. 2011 Mineral earthquake records (CESMD, 2012) No.

Station

Code/ID

Network

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

North Anna NPP VA, Corbin (Fredricksberg Obs.) VA, Charlottesville VA, Reston Fire Station 25 PA, Philadelphia – Drexel University NY, Albany – VA Med VT White River Junction VAMC VA Pearisburg – Giles County CH SC Columbia – VA Hospital SC Charleston – Cha Pla Hotel SC Summerville – Fire Station NY Buffalo – VA Medical Center MA Bedford – VA Hospital Manchester – VA Medical Center MA Boston – Jamaica Plains

CW026 CBN CVVA 2555 2648 2653 2655 2549 2554 2544 2552 2654 2602 2652 2649

– NEIC NMSN USGS USGS USGS USGS USGS USGS USGS USGS USGS USGS USGS USGS

63


Repic: km 18 .7 58 .2 53 .5 121 .6 326 .1 629 .9 787 .9 254 .5 519 .3 603 .1 584 .5 557 .3 759 .6 787 .0 758 .0

PGA: g

PGV: cm/s

0 .11; 0 .26 0 .14; 0 .08 0 .10; 0 .12 0 .09; 0 .04 , 0 .01 , 0 .01 , 0 .01 , 0 .01 , 0 .01 , 0 .01 , 0 .01 , 0 .01 , 0 .01 , 0 .01 , 0 .01

4 .0; 14 .0 7 .1; 5 .1 1 .1; 1 .7 3 .0; 1 .2 , 1 .0 , 1 .0 , 1 .0 , 1 .0 , 1 .0 , 1 .0 , 1 .0 , 1 .0 , 1 .0 , 1 .0 , 1 .0

LEE AND GREEN 77°00W

78°00W

79°00W

39°00N N

2555 station

39°00N

CBN station

CVVA station

CW026 station

38°00N Epicentre

0 5 10

79°00W

20

30

38°00N

40 km 77°00W

78°00W

Fig. 11. Locations of the 2011 Mineral earthquake epicentre and the recording stations within 200 km from the epicentre (CESMD, 2012)

sions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. The authors sincerely thank Francisco Ciruela-Ochoa of Arup for generating the GIS map.

relationships proposed herein differs from existing relationships in that the NLME regression technique was used and separate regression analyses were performed for each ky. The resulting relationships have simple functional forms and correlate permanent relative displacement to Amax and Vmax. The predicted median permanent relative displacements decrease with increasing ky /Amax but increase with increasing Vmax. In comparing predicted displacements for WNA and CENA, WNA motions produce larger permanent relative displacements than CENA motions. In comparing rock and soil sites, soil motions have larger displacements than rock motions. Lastly, the permanent relative displacements predicted by the proposed model are shown to be in good agreement (mostly in +/ one standard deviation) with the displacements computed from motions recorded during the Mineral, Virginia earthquake of 23 August 2011, giving credence to the validity of the proposed model.

NOTATION maximum ground acceleration (g) regression coefficient regression coefficient regression coefficient permanent relative displacement (cm) yield acceleration (g) moment magnitude regional-dependent parameters for the frequency-dependent quality factor Q( f ) R site-to-source distance Repic epicentral distance Vmax maximum ground velocity (cm/s) VS shear wave velocity 0 shear wave velocity of the crust at the source 1 regression coefficient 2 regression coefficient 3 regression coefficient

Amax C1 C2 C3 D ky Mw Q0, n

ACKNOWLEDGEMENTS This material is based upon work supported in part by the National Science Foundation under grant numbers CMMI 1030564 and CMMI 1435494. All support is gratefully acknowledged. However, any opinions, findings and conclu-

64


PREDICTIVE RELATIONSHIP FOR SEISMIC LATERAL DISPLACEMENT OF SLOPES 1000

1000 CW026 CH1

CW026 CH1

CW026 CH3

CW026 CH3

μ (cm)

μ (cm) μ 1σ (cm)

100

μ 1σ (cm)

D: cm

D: cm

μ 1σ (cm)

10

1

μ 1σ (cm)

100

10

0

0·05

ky

0·10

1000

1

0·15

0·05

ky

0·10

CBN – H2

CBN – H2

μ (cm)

μ (cm)

μ 1σ (cm)

100

μ 1σ (cm)

D: cm

D: cm

μ 1σ (cm)

100

μ 1σ (cm)

10

1

10

0

0·05

ky

0·10

1

0·15

0

0·05

ky

0·10

0·15

1000

1000 CVVA – H1

CVVN – H1

CVVA – H2

CVVN – H2

μ (cm)

μ (cm)

μ 1σ (cm)

100

μ 1σ (cm)

D: cm

D: cm

μ 1σ (cm)

100

μ 1σ (cm)

10

1

0·15

1000

CBN – H1

CBN – H1

0

10

0

0·05

ky

0·10

1

0·15

0

0·05

ky

0·10

0·15

Fig. 12. Permanent relative displacements (D) of the recorded motions in two horizontal components (left and right) at the three stations (CW026, CBN and CVVA) during the 2011 Mineral earthquake compared to the present study’s model for CENA – rock; only the records with D greater than 1 cm are shown Table 7. Point source parameters for WNA and CENA motions (McGuire et al., 2001)

˜t ˜ k

time interval (s) stress drop at source (bars) parameter representing damping in shallow crust directly below site (s) r0 crustal density in the source region (g/cm3) standard deviation of intra-event error total the standard deviation of total error standard deviation of inter-event error

˜: bars k: s Q0 n 0: km/s r0: g/cm3

APPENDIX Scaling procedure for CENA motions The scaling procedure used by McGuire et al. (2001) consists of the following computation processes (a) (b) (c) (d )

WNA

CENA

65 0 .040 220 0 .60 3 .50 2 .70

120 0 .006 351 0 .84 3 .52 2 .60

the stress drop at the source; Q0 and n are regional-dependent parameters for the frequency-dependent quality factor, Q( f ); r0 is crustal density in the source region; and 0 is shear wave velocity of the crust at the source. Next, random vibration theory (RVT) was used to generate response spectra from the FAS (e.g. Boore, 1983; Boore & Joyner, 1984; Silva & Lee, 1987). The ratio of these two response spectra is the spectral transfer function. The response spectral transfer functions were generated for each site condition; horizontal/vertical components; earthquake magnitudes of 5 .5, 6 .5 and 7 .5 (i.e. centre value of magnitude bins); and distances of 1, 5, 30, 75 and 130 km. In total, 60 different transfer functions were therefore developed. Example transfer functions for M6 .5 cases are

determination of response spectral transfer function computation of response spectrum for a given ground motion determination of target response spectrum spectral matching of the time history.

A response spectral transfer function was obtained by first using the single-corner frequency point source model (Brune, 1970, 1971) to compute smoothed Fourier amplitude spectra (FAS) for both the CENA and WNA. The values of the point source model parameters used are listed in Table 7, where k is a parameter that represents damping in the shallow crust directly below the site; ˜ represents

65


LEE AND GREEN 10

Horizontal, Mw6·5, rock

Amplification

Amplification

10

1 R 130 km R 75 km R 30 km R 5 km R 1 km

Horizontal, Mw6·5, soil

1 R 130 km R 75 km R 30 km R 5 km R 1 km 0·1

0·1 0·1

1

10

0·1

100

1

Vertical, Mw6·5, rock

100

Vertical, Mw6·5, soil

10

Amplification

Amplification

10

10 Frequency: Hz

Frequency: Hz

1 R 130 km R 75 km R 30 km R 5 km R 1 km

1 R 130 km R 75 km R 30 km R 5 km R 1 km 0·1

0·1 0·1

1

10

0·1

100

1

10

100

Frequency: Hz

Frequency: Hz

Fig. 13. Response spectral transfer functions for M6 .5, rock and soil sites, horizontal and vertical components, and each of the distance cases (from McGuire et al., 2001) Chapman, M. C. (2013). On the rupture process of the 23 August 2011 Virginia Earthquake. Bull. Seismol. Soc. Am. 103, No. 2A, 613–628. Franklin, A. G. & Chang, F. K. (1977). Earthquake resistance of earth and rock-fill dams. In Report 5: Permanent displacements of earth embankments by Newmark sliding block analysis, Miscellaneous Paper S-71-17. Vicksburg, MS: Soils and Pavements Laboratory, U. S. Army Engineer Waterways Experiment Station. Green, R. A. & Cameron, W. I. (2003). The influence of ground motion characteristics on site response coefficients. Proceedings of the 7th Pacific Conference on earthquake engineering, University of Canterbury, Christchurch, New Zealand, paper number 90. Wellington, New Zealand: New Zealand Society of Earthquake Engineering. Green, R. A., Lee, J., Cameron, W. & Arenas, A. (2011). Evaluation of various definitions of characteristic period of earthquake ground motions for site response analyses. Proceedings of the 5th international conference on earthquake geotechnical engineering, Santiago, Chile. London, UK: International Society of Soil Mechanics and Geotechnical Engineering. Hanks, T. C. & McGuire, R. K. (1981). The character of highfrequency strong ground motion. Bull. Seismol. Soc. Am. 71, No. 6, 2071–2095. Jibson, R. W. (2007). Regression models for estimating coseismic landslide displacement. Engng Geol. 91, No. 2–4, 209–218. Lee, J. (2009). Engineering characterization of earthquake ground motions. PhD thesis, University of Michigan, Ann Arbor, MI, USA. McGuire, R. K., Becker, A. M. & Donovan, N. C. (1984). Spectral estimates of seismic shear waves. Bull. Seismol. Soc. Am 74, No. 4, 1427–1440. McGuire, R. K., Silva, W. J. & Costantino, C. J. (2001). Technical basis for revision of regulatory guidance on design ground motions: Hazard-and risk-consistent ground motion spectra guidelines. Washington, DC, USA: US Nuclear Regulatory Commission.

shown in Fig. 13. The response spectrum (5% damping) of a WNA ‘seed’ acceleration–time history is then computed. Next, the CENA target response spectrum is obtained by multiplying the ‘seed’ motion’s response spectrum by the appropriate response spectral transfer function. Lastly, the ‘seed’ acceleration time history is scaled to match the target CENA response spectrum (Silva & Lee, 1987). In the spectral matching process, a sample time interval ˜t of 0 .005 s (the corresponding Nyquist frequency is 100 Hz) was used to avoid aliasing effects in the frequency range of interest.

REFERENCES Ambraseys, N. N. & Menu, J. M. (1988). Earthquake-induced ground displacements. Earthquake Engng Structl Dynam. 16, No. 7, 985–1006. Ambraseys, N. N. & Srbulov, M. (1994). Attenuation of earthquakeinduced ground displacements. Earthquake Engng Structl Dynam. 23, No. 5, 467–487. Boore, D. M. (1983). Stochastic simulation of high-frequency ground motions based on seismological models of the radiated spectra. Bull. Seismol. Soc. Am. 73, No. 6A, 1865–1894. Boore, D. M. (1986). Short-period P-and S-wave radiation from large earthquakes: implications for spectral scaling relations. Bull. Seismol. Soc. Am. 76, No. 1, 43–64. Boore, D. M. & Joyner, W. B. (1984). A note on the use of random vibration theory to predict peak amplitudes of transient signals. Bull. Seismol. Soc. Am. 74, No. 5, 2035–2039. Bray, J. D. & Travasarou, T. (2007). Simplified procedure for estimating earthquake-induced deviatoric slope displacements. J. Geotech. Geoenviron. Engng 133, No. 4, 381–392. Brune, J. N. (1970). Tectonic stress and spectra of seismic shear waves from earthquakes. J. Geophys. Res. 75, No. 26, 611–614. Brune, J. N. (1971). Correction. J. Geophys. Res. 76, No. 20, 1441– 1450. CESMD (2012) Center for Engineering Strong Motion Data. See http://strongmotioncenter.org (accessed 04/02/2012).

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PREDICTIVE RELATIONSHIP FOR SEISMIC LATERAL DISPLACEMENT OF SLOPES Richards, R. & Elms, D. G. (1979). Seismic behavior of gravity retaining walls. J. Geotech. Engng Div., ASCE 105, No. GT4, 449–464. Saygili, G. & Rathje, E. M. (2008). Empirical predictive models for earthquake-induced sliding displacements of slopes. J. Geotech. Geoenviron. Engng 134, No. 6, 790–803. Schneider, J. F., Silva, W. J. & Stark, C. (1993). Ground motion model for the 1989 M 6 .9 Loma Prieta earthquake including effects of source, path, and site. Earthquake Spectra 9, No. 2, 251–287. Silva, W. J. (1993). Factors controlling strong ground motion and their associated uncertainties. In Dynamic analysis and design considerations for high-level nuclear waste repositories (ed. Q. A. Hossain), pp. 132–161. New York, NY, USA: American Society of Civil Engineers. Silva, W. J. & Lee, K. (1987). WES RASCAL Code for Synthesizing Earthquake Ground Motions: State-of-the-art for assessing earthquake hazards in the United States, Report 24. Vicksburg, MS, USA: US Army Engineering Waterways Experiment Station.

NCHRP (2009). National Cooperative Highway Research Program (NCHRP) Report 611: Seismic analysis and design of retaining walls, buried structures, slopes, and embankments. Washington, DC, USA: The National Academies Press. Newmark, N. M. (1965). Effects of earthquakes on dams and embankments. Geótechnique 15, No. 2, 139–160, http:// dx.doi.org/10.1680/geot.1965.15.2.139. Pinheiro, J. C. & Bates, D. M. (2000). Mixed-effects models in S and S-PLUS. New York, NY, USA: Springer. R Foundation (2013). Program-R (version 3.0.2) A language and environment for statistical computing and graphics. Vienna, Austria: R Foundation. See http://www.r-project.org/ (accessed 07/10/2013). Rathje, E. M. & Antonakos, G. (2011). A unified model for predicting earthquake-induced sliding displacement of rigid and flexible slopes. Engng Geol. 122, No. 1–2, 51–60. Rathje, E. M. & Saygili, G. (2009). Probabilistic assessment of earthquake-induced sliding displacements of natural slopes. Bull. New Zealand Soc. Earthquake Engng 42, No. 1, 18–27.

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Tsinidis, G. et al. (2015). Geótechnique 65, No. 5, 401–417 [http://dx.doi.org/10.1680/geot.SIP.15.P.004]

Dynamic response of flexible square tunnels: centrifuge testing and validation of existing design methodologies G . T S I N I D I S , K . P I T I L A K I S , G . M A DA B H U S H I † a n d C . H E RO N †

A series of dynamic centrifuge tests were performed on a flexible aluminium square tunnel model embedded in Hostun dry sand. The tests were carried out at the centrifuge facility of the University of Cambridge in order to further improve knowledge regarding the seismic response of rectangular embedded structures and to calibrate currently available design methods. The soil–tunnel system response was recorded with an extensive instrumentation array, comprising miniature accelerometers, pressure cells and position sensors in addition to strain gauges, which recorded the tunnel lining internal forces. Tests were numerically analysed by means of full dynamic time history analysis of the coupled soil–tunnel system. Numerical predictions were compared to the experimental data to validate the effectiveness of the numerical modelling. The interpretation of both experimental and numerical results revealed, among other findings: (a) a rocking response of the model tunnel in addition to racking; (b) residual earth pressures on the tunnel side walls; and (c) residual internal forces after shaking, which are amplified with the tunnel’s flexibility. Finally, the calibrated numerical models were used to validate the accuracy of simplified design methods used in engineering practice. KEYWORDS: centrifuge modelling; earthquakes; numerical modelling; soil/structure interaction; tunnels & tunnelling

issues that significantly affect the seismic response (Pitilakis & Tsinidis, 2014). Seismic earth pressures and shear stresses distributions along the perimeter of the embedded structure and complex deformation modes during shaking for rectangular cross–sections (e.g. rocking and inward deformations) are, among other issues, still not entirely understood. The knowledge shortfall motivated a range of experimental (e.g. Chou et al., 2010; Shibayama et al., 2010; Chian & Madabhushi, 2012; Cilingir & Madabhushi, 2011a, 2011b, 2011c; Lanzano et al., 2012; Chen et al., 2013), numerical (e.g. Anastasopoulos et al., 2007, 2008; Amorosi & Boldini, 2009; Anastasopoulos & Gazetas, 2010; Kontoe et al., 2011; Lanzano et al., 2014) and analytical (e.g. Huo et al., 2006; Bobet et al., 2008; Bobet, 2010) research studies over recent years, investigating the effects of seismic shaking and earthquake-induced ground failures (e.g. liquefaction) on the response of embedded structures. In some cases, the efficiency of different design methods has been investigated by comparing the outcomes of the methods (e.g. tunnel distortions or dynamic internal forces) with each other (e.g. Hashash et al., 2005, 2010; Kontoe et al., 2014). This study presents a series of dynamic centrifuge tests that were performed on a flexible aluminium square tunnel model embedded in dry sand. The soil–tunnel system response was recorded with an extensive instrumentation array comprising miniature accelerometers, pressure cells and position sensors, in addition to strain gauges, which recorded the tunnel lining internal forces. The test case is also numerically analysed by means of a full dynamic time history numerical analysis of the coupled soil–tunnel system. Numerical predictions are compared to the experimental data to validate the effectiveness of the numerical modelling. The calibrated numerical models are finally used to validate the accuracy of available simplified design methods used in engineering practice.

INTRODUCTION Recent earthquake events have demonstrated that underground structures in soft soils may undergo extensive damage or even collapse (Dowding & Rozen, 1978; Sharma & Judd, 1991; Iida et al., 1996; Kawashima, 2000; Wang et al., 2001; Kontoe et al., 2008). These failures increased interest in further investigation of the seismic response of these types of structures. Generally, the seismic response of embedded structures is quite distinct from that of above-ground structures, as the kinematic loading induced by the surrounding soil prevails over inertial loads stemming from the oscillation of the structure itself (Kawashima, 2000). In addition, large embedded structures are commonly stiff structures to withstand static loads. Hence, during earthquake shaking, strong interaction effects are mobilised between the structure and the surrounding soil, especially for structures of rectangular cross-section. These interaction effects are mainly affected by two crucial parameters, namely: (a) the soil to structure relative flexibility and (b) the soil– structure interface characteristics. In general, both are changing with the amplitude of seismic excitation, as they depend on the soil shear modulus and strength, which are related to the ground strains and the non-linear behaviour of the soil. Several methods are available in the literature for the evaluation of the response of underground structures and tunnels under seismic shaking (e.g. St John & Zahrah, 1987; Wang, 1993; Penzien, 2000; AFPS/AFTES, 2001; Hashash et al., 2001; ISO, 2005; Anderson et al., 2008; FHWA, 2009). The results of these methods may deviate, even under the same design assumptions, especially in case of rectangular structures (e.g. cut and cover tunnels), owing to both inherent epistemic uncertainties and a knowledge shortfall regarding some crucial Manuscript received 31 March 2014; revised manuscript accepted 23 December 2014. Published online ahead of print 20 March 2015. Discussion on this paper closes on 1 October 2015, for further details see p. ii. Department of Civil Engineering, Aristotle University, Thessaloniki, Greece. † Schofield Centre, University of Cambridge, Cambridge, UK.

DYNAMIC CENTRIFUGE TESTING The test was carried out on the 10 m diameter Turner beam centrifuge of the University of Cambridge (Schofield,

71


TSINIDIS, PITILAKIS, MADABHUSHI AND HERON 1980) under a centrifuge acceleration of 50g (scale factor n ¼ 50). A large equivalent-shear-beam (ESB) container was used to contain the model (Zeng & Schofield, 1996). Scaling laws that are applied to convert the measured quantities from model to prototype scale are summarised in Table 1 (Schofield, 1981). The soil deposit was made of uniform Hostun HN31 sand with 90% relative density. The physical and mechanical properties of the sand are summarised in Table 2. Sand pouring was performed in layers using an automatic hopper system (Madabhushi et al., 2006), while the model tunnel and the instruments were properly positioned during construction. The model tunnel, manufactured using 6063A aluminium alloy, was 100 mm wide and 220 mm long, having a lining thickness of 2 mm (Fig. 1(a)). The aluminium alloy mechanical properties are summarised in Table 3. According to the scale factor, the model corresponds to a 5 3 5 (m) square tunnel having an equivalent concrete lining thickness equal to 0 .13 m (assuming Ec ¼ 30 GPa for the concrete). This thickness is obviously unrealistic in practice, as the design analysis for the static loads will result in a much thicker lining. However, this selection was made in order to study the effect of high flexibility on the tunnel response, as well as to obtain clear measurements of the lining bending and axial strains. To simulate more realistically the soil– tunnel interface, sand was stuck on the external face of the

Table 3. Model tunnel mechanical properties Unit weight, ªt: kN/m3 2 .7

Model/Prototype

Dimensions

Length Mass Stress Strain Force Time (dynamic) Frequency Acceleration Velocity

1/n 1/n3 1 1 1/n2 1/n n n 1

l m ml�1t�2 1 mlt�2 t t�1 lt�2 lt�1

Table 2. Hostun HN31 physical and mechanical properties rs: g/cm3

emax

emin

2 .65

1 .01

0 .555

d10: mm d50: mm d60: mm �cv: degrees 0 .209

0 .335

0 .365

69 .5

Poisson ratio, v

Tensile strength, fbk: MPa

0 .33

220

model tunnel, creating a rough surface. Two polytetrafluoroethylene (PTFE) rectangular plates were placed at each end of the tunnel to avoid the entry of sand into the tunnel. The plates, which were marginally larger than the model tunnel, were connected to each other by a rod which passed through the tunnel (Fig. 1(b)). A dense instrumentation array was implemented to monitor the soil–tunnel response (Fig. 2). Miniature piezoelectric accelerometers were used to measure the acceleration in the soil, on the tunnel and on the container. The soil surface settlements were recorded in two locations using linear variable differential transformers (LVDTs), while two position sensors (POTs) were attached to the upper edges of the tunnel walls to capture the vertical displacement and the possible rocking of the model tunnel. Both the LVDTs and the POTs were attached to gantries running above the ESB container. Two miniature total earth pressure cells (PCs) were attached to the left side wall of the tunnel, allowing the measurement of the soil earth pressures on the wall. Strain gauges were attached to the inner and outer faces of the tunnel to measure the lining bending moment and axial force at several locations (Fig. 2). Unfortunately, the bending moment strain gauge at the middle of the roof slab (SG-B3) malfunctioned during testing. All the instruments were properly calibrated before and checked after testing. The strain gauges were carefully calibrated for static loading patterns using the procedure outlined in Tsinidis et al. (2014a). The data were recorded at a sampling frequency of 4 Hz during the swing up of the centrifuge and at 4 kHz during shaking. A series of air hammer tests was performed to estimate the soil shear wave velocity profile (Ghosh & Madabhushi, 2002). A small air hammer was introduced close to the base of the soil layer, while a set of accelerometers (AH, Fig. 2) were placed above it, forming a vertical array, allowing a record of the arrival times of the waves emanating from the air hammer. To ensure that the arrival times were adequately recorded, the accelerometers along this array were attached to a different acquisition system that allowed for a sampling frequency of 50 kHz. The dynamic input was provided at the container base by a stored angular momentum actuator, designed to apply

Table 1. Centrifuge scaling laws (Schofield, 1981) Parameter

Young’s modulus, E: GPa

33

PTFE plate

PTFE plate (a)

(b)

(c)

Fig. 1. (a) Model tunnel; (b) model tunnel placement in the equivalent shear beam container; (c) completed model in the equivalent shear beam container

72


DYNAMIC RESPONSE OF FLEXIBLE SQUARE TUNNELS 145

45

96·5

85

LVDT1

AH5 A15

100

AH3

PC2

AH2

PC1

110 60

141·5

145

A14

A3

A8

A10

AH4

427 370

100

LVDT2 POT1 POT2

A16

Strain gauges set-up SG-A4 SG-B3

A7 SG-B4

A6

A13

A2

A5

A12

SG-A3 SG-B2

SG-A1 SG-B1 SG-A2

AH1 Air hammer

A11

(Dimensions in mm) A9

A1

A4

673 Pressure cell

Accelerometer

POT

LVDT

Strain gauge

Fig. 2. Model layout and instrumentation scheme (A: accelerometer; AH: accelerometer above air hammer; LVDT: linear variable differential transformer; POT: position sensor; SG-A: axial strain gauge; SG-B: bending moment strain gauge; PC: pressure cell)

ducted in the frequency domain. In particular, acceleration– time histories were filtered at 10 to 400 Hz using a band pass eighth-order Butterworth filter. All the other data were filtered using a low-pass eighth-order Butterworth filter at 400 Hz.

sinusoidal or sine-sweep wavelets (Madabhushi et al., 1998). The model was subjected to a total of eight earthquakes during two flights: EQ1 to EQ5 were fired during a first flight, whereas EQ6 to EQ8 were fired during a subsequent flight. Fig. 3 presents the input motion–time histories, while Table 4 tabulates their characteristics. During each flight, the centrifuge was spun up in steps until 50g and then the earthquakes were fired in a row, leaving some time between them to acquire the data. To interpret the experimental results, the data were windowed, neglecting the parts of the signals before and after the main shake duration, while a filtering procedure was con-

0·15

0·15

0·30

0·2

0·4 t: s

0·6

0·30

0·8

EQ5

A/50g

A/50g

0

0·30

0·15 0·30 0·45 0·60 t: s EQ6

0·15

t: s

0·7

0·8

0

0·30

0

0·30

0·15 0·30 0·45 0·60 t: s EQ7

0

0·30

0·15 0·30 0·45 0·60 t: s

0·15 0·30 0·45 0·60 t: s EQ8

0·15

0·15 0

0

0·30

0·15

0·15 0·6

0 0·15

0·30

0·15

0

0·30 0·5

0·15

0·30

0·15

0

A/50g

0

0·15

A/50g

0

EQ4

0·30

0·15

A/50g

0

EQ3

0·30

0·15

A/50g

A/50g

0·15

0·30

EQ2

0·30

A/50g

EQ1

0·30

NUMERICAL ANALYSIS Numerical model The test was numerically simulated by means of full dynamic time history analyses, using the finite-element code Abaqus (Abaqus, 2012). The analyses were performed in

0 0·15

0

0·35 t: s

0·70

0·30

0

0·175 0·350 0·525 t: s

Fig. 3. Input motion–time histories

Table 4. Input motions characteristics (bracketed values in prototype scale) EQ ID Frequency, f: Hz Amplitude, a: g

EQ1

EQ2

EQ3

EQ4

EQ5

EQ6†

EQ7†

EQ8†

30 (0 .6) 1 .0 (0 .02)

45 (0 .9) 4 .0 (0 .08)

50 (1) 6 .5 (0 .13)

50 (1) 12 .0 (0 .24)

60 (1 .2) 12 .0 (0 .24)

50 (1) 5 .8 (0 .116)

50 (1) 6 .0 (0 .12)

50 (1) 11 .0 (0 .22)

Sine sweep. † Fired during a second flight.

73


TSINIDIS, PITILAKIS, MADABHUSHI AND HERON (Bilotta et al., 2014). Several research groups have simulated the tests using different numerical codes and constitutive models of different complexity (Amorosi et al., 2014; Conti et al., 2014; Gomes, 2014; Hleibieh et al., 2014; Tsinidis et al., 2014b). Among the most interesting results of this comparative effort is that even sophisticated constitutive models produced results that deviated considerably from the recorded data. Part of the difference was attributed to calibration issues and determination of constitutive parameters. The input motion was introduced at the model base in terms of acceleration–time histories, referring to the motion recorded by the reference accelerometer (A1, Fig. 2). The analyses were performed in two steps: first the gravity loads were introduced, while in a second step the earthquake motions were applied in a row, replicating each test flight. To this end, the loading history for the sand was accounted for.

prototype scale assuming plane strain conditions. Fig. 4 presents the numerical model layout. The soil was meshed with quadratic plane strain elements, while the tunnel was simulated with beam elements. The element size was selected in a way that ensured efficient reproduction of the waveforms of the whole frequency range under study. The base boundary of the model was simulated as rigid bedrock (shaking table), while for the side boundaries kinematic tie constraints were introduced, forcing the opposite vertical sides to move simultaneously, simulating, in that simplified way, the container. The soil–tunnel interface was modelled using a finite sliding hard contact algorithm embedded in Abaqus (Abaqus, 2012). The model constrains the two media when attached, using the penalty constraint enforcement method and Lagrange multipliers, while it also allows for separation. The interface friction effect on the soil–tunnel system response was investigated by applying different Coulomb friction coefficients �, namely � ¼ 0 for the full slip and 0 .4 and 0 .8 for non-slip conditions. In a final series of analyses, the soil and the tunnel were fully bonded, assuming no slip conditions, precluding separation. The model tunnel was modelled using an elastic–perfectly plastic material model, with yield strength equal to 220 MPa, while the soil response under seismic shaking was simulated in two ways. In a first series of analyses, a viscoelastic model was implemented, introducing a degraded shear modulus distribution and viscous damping (e.g. following the equivalent linear approximation method). In the second series of analyses, a non-associated Mohr–Coulomb model was used to account for the permanent deformations of the soil. The latter model, embedded in Abaqus, allows for simulation of certain hardening or softening responses after yielding. Elastic properties were assumed the same with the visco-elastic analyses, following a similar procedure as in Amorosi & Boldini (2009). This elasto-plastic model has been implemented by several researchers (e.g. Pakbaz & Yareevand, 2005; Hwang & Lu, 2007), while it has been recently used by Cilingir & Madabhushi (2011a, 2011b, 2011c) for the simulation of similar dynamic centrifuge tests on model tunnels in dry sand, revealing reasonable comparisons between the recorded data and the numerical results. The implemented models were selected as they are proposed in guidelines for dynamic analysis of embedded structures (e.g. equivalent linear approximation in FHWA (2009)) and are commonly used in tunnelling design practice owing to their easy calibration and control. Recently, a series of dynamic centrifuge tests on a flexible circular model tunnel embedded in dry sand (Lanzano et al., 2012) has been used as a benchmark of a numerical round robin on tunnel tests

Sand stiffness and strength The sand small-strain shear modulus (Gmax) was described according to Hardin & Drnevich (1972), which fits reasonably well with the air hammer test results and also results of laboratory tests (resonant column) that were performed on the specific sand fraction (Pistolas et al., 2014). Fig. 5 compares the estimated small-strain shear wave velocity gradient from different methods and the distribution proposed according to Hardin & Drnevich (1972). It is worth noting that these results refer to the ‘free-field’ conditions away from the model tunnel. The exact properties of sand in the area close to the tunnel are not well known. The reason is that considering the model’s formation (i.e. sand pouring from a height to achieve the desired relative density of the soil specimen), the existence of the model tunnel may affect the density of the sand in the adjacent zone, thus affecting the mechanical properties of the sand at this location. However, it is believed that after the first shakes the soil in this particular region will have reached a reasonable degree of densification comparable to the rest of the soil sample. To estimate the real sand stiffness and viscous damping during shaking a trial-and-error procedure was applied. More specifically, one-dimensional (1D) equivalent linear (EQL) soil response analyses of the soil deposit were performed, using different sets of G–ª–D curves for cohesionless soils (e.g. Seed et al., 1986; Ishibashi & Zhang, 1993; Pistolas et al., 2014). The analyses were performed in the frequency domain using EERA (equivalent-linear earthquake site response analyses) (Bardet et al., 2000). The computed

0 Soil–tunnel interface

0

90

VSO: m/s 180

270

360

Displacement constraints

Depth: m

5

10 AH flight 1 15

AH flight 2 RC results Empirical formulation

20 a(t)

Fig. 5. Small-strain shear wave velocity profiles estimated from air hammer tests (AH) and resonant column tests (RC) compared to the Hardin & Drnevich (1972) empirical formulation

Fig. 4. Numerical model in Abaqus

74


DYNAMIC RESPONSE OF FLEXIBLE SQUARE TUNNELS where e is the void ratio, � 9 is the mean effective stress (in MPa), G is the degraded shear modulus (in MPa) and Æ is the reduction value for each shake, ranging between 0 .3 and 0 .4. For the computation of the mean effective stress the earth coefficient at rest (K0) was evaluated as (Jaky, 1948)

horizontal acceleration–time histories and amplification were compared to the recorded data of the free field array (sensors A4 to A8 in Fig. 2). The adopted G–ª–D curves were those that resulted in the best fitting of the numerical predictions with the experimental results (Ishibashi & Zhang (1993) for small confining pressure). Comparisons of the adopted G–ª–D curves with empirical ones (Seed et al., 1986) and laboratory results from resonant column and cyclic triaxial tests for the specific sand fraction (Pistolas et al., 2014) are provided in Fig. 6. The adopted numerical curves compare reasonably well with the laboratory test results over a wide range of strain amplitudes. One-dimensional equivalent linear soil response analyses for the finally selected Gmax and G–ª–D curves revealed that a reduced Hardin and Drnevich distribution adequately reproduced the degraded sand shear modulus during shaking. To this end, the following expression was used for the description of the degraded strain shear modulus G ¼ Æ 3 100

(3 � e)2 . (� 9)0 5 1þe

K 0 ¼ 1 � sin �

where � is the sand friction angle. The reduced values for the sand shear modulus come in agreement with the shear moduli computed from the stress– strain loops, estimated using the recorded acceleration–time histories across the free-field array (A4–A8 in Fig. 2), following Zeghal & Elgamal (1994). It is noteworthy that this high decrease of the soil stiffness and increase of damping in this type of test is also reported by other researchers (Kirtas et al., 2009; Pitilakis & Clouteau, 2010; Lanzano et al., 2010, 2014; Li et al., 2013). In the final two-dimensional (2D) full dynamic analysis, the degraded elastic stiffness of the sand material for each shake was introduced through a Fortran user subroutine, which correlates the stiffness with the confining pressure at each soil element integration point. To this end, the effect of the tunnel on the surrounding sand stiffness was explicitly accounted for. In both visco-elastic and visco-elasto-plastic analyses, viscous damping was introduced in the form of the frequency dependent Rayleigh type. ‘Target’ damping (15%) was estimated through the 1D equivalent linear response analyses, as discussed before. For the calibration of the Rayleigh parameters, the double frequency approach was implemented. The Rayleigh parameters were properly tuned for different ‘important frequencies’ (e.g. soil deposit dominant frequencies or signal dominant frequencies). The finally selected parameters were those that resulted in good comparisons between the computed and recorded acceleration data. The importance of proper calibration for the Rayleigh coefficients is discussed in Kontoe et al. (2011). In the elasto-plastic analyses, additional energy dissipation was introduced by the hysteretic soil response. Regarding the strength parameters of the sand, a friction angle � equal to 338 (critical friction angle for the specific sand fraction) was used, while the dilatancy angle ł was assumed equal to 38 (Schanz & Vermeer, 1996). These strength parameters correspond to the specific sand fraction and are found to give reasonable comparisons with the recorded response. A slight cohesion (c ¼ 1 kPa) was introduced to avoid numerical problems.

(1)

1·00

G/G0

0·75

0·50

0·25

0 104

103

102 γ: % (a)

101

100

40 Seed et al. (1986) Analysis 30

σ 25 kPa, RC σ 50 kPa, RC

NUMERICAL PREDICTIONS COMPARED WITH EXPERIMENTAL RESULTS Representative comparisons between the recorded and the computed response are presented in this section. Through the presentation of relevant data several crucial aspects of the soil–tunnel response are discussed. Results are generally shown at model scale, if not stated otherwise.

D: %

σ 200 kPa, RC σ 50 kPa, TX 20

(2)

σ 200 kPa, TX

10

0 104

103

102 γ: % (b)

101

Horizontal acceleration Figure 7 presents time windows of typical comparisons between the recorded and the computed acceleration–time histories at two representative locations (middle section of left side wall, A13; top receiver of tunnel accelerometer array, A10). In Fig. 8 representative comparisons between the computed and recorded horizontal acceleration amplification along the free-field and the tunnel vertical accelerometer arrays are depicted. Generally, both visco-elastic and elastoplastic analyses reveal similar responses and amplification, while numerical predictions are in good agreement with the

100

Fig. 6. Adopted G–ª–D curves compared to resonant column test results (RC) (Pistolas et al., 2014), cyclic triaxial test results (TX) (Pistolas et al., 2014) and empirical proposals (Seed et al., 1986): (a) G/G0 plotted against shear strain; (b) damping plotted against shear strain

75


TSINIDIS, PITILAKIS, MADABHUSHI AND HERON 0·4

0·2

A/50g

0·2

A/50g

A10 – EQ4

0 Test

0·2

A10 – EQ5

0·4

0·2

0 0·2

A10 – EQ7

0·2

A/50g

0·4

A/50g

0·4

A10 – EQ2

0 0·2

0 0·2

Analysis 0·4 0·20

0·30

A13 – EQ2

0·4

0·4 0·50

0·30

(a)

A13 – EQ4

0·4

0·2

A/50g

A/50g

0·2

0·25 t: s

0 0·2

0 0·2

0·4 0·20

0·25 t: s

0·4 0·20

0·30

0·55 t: s

0·4 0·20

0·60

A13 – EQ5

0·4

0

0·4 0·50

0·30

0·30

A13 – EQ7

0 0·2

0·2

0·25 t: s

0·25 t: s

0·2

0·2

A/50g

0·4

0·25 t: s

A/50g

0·4 0·20

0·55 t: s

0·60

0·4 0·20

0·25 t: s

0·30

(b)

Fig. 7. Time windows of representative acceleration–time histories recorded and computed for different earthquake input motions; experimental data compared with visco-elasto-plastic results: (a) accelerometer A10 at the soil surface above the model tunnel; (b) accelerometer A13 on the tunnel side-wall (notation according to Fig. 2) A50g – EQ2 0

0

0·175

0·350

0

0·2

0

0·175

A50g – EQ8 0·350

0

0·1

0·1

Depth: m

0·2

0·2 0·3

0·3

0·4

0·4

0·4

0·4

Analysis

A50g – EQ3 0·350

0

0·1

0·175

Depth: m

A50g – EQ8

A50g – EQ5 0·350

0

0·1 Tunnel level

0·2

0

0·2

0

0·175

0·350

0

0·2

0

0·175

0·350

0·1

0·1

Depth: m

0

0·175

0·350

(a)

A50g – EQ2 0

0·175

0·2

0·3

Test

0

0·1

0·3

Depth: m

Depth: m

0·1

Depth: m

A50g – EQ5

A50g – EQ3 0·350

Depth: m

0·175

Depth: m

0

0

0·2

0·3

0·3

0·3

0·3

0·4

0·4

0·4

0·4

(b)

Fig. 8. Horizontal acceleration amplification along (a) the soil free-field accelerometers vertical array, and (b) the tunnel accelerometers vertical array, for different earthquake input motions; experimental data compared with visco-elasto-plastic results

deformations, discussed in the following section, are likely to have caused a malfunction of the accelerometer at this location. It worth mentioning the higher frequencies of the signals observed in the Fourier spectra shown in Fig. 9. Significant energy content is associated with higher frequencies than with the predominant one. These higher frequencies, which are attributed to the experimental equipment’s

records both in terms of amplitude and frequency content (Fig. 9). The differences, generally minor, are attributed to the inevitable differences between the assumed soil mechanical properties (stiffness and damping) and their actual values during the test, especially near the tunnel. The larger deviation observed at the tunnel roof slab is attributed to an erroneous record at this location. Actually, the slab inward

76


DYNAMIC RESPONSE OF FLEXIBLE SQUARE TUNNELS A8 – EQ1

0·2

A8 – EQ2

0·6

A8 – EQ5

0·8

A8 – EQ8

1·0

0

0

3 f/50: Hz

0·3

0

6

0

3 f/50: Hz

Amplification

Analysis 0·1

Amplification

Amplification

Amplification

Test

0·4

0

6

0

3 f/50: Hz

0·5

0

6

0

3 f/50: Hz

6

(a)

0

3 f/50: Hz

6

0·3

0

0

A10 – EQ5

0·8

Amplification

Amplification

Amplification

0·1

0

A10 – EQ2

0·6

3 f/50: Hz

0·4

0

6

0

A10 – EQ8

1·0

Amplification

A10 – EQ1

0·2

3 f/50: Hz

6

0·5

0

0

3 f/50: Hz

6

(b)

Fig. 9. Fourier spectra of acceleration–time histories recorded and computed at locations of (a) accelerometer A8 and (b) accelerometer A10 for different earthquake input motions; experimental data compared with visco-elasto-plastic results

Tunnel initial shape

mechanical response (Brennan et al., 2005), are described quite efficiently by the numerical model. Tunnel deformed shapes Figure 10 presents time windows of typical comparisons between the recorded and computed vertical accelerations at the sides of the tunnel roof slab. Experimental results are slightly larger than the numerical predictions. The difference is attributed to the parasitic yawing movement of the whole model on the shaking table during shaking, which may amplify vertical acceleration and cannot be reproduced by the numerical analysis. The no-slip condition analysis results are closer to the recorded response. Generally, signals are out of phase, indicating a rocking mode of vibration for the tunnel, in addition to the racking mode. Fig. 11 presents typical computed deformed shapes of the tunnel during shaking, verifying this complex racking–rocking response. Owing to the high flexibility of the tunnel, inward deformations are also observed for the slabs and the walls.

Test

0·2

Analysis – Full slip

0·2

0·1

A/50g

A/50g

0·1

0

Tunnel deformed shape

Analysis – No slip

0·1

0

A15 A16

0

0·1

0·1

0·1

0·2 0·30

0·2 0·30

0·2 0·30

A15 0·33 t: s

0·36

(b)

Fig. 11. Shape of deformed tunnel for time steps of the computed maximum racking distortion; EQ4 earthquake, elasto-plastic analysis for no slip conditions: (a) motion towards left; (b) motion towards right (deformations scale 360)

A/50g

0·2

(a)

0·33 t: s

0·36

A16 0·33 t: s

0·36

Fig. 10. Time windows of recorded and computed vertical acceleration–time histories at the sides of the tunnel roof slab for EQ4 earthquake; experimental data compared with visco-elasto-plastic results

77


TSINIDIS, PITILAKIS, MADABHUSHI AND HERON efficiency of the constitutive models, and also to recording issues that are related to the response of the miniature earth pressures cells in the case of granular dry sands. Accurate measurement of earth pressures in sands with miniature pressure cells is always difficult, as the relative stiffness of the sensing plate may affect the readings, while there are also problems related to the grain size effect (Cilingir, 2009). Moreover, inward deformations of the tunnel wall may slightly change the recording direction (small inclination of the pressure cell) and therefore the recorded earth pressure may be different from the ‘normal’ value computed by the analysis. Considering the aforementioned points, the comparisons indicate a reasonable agreement. Figure 13 presents typical dynamic earth pressure distributions around the tunnel’s perimeter, referring to the time step of the tunnel maximum racking distortion. Soil yielding around the tunnel results in stress redistributions leading to a slightly different response between elasto-plastic and visco-

Dynamic earth pressures Typical comparisons between the computed and recorded dynamic earth pressures–time histories at the left side wall are presented in Fig. 12. The effect of the soil–tunnel interface characteristics on the computed earth pressures is also highlighted. Residual values are presented in records after shaking as a result of the soil yielding and densification around the tunnel. This post-earthquake residual response has also been reported during similar centrifuge tests (Cilingir & Madabhushi, 2011a, 2011b) and is amplified with the flexibility of the tunnel. In addition, dynamic pressure increments are found to be larger near the stiff corners of the tunnel. Generally, numerical predictions for no-slip conditions are closer to the recorded response. The comparison is more satisfactory, especially for the last shakes. Observed differences in amplitude can be attributed to the discrepancies between the assumed and the actual in test mechanical properties of the sand and the soil–tunnel interface, the

0 50

50

0

0·3 t: s

0·13 t: s

0

100

0·18

PC2 PC1

20

50

100 0·08

0·6

PC2 – EQ6

40

50

σ: kPa/m

σ: kPa/m

σ: kPa/m

0

PC2 – EQ6

100

50

50

100

PC1 – EQ6

100

σ: kPa/m

PC1 – EQ6

100

0 20

0

0·3 t: s

0·6

40 0·08

0·13 t: s

0·18

(a)

0 80

80

0

0·3 t: s

0·6

20

0 80

160 0·06

0·11 t: s

0·16

160

PC2 – EQ8

40

80

σ: kPa/m

σ: kPa/m

σ: kPa/m

0

PC2 – EQ8

160

80

80

160

PC1 – EQ8

160

σ: kPa/m

PC1 – EQ8

160

0 20

0

0·3 t: s

0·6

40 0·06

0·11 t: s

0·16

(b) Test



Fig. 12. Dynamic earth pressure–time histories recorded and computed by visco-elasto-plastic analyses on the left side wall for different earthquake inputmotions; effect of the soil–tunnel interface characteristics: (a) earthquake EQ6; (b) earthquake EQ8 120

120 C

A

B

60

σ: kPa/m

σ: kPa/m

60

D

0

0

Full slip, elasto-plastic

60

60

No slip, elasto-plastic No slip, elastic 120

A

B

C

D

120

A

(a)

A

B

C

D

A

(b)

Fig. 13. Effect of soil–tunnel interface characteristics and soil yielding response on the dynamic earth pressures distributions computed along the perimeter of the tunnel at the time step of maximum racking distortion: (a) earthquake EQ2; (b) earthquake EQ4

78


DYNAMIC RESPONSE OF FLEXIBLE SQUARE TUNNELS case. Fig. 17 illustrates this effect on the residual bending moments computed for different shaking scenarios. It is noteworthy that the recorded residual bending moments for EQ7 are much lower than EQ6, although both input motions share the same amplitude and frequency characteristics. This is attributed to the fact that the largest part of soil plastic strain that is induced by the specific input motion amplitude (the same for both shakes) is accumulated during the first loading circles of the first shake (EQ6). This phenomenon is simulated reasonably well by the implemented elasto-plastic model.

elastic analyses (effect on distributions). Moreover, soil– tunnel interface properties seem to affect the soil yielding response in the area adjacent to the tunnel (Fig. 14) and therefore the pressure distributions. This relation between the soil yielding response and the soil–tunnel interface properties is also reported by Huo et al. (2005). Soil dynamic shear stresses Figure 15 portrays representative soil dynamic shear stress distributions around the tunnel computed for the time step of maximum racking distortion. As for the earth pressures, soil yielding affects the soil shear stress around the tunnel. Generally, shear stresses tend to increase near the tunnel corners due to the higher earth pressures (confining pressures for the tunnel) at these locations. As expected, interface friction plays an important role on the shear stress distribution and magnitude. An increase of the soil–tunnel interface friction results in an increase of the soil shear stresses along the middle sections of the tunnel slabs and walls.

Lining dynamic axial force Similar to the dynamic bending moments, residual values were recorded for the lining axial forces (Fig. 18). Residuals were generally smaller than the ones of the bending moment, but were larger along the slabs. In addition, dynamic axial forces recorded on the side walls were out of phase, verifying the racking–rocking response of the tunnel during shaking (Tsinidis et al., 2014a). Numerical results revealed similar tendencies. The effect of the mobilised friction (along the interface) on the lining axial forces is quite important (Fig. 18). Similar to the dynamic earth pressures, recorded axial forces were found to be in better agreement with the numerical predictions assuming no-slip conditions. This observation may be attributed to the inward deformations of the model tunnel that are amplified by the tunnel’s high flexibility. The surrounding sand is actually squeezing the tunnel, leading to a more rigid soil–tunnel interface (no separation–no-slip conditions). Generally, both the visco-elastic and the elasto-plastic analyses reproduce the recorded dynamic internal forces increments (reversible component of force increments) reasonably well (Fig. 19). These increments, which are computed as the half of the amplitude of the maximum values of the loading cycles in the internal forces–time histories, are in both cases amplified near the tunnel corners (Fig. 19).

Lining dynamic bending moment Representative comparisons between recorded and computed by elasto-plastic analyses dynamic bending moment– time histories are presented in Fig. 16. Both experimental data and numerical predictions indicate a post-earthquake residual response, similar to that of the earth pressures. This residual response is highly affected by the tunnel’s flexibility. Different assumptions for the soil–tunnel interface characteristics may affect the computed bending moments both in terms of residuals and dynamic increments, mainly due to the different soil yielding response around the tunnel in each No slip

Full slip

0·018

0·009

SIMPLIFIED ANALYSIS METHODS Simplified methods are commonly used in design practice, especially during preliminary stages of design, mainly due to their simplicity and reduced computational cost compared to the non-linear full dynamic analysis. The majority of these methods rely on the assumption that the seismic load is introduced on the tunnel in a quasi-static manner, and therefore they do not account for the dynamic soil–structure interaction effects (Pitilakis & Tsinidis, 2014). In this section two of the most commonly used methods are discussed,

0

Fig. 14. Soil plastic deformations computed by the visco-elastoplastic numerical analyses around tunnel at end of first flight (deformations scale 310) 60

60

45

D

C

A

B

Full slip μ 0·4 45

μ 0·8

τ: kPa/m

τ: kPa/m

No slip 30

15

0

30

15

A

B

C (a)

D

0

A

A

B

C (b)

D

A

Fig. 15. Effect of the soil–tunnel interface properties on the soil dynamic shear stress distributions computed along the perimeter of the tunnel at the time step of maximum racking distortion: (a) earthquake EQ4; (b) earthquake EQ7

79


TSINIDIS, PITILAKIS, MADABHUSHI AND HERON

5

10

0

0·3 t: s

0

2·5

5·0 0·10

0·6

0·15 t: s

SG-B4 – EQ2

5

M: N mm/mm

0

SG-B1 – EQ2

2·5

M: N mm/mm

M: N mm/mm

5

0·20

0

5

10

0

0·3 t: s

SG-B4 – EQ2

2·5

M: N mm/mm

SG-B1 – EQ2

0·6

0

2·5

5·0 0·10

0·15 t: s

0·20

(a)

0

5

10

0

0·3 t: s

0·17 t: s

10

0

0·3 t: s

5

0·18

0·6

SG-B4 – EQ8

5

0

0·13 t: s

5

(b)

SG-B1 – EQ8

10 0·08

0·6

0·22

0

0

5

10

0

0·3 t: s

SG-B4 – EQ3

2·5

M: N mm/mm

2·5

5

M: N mm/mm

0

0

5·0 0·12

0·6

SG-B1 – EQ8

5

M: N mm/mm

0·3 t: s

M: N mm/mm

5

SG-B4 – EQ3

5

M: N mm/mm

0

10

2·5

M: N mm/mm

M: N mm/mm

5

SG-B1 – EQ3

SG-B4 0

2·5 SG-B1 5·0 0·12

0·6

0·17 t: s

0·22

SG-B4 – EQ8

5

M: N mm/mm

SG-B1 – EQ3

0

5

10 0·08

0·13 t: s

0·18

(c) Test



Fig. 16. Dynamic bending moment–time histories recorded and computed by visco-elasto-plastic analysis for different earthquake input motions, effect of the soil–tunnel interface properties: (a) earthquake EQ2; (b) earthquake EQ3; (c) earthquake EQ8

EQ2

2·5 0 2·5 5·0

A

B

C (a)

EQ4

12 6

M: N mm/mm

M: N mm/mm

5·0

D

A

0 6 12

A

B

2

5

1

0 5 10

A

B

C

A

D

A

D

C

A

B

0 1 2

A

B

(c) Full slip

D

EQ7

10

M: N mm/mm

M: N mm/mm

EQ6

C (b)

C

D

A

(d) μ 0·4

μ 0·8

No slip

Test

Fig. 17. Residual dynamic bending moment distributions along the perimeter of the tunnel, recorded and computed by visco-elasto-plastic analyses at the end of shaking: (a) earthquake EQ2; (b) earthquake EQ4; (c) earthquake EQ6; (d) earthquake EQ7

80


DYNAMIC RESPONSE OF FLEXIBLE SQUARE TUNNELS 2·50

2·50

1·25 2·50

0

0

0·3 t: s

2·50 0·11

0·6

0·25

0 0·5

1·25

0·16 t: s

1·0

0·21

SG-A4 – EQ3

0·50

0·5

N: N/mm

0

SG-A4 – EQ3

1·0

1·25

N: N/mm

N: N/mm

1·25

SG-A3 – EQ3

N: N/mm

SG-A3 – EQ3

0

0·25

0

0·3 t: s

0·50 0·11

0·6

0·16 t: s

0·21

(a) 2·50

2·50

1·25 2·50

0 1·25

0

0·35 t: s

0·25

0

0·12 t: s

1·0

0·17

SG-A4

0

SG-A3

0·25

0·5

2·50 0·07

0·70

SG-A4 – EQ7

0·50

0·5

N: N/mm

0

SG-A4 – EQ7

1·0

1·25

N: N/mm

N: N/mm

1·25

SG-A3 – EQ7

N: N/mm

SG-A3 – EQ7

0

0·35 t: s

0·70

0·50 0·07

0·12 t: s

0·17

(b) 2·50

0 1·25

1·25

0

0·3 t: s

0·3

0 0·5

2·50 0·07

0·6

0·12 t: s

0·17

1·0

SG-A4 – EQ8

0·6

0·5

N: N/mm

N: N/mm

N: N/mm

0

SG-A4 – EQ8

1·0

1·25

1·25

2·50

SG-A3 – EQ8

N: N/mm

SG-A3 – EQ8

2·50

0 0·3

0

0·3 t: s

0·6

0·60 0·07

0·12 t: s

0·17

(c) Analysis – No slip

Test


Fig. 18. Dynamic axial force–time histories recorded and computed by visco-elasto-plastic analysis for different earthquake input motions, effect of the soil–tunnel interface properties: (a) earthquake EQ3; (b) earthquake EQ7; (c) earthquake EQ8 6·0

6·0 C

A

B

Elasto-plastic analysis Visco-elastic analysis

2|ΔM|

4·5

|ΔN|: N mm/mm

4·5

|ΔM|: N mm/mm

D

3·0

1·5

0

Test 2|ΔN|

3·0

1·5

A

B

C (a)

D

0

A

A

B

C (b)

D

A

Fig. 19. Internal forces dynamic increments along the tunnel perimeter: (a) bending moment for EQ3; (b) axial force for EQ4

modelled as an equivalent static load or pressure that is imposed on the frame (Fig. 20(a)). This ‘structural’ racking distortion is evaluated by the free-field ground racking distortion, which is properly adjusted, through the so-called racking ratio (structural to ground racking distortions), in order to account for the soil–tunnel interaction effects. The racking ratio is correlated with relative flexibility of the soil

namely, the design procedure proposed by Wang (1993) and the pseudo-static seismic coefficient deformation method (FHWA, 2009) or detailed equivalent static analysis method (ISO, 2005). According to the first methodology, the tunnel seismic response is evaluated through a simple static frame analysis. The structural racking distortion due to ground shaking is

81


TSINIDIS, PITILAKIS, MADABHUSHI AND HERON Δstr P

Finertia

αff

δ

Δstr

H

H

Δstr

Δff

Δstr

P B

B x (a)

(b)

(c)

Fig. 20. Schematic representation of the simplified analysis methods: (a) Wang (1993) simplified method, (b) detailed equivalent static analysis method, distributed inertial loads, (c) detailed equivalent static analysis method, imposed deformations at model boundaries

separately for each earthquake scenario, using the numerical model presented in Fig. 4. Although simplified methods propose an equivalent linear approximation (e.g. degraded shear modulus computed from site response analysis) to account for the soil non-linear response under ground shaking (e.g. FHWA, 2009), both elastic and elasto-plastic analyses are performed, using the constitutive models presented before, in order to check the effect of the soil yielding response on the results. Moreover, to study the effect of the soil–tunnel interface properties, the analyses are carried out under full slip and no-slip conditions. Sand mechanical properties (e.g. stiffness and strength) are selected in order to correspond with those of the dynamic analysis, while the equivalent seismic loads (e.g. inertia forces or ground displacements) are computed from the dynamic analysis, referring to the free field and for the time step of maximum tunnel racking distortion. To investigate the effect of the input motion amplitude, the analyses are performed for EQ3 (0 .13g) and for EQ4 (0 .24g) according to Table 4, whereas to study the input motion frequency content on the response, a final set of analyses is performed using the Japanese Meteorological Agency (JMA) record from the 1995 Kobe earthquake scaled down to 0 .24g. The following presented results refer to extreme scenarios regarding the tunnel flexibility and therefore they should be interpreted as limit cases. Soil strength parameters may affect the soil yielding response and therefore may alter the results of non-linear analyses. Considering the relatively low strength estimated in the examined cases and the associated increased yielding response, the results may be considered conservative. Table 5 presents representative comparisons of racking ratios estimated from different approaches for EQ4, assuming elastic soil response. Generally, the numerical results for no-slip conditions resulted in larger racking ratios (12–35% larger) compared to the full slip conditions. Moreover, racking ratios computed from the equivalent static analyses seem to be slightly lower (15–20%) compared to the dynamic analysis results. The NCHPR611 analytical relation (Anderson et al., 2008) overestimates the racking ratio for the flexible tunnel, while for the rigid tunnel, assuming no-slip

to the tunnel that is expressed through the flexibility ratio F (Wang, 1993) F¼

G3B S3H

(3)

where G is the soil shear modulus, B and H are the width and the height of the structure, respectively, and S is the required force to cause a unit racking deflection of the structure. According to NCHPR611 regulations (Anderson et al., 2008) the racking ratio can be computed as R¼

˜str 2F ¼ ˜ff (1 þ F)

(4)

In the detailed equivalent static analysis method, a 2D soil–tunnel numerical model is proposed for the analysis, similar to the dynamic analysis (ISO, 2005; FHWA, 2009). The seismic load is introduced in a pseudo-static manner, as equivalent inertial load throughout the entire model that corresponds to the ground free-field acceleration amplification profile (Fig. 20(b)). In an alternative to this method, equivalent seismic load is introduced as a ground deformation pattern on the numerical model boundaries (Fig. 20(c)), corresponding to the free-field ground response (Kontoe et al., 2008; Hashash et al., 2010). The test case presented herein is used as a case study to verify the effectiveness of the aforementioned simplified methods. More specifically, the results of the implemented simplified methods are compared to the calibrated dynamic analysis that is used as the benchmark case. The comparisons are made in terms of computed racking ratio and dynamic bending moment in the lining, which are considered to be representative parameters for the validation. The flexibility ratio for the given case is estimated equal to F ¼ 62 .5, indicating a quite flexible structure compared to the surrounding soil. To further extend the comparisons, a second series of analyses are performed, increasing the tunnel lining thickness, in order to model a rigid tunnel (F ¼ 0 .29). Both static and dynamic analyses are performed

Table 5. Racking ratios estimated by different methods under the assumption of elastic soil response for EQ4 Case Flexible tunnel – full slip Flexible tunnel – no slip Rigid tunnel – full slip Rigid tunnel – no slip

Dynamic analysis 1 .3 1 .46 0 .5 0 .74

Equivalent static analysis – force

Equivalent static analysis – displacement

NCHPR611 Anderson et al. (2008) – (R ¼ 2F/(1 + F))

1 .27 1 .42 0 .47 0 .72

1 .22 1 .40 0 .40 0 .65

1 .96 1 .96 0 .45 0 .45

82


DYNAMIC RESPONSE OF FLEXIBLE SQUARE TUNNELS conditions, numerical analyses result in a ratio larger than the analytical estimation. An underestimation of the racking ratio will result in underestimation of the lining forces (e.g. implementing Wang’s method). On the contrary, an overestimation of the racking ratio may lead to an overdesign that may be considered as a conservative ‘safe’ design concept. However, overdesign is not only needlessly expensive but may lead to the stiffening of the structure, which may in turn change the whole response pattern in a detrimental way. Figure 21 presents representative comparisons of the dynamic bending moment distributions along the tunnel’s perimeter, computed with different design methods, assuming no-slip conditions. The elasto-plastic analyses numerical results for the flexible tunnel case are also compared with the experimental data (Fig. 21(b)). Table 6 tabulates similar comparisons between the recorded and the computed dynamic bending moment at the locations of the strain gauges. Numerical results correspond to full-slip conditions in this case. Generally, for the assumption of elastic soil response, the equivalent static analyses reproduce well the computed bending moment distribution from the dynamic analysis. However, the maximum bending moment is underestimated for both the flexible and the rigid tunnel, especially when the equivalent seismic load is introduced in terms of deformation at the model boundaries. In the case of the elasto-plastic analyses, bending moment distributions are more complex, especially for the flexible tunnel, due to the associated larger soil yielding. Experimental data are generally closer to the dynamic analysis results (Fig. 21(b) and Table 6). Actually, equivalent static analyses results barely follow the experimental data and the bending moment distribution computed by the dynamic analysis, exhibiting values which are considerably lower. For the rigid tunnel case, simplified analyses results are closer to the dynamic analysis, but again the differences are quite noticeable. It is obvious that simplified methods cannot reproduce the soil loading history during shaking as efficiently as the dynamic analysis. This loading history affects significantly

Table 6. Comparisons between recorded and computed – from different design methods – bending moments at receivers’ positions (EQ3 elasto-plastic analyses for full slip conditions) Position

M: N mm/mm Full dynamic analysis 3 .90 1 .59 4 .00

SG-B1 SG-B2 SG-B4

2

6

0 2 B

C (a)

D

12

A

A

30

M: N mm/mm

M: N mm/mm

30 0 30 C (c)

B

C (b)

D

A

Elasto-plastic analyses, F 0·29 60

B

4 .16 3 .59 4 .21

6

Elastic analyses, F 0·29

A

1 .74 0 .20 0 .25

0

60

60

Test

Elasto-plastic analyses, F 62·5 12

M: N mm/mm

M: N mm/mm


A

2 .55 0 .25 1 .10

Equivalent static analysis – deformation

the soil permanent response. Similar to the elastic analyses, the differences are higher for the cases where the equivalent seismic loads are introduced in terms of imposed ground displacement at the boundaries. Local yielding at these boundary locations may affect the tunnel loading. Figure 22 plots static to dynamic bending moment ratios that are computed at a crucial lining section (joint C, Fig. 21) under different assumptions regarding the soil–tunnel interface properties, the soil response (elastic and elastoplastic) and the input motion characteristics. Generally, equivalent static analyses underestimate the bending moment compared to the full dynamic analysis. For the elastic analyses, the differences may reach 20 to 40%. The discrepancies are even higher for the elasto-plastic analyses (differences up to 60%), especially for the flexible tunnel case. The differences are generally higher for the cases where the equivalent seismic load is introduced in terms of ground displacements at the model boundaries. This may be attributed to the relatively large distance between the tunnel and the numerical model boundaries (14 .3 m for the side boundaries), where the ground deformation is imposed. By increasing this distance it is possible that a greater amount of induced ground strain is artificially absorbed by the soil elements, thus ‘relieving’ the structure and altering the

4

4

Equivalent static analysis – force

D

30 60

A

Dynamic analysis

0

A

EQL static analysis – force Test

EQL static analysis – deformation

B

C (d) D

C

A

B

D

A

Fig. 21. Dynamic bending moment distributions along the tunnel perimeter computed from different methods for EQ3: (a) flexible tunnel–elastic analysis; (b) flexible tunnel–elasto-plastic analysis; (c) rigid tunnel–elastic analysis; (d) rigid tunnel– elasto-plastic analysis

83


TSINIDIS, PITILAKIS, MADABHUSHI AND HERON Elasto-plastic analyses, F 62·5

1·0

1·0

0·8

0·8

Mstatic /Mdynamic

Mstatic /Mdynamic


0·6 0·4 0·2 0

EQ3

0·4 0·2 0

JMA

EQ4 (a)

0·6

EQ3

Elasto-plastic analyses, F 0·29

1·0

1·0

0·8

0·8

Mstatic /Mdynamic

Mstatic /Mdynamic


0·6 0·4 0·2 0

EQ3 Full slip, force

EQ4 (c)

JMA

EQ4 (b)

0·6 0·4 0·2 0

JMA Full slip, displacement

EQ3 No slip, force

EQ4 (d)

JMA No slip, displacement

Fig. 22. Static to dynamic bending moment ratios computed at the right side-wall–roof slab corner: (a) flexible tunnel– elastic analysis; (b) flexible tunnel–elasto-plastic analysis; (c) rigid tunnel–elastic analysis; (d) rigid tunnel–elasto-plastic analysis

earth pressures on the side walls and the lining forces, which were amplified by the increased flexibility of the tunnel. This complex response associated with residual deformations and internal forces in the lining cannot be reproduced by the equivalent linear approximation method that is often proposed in regulations and used in engineering practice. Therefore, this approach should be used with caution, especially when the tunnel is quite flexible and high soil non-linearity is expected, as in the case of strong earthquakes. The calibrated dynamic numerical models were finally used as a benchmark to validate the accuracy of currently used simplified methods. Racking ratios computed from the equivalent static analyses were found to be slightly lower compared to the dynamic analysis results, while the NCHPR611 analytical relation (Anderson et al., 2008) was found to overestimate the racking ratio for the flexible tunnel case. In general, simplified methods underestimated the tunnel lining forces compared to the full dynamic analysis. Assuming an elastic soil response, the differences were up to 30%, and the discrepancies were much higher for the cases when the soil permanent deformation was accounted for. Equivalent static analyses, where the load is introduced in terms of distributed inertial loads throughout the model, were found to be more efficient. The main conclusion drawn is that simplified methods should be used with caution, mainly during preliminary stages of design, and for cases where high soil non-linearity is not expected (e.g. rather low to medium seismic intensities).

analysis results (Pitilakis & Tsinidis, 2014). It is worth noting that Hashash et al. (2010) propose this distance to be significantly smaller. Soil–tunnel interface properties and input motion characteristics seem to have a minor effect on the computed ratios in case of the elastic analyses, whereas these parameters become more important in the case of the elasto-plastic analyses (especially in the case of the flexible tunnel), owing to their effect on the soil yielding response. CONCLUSIONS The paper presented representative experimental results from a series of dynamic centrifuge tests on a flexible model tunnel embedded in dry sand, along with results from numerical simulations of the tests. Numerical models were found capable of reproducing the recorded response with reasonable engineering accuracy. Some inevitable differences between the recorded and the computed response are attributed to the difficulties in ascertaining precisely the soil, tunnel and soil–tunnel interface mechanical properties of the centrifuge model. To a certain degree this also depends on the constitutive models used; however, these models are adequately calibrated. All constitutive models actually constitute an approximation of the actual sand behaviour under seismic loading. Their accuracy depends on numerous parameters, which are mainly affected by the typology and the complexity of the problem modelled. Sometimes modelling very complex problems, such as the one in this paper, using complicated constitutive models for the soil, which could not be well calibrated, may increase considerably the uncertainties and reduce the accuracy of the results. The use of the models implemented herein and the comparisons to the experimental data are an additional verification of their efficiency to model complicated problems such as the one presented in this paper. With regard to the tunnel’s response: both the experimental and the numerical data revealed a rocking mode of vibration for the tunnel in addition to the racking distortion. Inward deformations were also observed due to the high flexibility of the tunnel. Post-earthquake residual values were recorded experimentally and predicted numerically for the

ACKNOWLEDGEMENTS The research leading to the presented experimental results has received funding from the European Community’s Seventh Framework Programme (FP7/2007–2013) for access to the Turner beam centrifuge, Cambridge, UK, under grant agreement no. 227887 (SERIES – Seismic Engineering Research Infrastructures for European Synergies, http:// www.series.upatras.gr/). The excellent technical support received by the technicians at the Schofield Centre is gratefully acknowledged.

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DYNAMIC RESPONSE OF FLEXIBLE SQUARE TUNNELS NOTATION A a aff B c D d10 d50 d60 E Ec e emax emin F Finertia f fbk G Gmax H K0 l M Mdynamic Mstatic m N n P R RC S t TX Vso Æ ª ªt ˜ff ˜str |˜M| |Ñ| � � � rs � �9 � � �cv ł

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acceleration amplitude input motion amplitude soil free-field horizontal acceleration tunnel width cohesion damping of sand sand grain diameter at 10% passing sand grain diameter at 50% passing sand grain diameter at 60% passing aluminium alloy Young’s modulus concrete Young’s modulus sand void ratio maximum sand void ratio minimum sand void ratio soil to tunnel flexibility ratio equivalent to acceleration inertial load input motion dominant frequency aluminium alloy tensile strength sand reduced shear modulus sand small-strain shear modulus tunnel height earth coefficient at rest length lining bending moment per unit length lining bending moment evaluated through dynamic analysis lining bending moment evaluated through equivalent static analysis mass lining axial load per unit length scale factor equivalent to tunnel racking distortion force racking ratio resonant column tests required force to cause a unit racking deflection of the tunnel time cyclical triaxial tests small-strain shear velocity gradient of sand reduction coefficient for sand shear modulus during shaking shear strain aluminium alloy unit weight free-field ground racking distortion tunnel racking distortion lining bending moment dynamic increment lining axial force dynamic increment horizontal deformation at soil surface soil–tunnel interface friction coefficient aluminium alloy Poisson ratio sand density dynamic earth pressure per unit length mean effective stress dynamic shear stress per unit length sand friction angle sand critical friction angle sand dilatancy angle

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DYNAMIC RESPONSE OF FLEXIBLE SQUARE TUNNELS Earthquake Engineering, vol. 26, pp. 389–406, http://dx.doi.org/ 10.1007/978-3-319-00458-7_22. Geneva, Switzerland: Springer. Tsinidis, G., Pitilakis, K. & Trikalioti, A. D. (2014b). Numerical simulation of round robin numerical test on tunnels using a simplified kinematic hardening model. Acta Geotechnica 9, No. 4, 641–659, http://dx.doi.org/10.1007/s11440-013-0293-9. Wang, J. N. (1993). Seismic design of tunnels: A simple state of the art design approach. New York, NY, USA: Parsons Brinckerhoff. Wang, W. L., Wang, T. T., Su, J. J., Lin, C. H., Seng, C. R. & Huang, T. H. (2001). Assessment of damage in mountain tunnels

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Bertalot, D. & Brennan, A. J. (2015). Geótechnique 65, No. 5, 418–428 [http://dx.doi.org/10.1680/geot.SIP.15.P.002]

Influence of initial stress distribution on liquefaction-induced settlement of shallow foundations D. B E RTA L OT a n d A . J. B R E N NA N †

During earthquakes, saturated sandy soils may generate significant excess pore pressures and approach a state of liquefaction. Structures founded on shallow foundations above such soils may consequently undergo large settlements. Recent case history analysis has shown that the stress imposed by the foundation is a key factor in the estimation of such settlements. However, the case history data showed that although increasing bearing pressure caused an increase in settlements as expected, this was only true up to a point, and that very heavy structures appeared to settle less than some lighter structures. This work aims to investigate these counter-intuitive results by means of controlled experimental testing using a geotechnical centrifuge. Results of the centrifuge tests show that the trend derived from case histories is correct and that liquefaction-induced settlements peak for a given bearing stress (90 kPa for the models tested) and reduce for greater applied stresses. Further, by analysis of excess pore pressure distributions beneath the foundations it is shown that the main factor inhibiting pore pressure generation beneath the footings is not so much the confining pressure as the in-situ static shear stress around the edge of the foundation. This is supported by element test data from the literature. When this initial static shear stress is so high that the applied cyclic shear stress cannot exceed it (i.e. the direction of shear stress does not reverse) then pore pressure generation is greatly reduced, thus causing the observed reduction in expected settlements. KEYWORDS: centrifuge modelling; footings/foundations; liquefaction

motions and interpretation, meaning that further understanding would be better derived from controlled model testing. Most of the current understanding of the mechanics governing soil liquefaction has been derived by laboratory element testing (mainly cyclic triaxial and cyclic simple shear tests), leading to the recognition of three main initial state variables controlling the cyclic resistance of liquefiable soils: relative density, confining pressure and static shear stress. The effect of static shear stress in particular has been the object of debate over the last 40 years, and as the stress fields beneath shallow foundations contain such static shear stresses, an understanding of this is required to capture the behaviour of such systems. Initially, cyclic triaxial tests on isotropic consolidated samples or cyclic simple shear tests on one-dimensionally consolidated samples were commonly used to assess the behaviour of saturated sands. However, despite satisfactorily reproducing the behaviour of level saturated soil deposits (where static shear stress is zero), these techniques could not capture the soil behaviour in sloping ground or underneath a footing, where significant static shear stresses are induced (Vaid & Finn, 1979). Since then, many authors have recognised that static shear stresses may, under given conditions, increase the soil’s resistance to liquefaction by limiting the generation of excess pore pressure during cyclic loading (Lee & Seed, 1967; Vaid & Finn, 1979; Boulanger & Seed, 1995; Vaid et al., 2001; Kammerer, 2002). The aim of the current study is therefore to present an explanation of the apparently counter-intuitive relationship between settlement and footing bearing stress by means of a series of dynamic centrifuge tests. This will be achieved by first validating measured settlements against the case study data, then looking more closely at developed excess pore pressures, and consequently determining whether existing knowledge about static shear stresses can explain the observed pore pressures and settlement trends.

INTRODUCTION During earthquakes, an increase in pore pressure may occur in saturated sandy/silty soils as a result of the soil structure collapse and consequent transfer of load to the pore fluid. This phenomenon results in a reduction of the effective stresses acting on the soil and a consequent degradation of the soil’s shear stiffness and strength, facilitating settlement of structures having shallow foundations. If the generated excess pore pressure is high enough to induce full liquefaction underneath the footing (i.e. equals the effective overburden stress), very large deformations may take place, with one resultant phenomenon being the excessive settlement of structures with shallow foundations. A number of examples of this exist through the literature. Data from this earthquake helped to identify an omission from the existing methods of assessing such settlements. Current practice is based primarily on the variables of footing width B and the depth of liquefiable soil DL (Liu & Dobry, 1997). Analysis of both new and previously published case histories by Bertalot et al. (2013) led to the identification of footing bearing pressure q as a significant additional variable. Further, it was also identified from this analysis that, although observed settlements increased with q, this was only true for lower imposed stresses and that structures imposing stresses of the order of 100 kPa and above on the soil appeared to undergo comparatively reduced settlements. However, case history data contain significant variability in local site factors, Manuscript received 27 March 2014; revised manuscript accepted 9 January 2015. Published online ahead of print 7 April 2015. Discussion on this paper closes on 1 October 2015, for further details see p. ii. D’Appolonia S.p.A., Genova, Italy; also Civil Engineering Department, University of Dundee, UK. † Civil Engineering Department, University of Dundee, UK.

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BERTALOT AND BRENNAN action and resultant ratcheting of the rigid footing into the soil, footings have identical low heights and bearing pressures achieved by controlling material density (aluminium, steel and lead). All of the footing models tested are placed on the ground surface, with an embedment of 0 .5 m. All three models consist of 15 m deep deposits of clean sand with a top loose liquefiable layer (DR ¼ 40%) and a bottom dense layer (DR ¼ 80 %). The thickness of the liquefiable layer varies in each model as the analysed tests are part of a broader testing campaign whose goal is, among others, the investigation of the role of DL on footing settlement. The presence of the bottom dense sand layers is necessary to achieve the same overall depth of the soil deposit in the three centrifuge models. The effect of these layers was seen to have limited effects both on the overall vertical strain of the soil deposit and on the vertical propagation of the base motion. In test BD8 the top liquefiable sand layer has a depth of 8 .25 m in prototype scale, whereas in test BD5 the liquefiable layer depth is reduced to 4 m measured from the foundation plan. In test BD7 the container was divided into different sectors by means of thin aluminium walls to achieve different stratification profiles

CENTRIFUGE MODELLING To achieve this aim, three centrifuge tests have been performed on the University of Dundee geotechnical centrifuge. Centrifuge modelling enables small-scale models to be tested at stress levels equivalent to a larger prototype. In this case, a length scale factor of 50 has been adopted, and a corresponding centrifugal acceleration of 50g applied. All data are here reported in prototype scales; a full discussion of scaling may be found in, for example, Muir Wood (2004) or Schofield (1980). This is a 3 m diameter Actidyn C-67-2 centrifuge, equipped with the Actidyn QS-67-2 onedimensional servo-hydraulic shaker capable of applying userdefined motions up to 0 .4g within the range of 0 .6–8 Hz prototype scale at the 50g acceleration considered (Brennan et al., 2014). Each centrifuge model tested consists of four rigid square footings (B ¼ 2 .75 m at prototype scale) resting on level loose liquefiable sand (DR ¼ 40%) deposits of different thickness. Despite having the same dimensions, each model footing exerts a different bearing pressure, namely 30, 60, 90 and 130 kPa (Figure 1). To minimise the component of the settlement due to moment-induced soil–structure inter-

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INITIAL STRESS DISTRIBUTION AND SETTLEMENT OF SHALLOW FOUNDATIONS within the same model. Thin aluminium walls (i.e. 0 .3 mm Table 1. Properies of HST95 Congleton sand thick) were used for this purpose in order to create a low Property Measured Lauder (2010) lateral stiffness impermeable boundary between the different compartments, able to stop pore fluid migration without Gs: g/cm3 – 2 .63 imposing constraints to the soil model. Because of the 3 .34 .59 ª : kN/m 14 14 d,min limited bending stiffness of the walls, lateral deformations at ªd,max: kN/m3 17 .6 17 .58 these boundaries are assumed to be controlled by the soil. In �crit: degrees 33 32 particular, liquefiable layers of 6 and 10 m have been �peak: degrees� 46 44 reproduced in this test. � Indicates a property evaluated for soil at D ¼ 90%. In order to compare the results from different tests, R normalised settlements (S/DL) are considered. This normalisation has been previously used to compare field cases of Soil models are contained in an equivalent shear beam liquefaction-induced foundation settlement (Yoshimi & Tokibox, consisting of stacked aluminium rings separated by thin matsu, 1977; Liu & Dobry, 1997). Dashti et al. (2010) and rubber layers providing the desired flexibility (Bertalot Bertalot et al. (2013) pointed out that the assumption of a (2012), and shown schematically in Fig. 1). The container is linear relationship between the induced settlement and liquedesigned so that the shear stiffness of the end walls matches fied soil thickness may be misleading for small B/DL ratios. that of the soil model before soil softening due to shakingThis limits the value of quantitative comparison of normalinduced excess pore pressure build-up, thereby minimising ised footing settlement in different tests. dynamic boundary effects. Upon liquefaction the soil stiffBesides footing settlement, pore pressure and acceleration ness drops significantly and this condition is no longer met; are measured within the soil. In each model the pore however, such stiffness reduction also inhibits the propagapressure measurements were concentrated underneath a diftion of parasitic waves generating at the model boundaries ferent footing, providing detailed information about the due to the marked stiffness contrast. excess pore pressure generation pattern in an area with width The input motion chosen was that experienced at the B from the footing axis, and depth B from the foundation primary case study site during the 2010 Maule earthquake. plan. A schematic layout of these tests is shown in Fig. 1. The initial reason is to better compare data obtained against The soil used for these models was HST95 Congleton that from the case study site (Brennan et al., 2014). This silica sand; this is a fine, uniformly graded sand with motion was also seen as useful due to its long duration and rounded particle shape (Table 1). In order to achieve a large magnitude (Mw ¼ 8 .8), meaning that settlements correct scaling of time in both inertial and seepage controlled phenomena, a pore fluid with a viscosity 50 times achieved are likely to be an upper bound on those experihigher than that of water was required (Stewart et al., 1998). enced in lower magnitude events, while also being less This was achieved by using a solution of methylcellulose in damaging than the sinusoidal shaking applied in older water as pore fluid, at a concentration of 2 .3% of methylcelcentrifuge based investigations. Fig. 2 shows the target lulose by mass. motion record used in both time and frequency domain, Test BD3

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BERTALOT AND BRENNAN EXCESS PORE PRESSURE GENERATION The effective confining stress acting on the soil has been observed to influence its potential for excess pore pressure (Epp) generation. Steedman et al. (2000) performed a series of centrifuge tests investigating the cyclic behaviour of saturated sands under high confining stress (� v9 ) by testing deep level soil models with a uniform, freely draining surcharge. A reduction of the measured excess pore pressure ratio (ru) with depth (i.e. with increasing � v9 ) was observed. Similar conditions are present beneath a footing resting on a liquefiable sand deposit. However, in this case the confining effect exerted by the footing is localised to the soil underneath it, generating a more complex stress distribution. Beside the horizontal dis-homogeneity of confining stress, shallow foundations also generate shear stresses in the foundation soil, significantly influencing its cyclic response. In order to investigate the excess pore pressure generation in such conditions, in each of the centrifuge tests performed the soil underneath one of the footing models (test BD8– 60 kPa footing; test BD7–90 kPa footing; test BD5–130 kPa footing) has been instrumented with an array of pore pressure transducers (PPTs). As instruments located at shallow depth underneath a footing are displaced proportionally to footing settlement during testing, and excess pore pressure requires reference to a measurement location, the pore pressure readings have been corrected for instrument displacement. The positions of the instruments prior to and after liquefaction were carefully measured. Correction of the excess pore pressure measurement (Epp) was then carried out by subtracting the hydrostatic surplus due to the instrument being pushed deeper into the foundation soil

together with the motion recorded on the centrifuge models, indicating a very good match between target motion and achieved motion, as well as good repeatability. SETTLEMENTS The initial objective of this work was to validate the trend observed in field data concerning the liquefaction-induced settlement of buildings having shallow foundations. Observed settlements of such buildings from the Chile earthquake, as well as past events in Niigata (Yoshimi & Tokimatsu, 1977) and Luzon in the Philippines (Adachi et al., 1992; Acacio et al., 2001), are plotted as a function of bearing pressure in Fig. 3. The analysis of the documented case histories of this specific phenomenon suggests that high foundation bearing pressure may result in reduced settlement in the case of liquefaction of the foundation soil (Bertalot et al., 2013). However, it is seen that there are limited field data for such heavy structures, necessitating the confirmation of this with the centrifuge models. Figure 3 therefore also includes the total normalised settlements recorded during tests BD5, BD7 and BD8 plotted against the footing model bearing pressure. The hypothesised bearing pressure dependence of the liquefaction-induced settlement is confirmed by the experimental results. Footing settlement was indeed seen to be directly proportional to the footing’s bearing pressure for bearing pressure up to about 100 kPa, as had been expected based on the field data. However, the heaviest (130 kPa) footing also experiences less settlement than the lighter ones. Data from similar experimental works from Hausler (2002) and Dashti et al. (2010) have also been included in Fig. 3 for comparison. In particular, Dashti et al. tested two types of footing models characterised by different bearing pressure (namely 76 and 120 kPa). In line with what was observed in this study, increasing the footing bearing pressure from 76 to 120 kPa did not result in further footing settlement in Dashti et al.’s experiment. Evidence from centrifuge test results implies that the trend identified by Bertalot et al. (2013) is due to a real physical phenomenon rather than arising by inconsistencies and omissions in the field data set. The remainder of this work therefore attempts to explain this trend by investigating the pore pressures beneath the footings.

Epp(t) ¼ Epp0 (t) � (z � z0 )[S av (t)=S av,final ]ªw where Epp0 represents the uncorrected measurement, z and z0 the final and initial instrument positions, respectively (below foundation plan), Sav(t ) and Sav,final are the settlement at time t and the final settlement, respectively, and ªw is the unit weight of water. It is noted that instruments did not displace laterally during testing. Figure 4 shows the instrument initial positions and the excess pore pressure measurements during shaking from the centrifuge tests. It is acknowledged that the thickness of the liquefiable layer was different for different models.

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Fig. 4. During shaking Epp measurements in the soil underneath: (a) 130 kPa footing; (b) 90 kPa footing; (c) 60 kPa footing

Comparing the settlement–, acceleration– and pore pressure– time histories during shaking relative to the 130 kPa footing and the underlying soil, four main phases can be identified (Figure 5).

Although this is more likely to affect the long-term pore pressure dissipation rather than the short-term pore pressure generation, there may still be variation between tests, preventing direct comparison of quantitative values. However, inspecting the distribution of pore pressures during shaking in each case is still instructive. The excess pore pressure traces in Fig. 4 show that, for the heavier footings, the lowest excess pore pressures are recorded not directly underneath the footing, but below the footing’s edge. Significant positive excess pore pressures were initially generated beneath the central axis of the 130 kPa footing, while small negative excess pore pressures were recorded below the footing edge and at a distance of B/2 outside the footing (Figure 4(a)). Because of the pressure gradient generated as a consequence of the dis-homogeneous excess pore pressure, cross-drainage occurred during cyclic loading between different areas of the foundation soil. All of the excess pore pressures recorded show their maximum absolute value after 15 to 20 s from the start of earthquake shaking; during this time interval a maximum pressure difference of approximately 55 kPa exists between PPT5 (footing axis) and PPT7 (B/2 metres outside footing). As a consequence of this gradient, drainage occurs from the area below the centre of the footing towards the soil below the footing edge, leading to an equalisation of excess pore pressure in the foundation soil (after 30 s for the 130 kPa footing in Fig. 4(a)). Similar behaviour is observed underneath the 90 kPa footing; however, in this case the pressure gradients generated between the axis area and the edge area are smaller. Negative excess pore pressures were recorded only below the footing edge (PPT6), determining a less marked post-peak drainage effect (Figure 4(b)). For the lighter, 60 kPa footing, this effect seems to have been suppressed and only small differences in measured excess pore pressures are observed. In particular no negative excess pore pressures were observed below the footing edge, resulting in a fairly homogeneous excess pore pressure response across the foundation soil (Figure 4(c)).

(a) Significant excess pore pressures are generated underneath the central axis of the footing, exceeding the vertical effective stress in the free-field (,12 kPa at z ¼ 0 .75 m below foundation plan). On the contrary, zero or negative excess pore pressures are generated in the proximity of the footing edges. Footing settlement starts as soon as excess pore pressures are generated in the foundation soil and reaches its maximum rate during this phase. (b) Significant pressure gradient exists between the axis and the edge of the footing. Cross-drainage towards the area underneath the footing edge, and possibly dilative behaviour in the foundation soil caused by the settling footing, act to reduce the excess pore pressure under the centre of the footing. During this phase the seismic loading on the soil reduces because of excess-porepressure-induced soil softening. During this phase, the rate of footing settlement also reduces. (c) Following cross-drainage, a more homogeneous excess pore pressure distribution in the foundation soil has been reached. However, a steady and relatively slow increase in the pore pressure is observed underneath the footing. A possible explanation for this is the vertical dissipation of excess pore pressure from the underlying soil. It is hypothesised that, in the short term, fluid would migrate toward areas characterised by higher effective stresses, which can accommodate higher pore pressure with respect to the free field. Footing settlement rate further reduces during this phase, reaching its final equilibrium. (d ) This phase corresponds to the post-shaking behaviour. The drainage toward the foundation soil observed in phase (c) does not stop at the end of shaking, but rather a

93


BERTALOT AND BRENNAN PPT2 – footing axis (z 3 m) PPT3 – footing edge (z 3 m)

Excess pore pressure: kPa

50

PPT4 – footing axis (z 1·875 m) PPT5 – footing axis (z 0·75 m)

40

PPT5 – footing edge (z 0·75 m) PPT5 – near footing (z 3 m)

30 20 (b) 10 0 (c)

(a)

(d)

10 0

S: m

Footing D 0·5 1·0

a: g

1·5 0·2

ACC11 (Footing axis; z 1·875 m)

0 0·2 0

10

20

30

40

50

60 Time: s

70

80

90

100

110

120

Fig. 5. During shaking Epp, settlement and horizontal acceleration measurements in the soil underneath the 130 kPa footing in test BD5

& Finn (1979) and Vaid & Chern (1983, 1985), who showed the effect of static shear to be strongly dependent on the initial confining stress and soil density. The results of these studies show that the initial state variables considered (static shear, confining stress, relative density) control the cyclic behaviour of saturated sands, and that their effects are mutually dependent. The soil relative density in particular will determine the mechanism of strain development, strain softening taking place in loose and strongly contractive soils and ‘cyclic mobility’ (or limited strain liquefaction) in denser soils (Castro, 1975). Again, the threshold density between these two mechanisms, other than on the soil type, depends on the confining stress and static shear acting on the soil. The addition of an initial shear stress in loose contractive materials would therefore lower its resistance to liquefaction as it moves it closer to the failure envelope. However, if such initial shear stress (�i) is higher than the cyclic shear stresses (�cyc) applied, than no stress reversal takes place in the soil, significantly increasing its cyclic resistance (Vaid & Finn, 1979; Boulanger & Seed, 1995; Kammerer, 2002). In order to adapt the empirical liquefaction triggering curves used in current practice, which are based on level ground conditions case histories, to sloping ground conditions (i.e. presence of static shear), Seed (1983) first proposed a correction factor KÆ dependent on the static shear. Several sets of curves relating KÆ values to static shear stress ratio (Æ ¼ �i/� v9 ) have subsequently been published (Seed & Harder, 1990; Harder & Boulanger, 1997). Similarly, an independent factor (K�) is used in current practice to account for the influence of confining stress on the soil liquefaction resistance. Based on the collective dependency of cyclic resistance on all initial state variables, Vaid et al. (2001) question the use of independent factors in order to account for confining stress and static shear. Results of a series of cyclic triaxial tests on Fraser River sand, presented by the authors, show that the empirical method

slight increase in its rate is observed, possibly due to the ceasing of the cyclic dilative response associated with the co-seismic soil behaviour. Pore pressures in the free field seem unaffected from phase (c). Footing settlement relative to the soil ceases with the end of shaking; however, further vertical footing displacement occurs in the longer term due to the dissipation of excess pore pressure associated with the post-liquefaction reconsolidation of sands (i.e. volumetric settlement). These data therefore show that the main cause for limiting excess pore pressure generation is associated with the soil beneath the edge of the footing. As this is where initial static shear stresses are at a maximum, and as static shear stresses have previously been identified with increased cyclic resistance under certain conditions, their potential correlation with these results is examined further in the next section. This mechanism has not been previously identified in similar works (Whitman & Lambe, 1982; Liu & Dobry, 1997; Hausler, 2002; Dashti et al., 2010) as the instrument distribution was not targeted to pick up variations in pore pressure within the space beneath the foundations. INFLUENCE OF INITIAL SHEAR STRESS The effect of static shear stress on the cyclic resistance to liquefaction has been investigated by several authors since the late 1960s. Most of the experimental work on this topic is based on soil element testing, in particular cyclic triaxial testing. Lee & Seed (1967) first hypothesised that the presence of static shear increases the soil’s cyclic resistance to liquefaction by performing a series of cyclic triaxial tests on anisotropically consolidated samples. Subsequently, Castro (1969, 1975) and Castro & Poulos (1977) verified that the presence of static shear may result in a reduction of the cyclic resistance of sand. These apparently contradictory results were unified in a more complex framework by Vaid

94


INITIAL STRESS DISTRIBUTION AND SETTLEMENT OF SHALLOW FOUNDATIONS The mechanical parameters required by the model are shown in Table 2. The computed stress distribution describes the static case before earthquake shaking, and is shown in Fig. 7 (to the right of the axis of each footing in the figure). During shaking the soil–structure interaction may result in a different, time-dependent, distribution of such stresses, reflecting the foundation tendency to rock and translate horizontally, and thus cyclically extending both laterally and downward. Therefore, the computed stress distribution has to be considered only indicative. The typical Æ distribution induced by a shallow foundation consists of ‘bulbs’ of high Æ beneath the footing edges, the extension of such bulbs being proportional to the foundation bearing pressure. Also plotted on Fig. 7 (to the left of the axis of each footing in the figure) are contours of the peak excess pore pressure before drainage, measured in the foundation soil below the 60, 90 and 130 kPa footings, respectively. Contours are obtained by interpolating between data points using a linear interpolation scheme. Contours are plotted such that darker colours represent lower pore pressures. Comparison between excess pore pressures and static shear stresses in Fig. 7 shows that the observed reduction in the generated excess pore pressure below the footing edge corresponds, as suggested above, with areas characterised by high values of Æ. In particular, the minimum excess pore pressures were recorded in portions of the foundation soil characterised by Æ . 0 .4 (Figs 7(a) and 7(b)). The heavier footing (q ¼ 130 kPa) generates significant shear stresses in a broad portion of the foundation soil. The ‘bulb’ of soil characterised by a value of Æ higher than 0 .4 extends to a depth of , B (footing width) and has a maximum width of , B/2 (Fig. 7(a)). Pore pressure measurements in correspondence with this area show no generated excess pore pressures. On the contrary, small negative peak excess pore pressures have been recorded by PPT6 (footing edge) and PPT7 (near footing), possibly due to dilation induced by the footing settlement. Similar observations have been made for the 90 kPa footing; in this case, only records from PPT6 (footing edge) show near-zero excess pore pressure, while significant excess pore pressure was measured by PPT7, in line with those recorded at the same depth in correspondence with the footing axis (Fig. 4). A possible explanation for these observations may be that the bulb of soil with Æ . 0 .4 induced by the 90 kPa footing is smaller; as a consequence the soil near PPT7 is subjected to lower shear stresses, resulting in lower cyclic resistance (Fig. 7(b)). Excess pore pressures generated at a depth of 0 .75 m underneath the lighter footing analysed (q ¼ 60 kPa) show little or no horizontal variability from the footing axis to a distance of B/2 from the footing edge (Fig. 7(c)). In particular, the excess pore pressure generation pattern observed is similar to what is observed in the free-field case (i.e. �i ¼ 0), suggesting a reduced influence of the footinginduced stresses for lower bearing pressure. In all cases the highest excess pore pressure values were recorded beneath the footing axis, where the shear stresses are a minimum. Despite full liquefaction (i.e. ru ¼ 100 %) being observed in the free field in all of the models tested,

currently used underestimates the cyclic resistance, the degree of conservatism being more dramatic for loose soils (Fig. 6). Effect of initial shear stress in the centrifuge tests Despite significant differences between the loading and boundary conditions of an element test and those of an element of soil in a centrifuge model (or in the field), these findings can still be used to inform interpretation of the soil behaviour observed in the centrifuge tests. One of the main differences in soil behaviour is that element tests are usually performed in fully undrained conditions, while results show that partially drained conditions apply to centrifuge tests (Madabhushi & Haigh, 2012; Lakeland et al., 2014). Stress distribution in the two cases may also vary significantly. In particular, in cyclic triaxial tests the soil is subjected to different admissible soil deformations and the total stress is kept constant, while in centrifuge tests it may vary during cyclic loading. The effect of static shear stresses on the cyclic response of the soil depends on the ranges of state variables represented in the centrifuge tests (� v9 varying between approximately 30 and 160 kPa between foundation plan and a depth equal to B metres below foundation plan; and DR ¼ 40%). According to the KÆ values proposed by Seed & Harder (1990), the presence of the footing-induced static shear stress within these ranges of state variables should result in a reduction of the foundation soil’s cyclic resistance (Fig. 6). On the contrary, in the centrifuge tests performed, reduced excess pore pressure generation was observed in soil associated with high static shear stresses (i.e. increased cyclic resistance). This is in accordance with the KÆ curves proposed by Vaid et al. (2001) for relative density of 40% and a similar initial confining stress. Results from Vaid et al. (2001) may be considered more reliable as, unlike the Seed & Harder (1990) formulation, they account for the combined effect of confining stress and relative density. In order to identify the initial distribution of static shear stress present in the centrifuge model, the contours of Æ along a longitudinal section of the model have been computed based on a finite-element simulation performed using the software Plaxis2D. The built-in elastic perfectly-plastic Mohr–Coulomb material model was used for this purpose. 2·25 DR 55–70%

2·00 1·75 1·50 1·25 Kα

DR 45–50%

1·00 0·75 Vaid et al. (2001)

0·50

DR 25% DR 35% DR 40% DR 60%

0·25 0 0

Table 2. Plaxis 2D linear-elastic perfectly-plastic Mohr–Coulomb model parameters (HST95 Congleton sand at DR 40%)

DR 35% Note: Shaded areas from Seed & Harder (1990)

E: MPa ı ªd: kN/m3 �crit: degrees ł: degrees�

0·05 0·10 0·15 0·20 0·25 0·30 0·35 0·40 0·45 0·50 α τi /σv,0

Fig. 6. KÆ plotted against Æ for Fraser River sand at different density states and �9v of 100 kPa (Vaid et al., 2001)

� Measured at D ¼ 40%. R

95


25 0 .3 15 .49 33 1

BERTALOT AND BRENNAN Excess pore pressure: kPa 0

10

30

40

Footing D q 130 kPa

t 16 s 0


20

α τi /σv

1 0·5 2

0·4 0·3

3

0·1 0·2

4 0·1 5

5

4

3

2

1

1

3

2

4

5

Footing C q 90 kPa

t 18 s 0


0 (a)

1

0·5 0·4

2

0·3 0·2

3

0·1 4 α τi /σv 5

5

4

3

2

1

0 (b)

1

3

2

4

5

Footing B q 60 kPa

t 18 s 0


0·5 0·4

1

0·3 0·2

2

0·1

3

α τi /σv

4

5

5

4

3

0 2 1 1 2 Distance from foundation axis: m (c)

3

4

5

Fig. 7. Contours of peak Epp before drainage against contours of footinginduced static shear stress ratios: (a) 130 kPa footing; (b) 90 kPa footing; (c) 60 kPa footing

96


INITIAL STRESS DISTRIBUTION AND SETTLEMENT OF SHALLOW FOUNDATIONS pressure generation (Vaid & Finn, 1979; Boulanger & Seed, 1995; Kammerer, 2002). Shear stress reversal occurs when the oscillating earthquake-induced shear stress (�cyc) exceeds the static shear stress (�i), crossing a state of zero shear and thus reversing direction. However, the magnitude of �i and �cyc is not sufficient information to evaluate the degree of stress reversal, as the direction of such stresses has to be taken into consideration as well. Kammerer (2002) points out that, in the case of sloping ground, �i will be oriented parallel to the dip direction of the slope, while the direction of �cyc is variable depending on the ground motion. In this scenario, the potential for stress reversal is higher when the principal direction of seismic loading is parallel to the dip direction of a slope. However, the author observes that significant excess pore pressures are generated in the soil even when the direction of loading is parallel to the strike direction of the slope. This is ascribed to the correspondent complementary shear stresses acting perpendicularly to the loading direction. Unlike those generated in slopes, the direction of the maximum shear stresses induced by a footing in the foundation soil changes with depth. This increases the complexity in the evaluation of the effect of shear stress reversal on the excess pore pressure generation below shallow foundations. Figure 9 compares the estimated stress paths in two soil elements located at the same depth below the foundation plan (z ¼ 0 .75 m), the first corresponding to the central axis of the 130 kPa footing (PPT5) and the second beneath the footing edge (PPT6). Since no stress measurement is available at the selected locations, the proposed stress paths represent an estimate based on the following assumptions.

all the PPTs positioned in the soil below or in the proximity of the footings recorded peak excess pore pressure ratios (ru,max ¼ Eppmax/� v9 ) significantly lower than 100%. This is plotted in Fig. 8, which shows the ru,max recorded beneath the footings against the estimated initial shear stress ratio at the measurement location. Also in Fig. 8, the results from the centrifuge tests are compared to those from cyclic shear tests performed by Kammerer (2002) and Boulanger et al. (1991) for different values of Æ. For higher Æ, all three sets of data show a decrease of the recorded excess pore pressure ratio for increasing initial static shear stress ratio. However, a substantial difference exists between centrifuge and cyclic shear data for low values of initial static shear stress ratio, which is ascribed to the different boundary conditions. Theoretically, applying horizontal fixity and no-flow boundary conditions to the soil column below the footing, if liquefaction is triggered the entire weight of the footing would be transferred to the pore fluid, reaching a state of full liquefaction. These idealised boundary conditions are unrepresentative of reality where significant cross-drainage occurs during shaking, resulting in peak ru values lower than unity. A threshold value of Æ exists, above which excess pore pressure generation is impeded. Such a threshold value is not unique but depends, other than soil type, on relative density and confining stress. For the range of confining stress and relative density considered in the models tested, Æ values higher than ,0 .4 showed zero or negative excess pore pressure generation. Results from a series of cyclic bi-directional simple shear tests performed by Kammerer (2002) on Monterey #0/30 sand show a similar threshold value (Fig. 8). These observations suggest that, in all of the analysed cases, the foundation soil underwent excess pore pressureinduced softening without reaching a condition of full liquefaction (i.e. ru ¼ 100%). Moreover, in the presence of significant static shear stress, areas of non-softened soil may initially exist below the edges of the footing during earthquake shaking, even in the case where the motion is strong enough to trigger liquefaction in the free field.

(a) The initial static shear stress (from static finite-element simulation) is assumed to be maintained during the entire earthquake. (b) Total vertical stress variations in the soil during earthquake shaking (˜�v) are accounted for based on the measured vertical footing accelerations. A vertical ‘piston-like’ footing movement is assumed and total vertical stress variations in the soil calculated according to Newton’s law and Boussinesq theory. (c) The static (�i) and cyclic (�cyc) shear stresses are assumed to act on the same plane (condition yielding maximum expected amount of shear stress reversal).

Maximum excess pore pressure ratio, ru,max: %

Effect of shear stress reversal As mentioned above, many authors have stressed the influence of shear stress reversal on cyclic excess pore

The proposed linear failure envelope (Fig. 9) corresponds to a friction angle of ,328 (Lauder, 2010), yielding a critical Æ value of 0 .61. Under these premises, the stresses in a soil element at the selected locations can be estimated according to

Kammerer (2002) Boulanger et al. (1991) 130 kPa footing (centrifuge test BD5) 90 kPa footing (centrifuge test BD7) 60 kPa footing (centrifuge test BD8)

100

˜� v (t) ¼ Bq ˜q(t) ¼ Bq

80

[M f av (t)] Af

60

� v9 (t) ¼ � v9,0 (t) þ ˜� v (t) � Epp

40

�cyc (t) ¼ ah (t) rd [� v9 (t) þ Epp(t)] �(t) ¼ �i þ �cyc (t)

20 0

where Bq represents the ratio ˜�v /˜q evaluated at the selected locations according to Boussinesq theory, ˜q is the variation in footing bearing pressure due to the footing vertical acceleration, av. ah is the horizontal acceleration measured in the ground at 0 .75 m below the foundation plane and rd is a stress reduction factor evaluated according to the formulation of Idriss & Boulanger (2004). According to the computed stress paths, no stress reversal occurs below the footing edge, whereas beneath the footing axis shear stress reverses direction in most of the loading cycles. It is also worthwhile to notice that the magnitude of the cyclic shear stress imposed to the foundation soil depends on the acting

20 40 60 80

0

0·05 0·10 0·15 0·20 0·25 0·30 0·35 0·40 0·45 0·50 Normalised initial shear stress, α τi /σv

Fig. 8. Æ values at Epp measurement point against ru,max together with data from bi-directional direct shear tests from Kammerer (2002) and Boulanger et al. (1991)

97


BERTALOT AND BRENNAN 60 50 40

PPT5 footing axis (z 0·75 m)

30 20

τ: kPa

10 0 10

PPT6 footing edge (z 0·75 m)

20 30 40

Pre-earthquake stresses

50

Stresses after cross-drainage (t 30 s)

Stresses at peak Epp (t 16 s) Failure envelope 60 50

60

70

80

90

100 110 σv : kPa

120

130

140

150

160

Fig. 9. Approximate stress paths in the –9v plane, occurring during shaking beneath the footing’s central axis (PPT5) and below the footing edge (PP6) (see Fig. 5(a))

and below the footing edge suggests that the absence of shear stress reversal when initial shear stress is greater than cyclic shear stress (verified if i . cyc) may provide an explanation to the limited excess pore pressure generation observed below the edges of the footing. These results clarify the hypothesis derived previously from case study data that, although increasing the bearing pressure of shallow foundations on liquefiable soil causes an increase in likely settlement, this is only true up to a certain stress level (here ,100 kPa), above which settlements may no longer increase and may even reduce. It appears from the testing performed that this is attributable to the initial shear stresses in the soil inhibiting pore pressure generation, apparently through a lack of shear stress reversal. Moreover, the soil tendency to dilate at high stress ratio may also contribute to inhibiting pore pressure generation. The generality of the observed behaviour would be further tested by consideration of very wide footings, where the areas of high initial shear stress are proportionally less than the more uniformly loaded soil beneath the footing, and also by consideration of a wider range of input motions to investigate the relationship between i and cyc, and whether this affects the bearing pressure at which peak settlement occurs.

vertical effective stress; therefore it is higher below the centre of the footing where v9 is maximum. High i brings the soil’s stress state closer to failure, but at the same time reduces its potential for softening as a consequence of excess pore pressure generation. The stress path corresponding to PPT6, despite starting closer to the failure envelope than PPT5, shows no initial softening as excess pore pressure generation is impeded. Instead a slight hardening behaviour is initially observed, possibly due to dilation occurring in the soil and consequent development of negative excess pore pressure, increasing the v9 : However, significant softening occurs following cross-drainage, bringing the soil state closer to failure (Fig. 9). On the contrary, PPT5 shows significant softening taking place from the first cycles, reducing the acting v9 of about 35%. Cross-drainage in this case takes place from the footing axis toward the areas below the edge of the footing, resulting in a reduction in excess pore pressure and hence in a regain of v9 (Fig. 8). CONCLUSIONS Excess pore pressure measurements from a series of three centrifuge tests showed that, in the presence of a footing resting directly on liquefiable soil, limited or no softening occurred in the soil below the footing edge during earthquake shaking. In particular, these observations were seen to be valid for footings exerting a relatively high bearing pressure (i.e. . ,100 kPa) on the soil. Areas of the foundation soil which experienced least pore pressure increase were seen to correspond with those where the footing-induced initial shear stresses i are a maximum. Partially drained loading conditions were observed in all of the centrifuge tests performed, with significant drainage taking place during shaking to equalise the initial inhomogeneous excess pore pressure generation. The comparison of the approximate stress paths between soil beneath the footing’s central axis

NOTATION

ah horizontal acceleration measured in ground at 0 .75 m below foundation plane amax peak ground acceleration (g) av footing vertical acceleration B building width (m) Bq ratio ˜v /˜q evaluated at selected locations according to Boussinesq theory DL liquefied soil thickness (m) DR relative density (%)

98


INITIAL STRESS DISTRIBUTION AND SETTLEMENT OF SHALLOW FOUNDATIONS E Epp Eppmax Epp0 KÆ K� q rd ru ru,max Sav Sav(t ) Sav,final � z z0 Æ ªd ªw ˜q � v9 � v9,0 �cyc �i �xz �crit �P �R ł

ings with shallow foundation on liquefiable soil. J. Geotech. Geoenviron. Engng, ASCE 136, No. 1, 151–164. Harder, L. F. Jr & Boulanger, R. W. (1997). Application of K� and KÆ correction factors. Proceedings of the NCEER workshop on evaluation of liquefaction resistance of soils, report NCEER-970022, pp. 167-190. Buffalo, NY, USA: National Center for Earthquake Engineering Research, State University of New York. Hausler, E. A. (2002). Influence of ground improvement on settlement and liquefaction: a study based on field case history evidence and dynamic geotechnical centrifuge tests. PhD thesis, University of California, Berkeley, CA, USA. Idriss, I. M. & Boulanger, R. W. (2004). Semi empirical procedures for evaluating liquefaction potential during earthquakes. In Proceedings of the 11th international conference on soil dynamics and earthquake engineering, and 3rd international conference on earthquake geotechnical engineering (eds D. Doolin, A. Kammerer, T. Nogami, R. B. Seed and I. Towhata), pp. 33– 56. Singapore: Stallion Press. Kammerer, A. M. (2002). Undrained response of Monterrey 0/30 sand under multidirectional cyclic simple shear loading conditions. PhD thesis, University of California, Berkeley, CA, USA. Lakeland, D. L., Rechenmacher, A. & Ghanem, R. (2014). Towards a complete model of soil liquefaction: the importance of fluid flow and grain motion. Proc. A, R. Soc. 470, No. 2165, 20130453, http://dx.doi.org/10 .1098/rspa.2013 .0453. Lauder, K. D. (2010). The performance of pipeline ploughs. PhD thesis, University of Dundee, UK. Lee, K. L. & Seed, H. B. (1967). Dynamic strength of anisotropically consolidated sand. J. Soil Mech. Found. Div., ASCE 93, No. SM5, 169–190. Liu, L. & Dobry, R. (1997). Seismic response of shallow foundation on liquefiable sand. J. Geotech. Geoenviron. Engng, ASCE 123, No. 6, 557–567. Madabhushi, S. P. G. & Haigh, S. K. (2012). How well do we understand liquefaction. Indian Geotech. J. 42, No. 3, 150–160. Muir Wood, D. (2004). Geotechnical modelling. London, UK: Spon Press, Taylor and Francis. Schofield, A. N. (1980). Cambridge geotechnical centrifuge operations. Geótechnique 30, No. 3, 227–268, http://dx.doi.org/ 10 .1680/geot.1980 .30 .3 .227. Seed, H. B. (1983). Earthquake resistant design of dams. In Seismic design of embankments and caverns (ed. T. R. Howard), pp. 41– 64. New York, NY, USA: American Society of Civil Engineers. Seed, R. B. & Harder, L. F. (1990). SPT-based analysis of cyclic pore pressure generation and undrained residual strength. In Proceedings of the Seed memorial symposium (ed. J. M. Duncan), pp. 351–376. Vancouver, BC, Canada: BiTech Publishers. Steedman, R. S., Ledbetter, R. H. & Hynes, M. E. (2000). The influence of high confining stress on the cyclic behavior of saturated sand, ASCE Geotechnical Special Publication No. 107, pp. 35–57. Reston, VA, USA: American Society of Civil Engineers. Stewart, D. P., Chen, Y. R. & Kutter, B. L. (1998). Experience with the use of methylcellulose as a viscous pore fluid in centrifuge models. Geotech. Testing J. 21, No. 4, 365–369. Vaid, Y. P. & Chern, J. C. (1983). Effect of static shear on resistance to liquefaction. Soils Found. 23, No. 1, 43–60. Vaid, Y. P. & Chern, J. C. (1985). Cyclic and monotonic undrained response of sands. Advances in the art of testing soils under cyclic loading conditions. In Advances in the art of testing soils under cyclic conditions (ed. V. Khosla), pp. 171–176. New York, NY, USA: American Society of Civil Engineers. Vaid, Y. P. & Finn, W. D. L. (1979). Static shear and liquefaction potential. J. Geotech. Engng Div., ASCE 105, No. GT10, 1233– 1246. Vaid, Y. P., Stedman, J. D. & Sivathayalan, S. (2001). Confining stress and static shear effects in cyclic liquefaction. Can. Geotech. J. 38, No. 3, 580–591. Whitman, R. V. & Lambe, P. C. (1982). Liquefaction: consequences for a structure. In Soil dynamics and earthquake engineering: proceedings of the conference on soil dynamics and earthquake engineering (eds A. S. Cakmak, A. M. Abdel-Ghaffar and C. A. Brebbia), vol. 2, pp. 941–949. Rotterdam, the Netherlands: Balkema. Yoshimi, Y. & Tokimatsu, K. (1977). Settlement of buildings on saturated sands during earthquakes. Soils Found. 17, No. 1, 23–28.

Young’s modulus (MPa) excess pore pressure (kPa) peak excess pore pressure before cross-drainage (kPa) uncorrected excess pore pressure (kPa) static shear correction factor overburden correction factor foundation bearing pressure (kPa) stress reduction factor evaluated according to formulation of Idriss & Boulanger (2004) excess pore pressure ratio, Epp/� v9 (%) peak excess pore pressure ratio before cross-drainage, Eppmax/� v9,0 (%) average liquefaction induced settlement (m) settlement at time t final settlement Poisson ratio depth from foundation plane (m) initial instrument depth (m) static shear stress ratio (or normalised initial shear stress), �i/� v9,0 dry unit weight (kN/m3) unit weight of water variation in footing bearing pressure due to footing vertical acceleration, av vertical effective stress (kPa) initial vertical effective stress (kPa) cyclic shear stress (kPa) initial (or static) shear stress (kPa) shear stress acting on the x–z plane (�i + �cyc) (kPa) friction angle at critical state (degrees) peak friction angle (degrees) residual friction angle (degrees) dilation angle (degrees)

REFERENCES Acacio, A., Kobayashi, Y., Towhata, I., Bautista, R. T. & Ishihara, K. (2001). Subsidence of building foundation resting upon liquefied subsoil: case studies and assessment. Soils Found. 41, No. 6, 111–128. Adachi, T., Iwai, S., Yasui, M. & Sato, Y. (1992). Settlement and inclination of reinforced-concrete buildings in Dagupan-City due to liquefaction during the 1990 Philippine earthquake. Proceedings of the 10th world conference on earthquake engineering, vol. 1, pp. 147–152. Rotterdam, the Netherlands: Balkema. Bertalot, D. (2012). Behaviour of shallow foundations on layered soil deposits containing loose saturated sands during earthquakes. PhD thesis, University of Dundee, UK. Bertalot, D., Brennan, A. J. & Villalobos, F. (2013). Influence of bearing pressure on liquefaction-induced settlement of shallow foundations. Geótechnique 63, No. 5, 391–399, http://dx.doi.org/ 10 .1680/geot.11.P.040. Boulanger, R. W. & Seed, R. B. (1995). Liquefaction of sand under bi-directional monotonic and cyclic loading. J. Geotech. Engng, ASCE 121, No. 12, 870–878. Boulanger, R. W., Seed, R. B., Chan, C. K., Seed, H. B. & Sousa, J. (1991). Liquefaction behaviour of saturated sands under unidirectional and bi-directional monotonic and cyclic simple shear loading, Geotechnical Engineering Report No. UCB/GT/ 91-08. Berkeley, CA, USA: University of California. Brennan, A. J., Knappett, J. A., Bertalot, D., Loli, M., Anastasopoulos, I. & Brown, M. J. (2014). Dynamic centrifuge modelling facilities at the University of Dundee and their application to studying seismic case histories. In ICPMG2014 – physical modelling in geotechnics (eds C. Gaudin and D. White), pp. 227–234. Boca Raton, FL, USA: CRC Press. Castro, G. (1969). Liquefaction of sands. PhD thesis, Harvard University, Cambridge, MA, USA. Castro, G. (1975). Liquefaction and cyclic mobility of saturated sands. J. Geotech. Engng, ASCE 101, No. 6, 551–569. Castro, G. & Poulos, S. J. (1977). Factors affecting liquefaction and cyclic mobility. J. Geotech. Engng Div., ASCE 103, No. GT6, 501–516. Dashti, S., Bray, J. D., Pestana, J. M., Riemer, M. & Wilson, D. (2010). Mechanism of seismically induced settlement of build-

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Knappett, J. A. et al. (2015). Geótechnique 65, No. 5, 429–441 [http://dx.doi.org/10.1680/geot.SIP.14.P.059]

Seismic structure–soil–structure interaction between pairs of adjacent building structures J. A . K NA P P E T T , P. M A D D E N a n d K . C AU C I S †

Structure–soil–structure interaction between adjacent structures, which may occur in densely populated urban areas, has received little attention compared to the soil–structure interaction of single isolated structures. Additionally, recent earthquakes in/near such areas (e.g. the Christchurch series, 2010– 2011) have shown that large motions can be followed by strong aftershocks. In this paper, the seismic behaviour of isolated structures and pairs of adjacent structures under a sequence of strong ground motions has been investigated using a combination of centrifuge and finite-element modelling. The latter utilised an advanced constitutive model that can be parameterised from routine test data, making it suitable for use in routine design. The finite-element models were shown to accurately simulate the centrifuge-measured response (in terms of surface ground motion and structural sway, settlement and rotation) even after multiple strong aftershocks, so long as the buildings’ initial conditions were reproduced accurately. For the case of a building structure with a close neighbour, structural drift and co-seismic settlement could be reduced or increased as a result of structure–soil–structure interaction, depending chiefly on the properties of the adjacent structure. This suggests that careful arrangement of adjacent structures and specification of their properties could be used to control the effects of structure–soil–structure interaction. In all cases where adjacent structures were present, permanent rotation (structural tilt) was observed to increase significantly, demonstrating the importance of considering structure–soil–structure interaction in assessing the seismic performance of structures. KEYWORDS: centrifuge modelling; earthquakes; numerical modelling; sands

they are limited in that the soil–structure interaction is always linear elastic. This may be an acceptable assumption in a very small earthquake, but when the ground motions become large (such as in the strong recent earthquakes mentioned previously) the soil would be expected to be highly non-linear with significant plastic strains. Under these circumstances, not only would the effects of the SSSI on the dynamic structural response (e.g. inter-storey drift or spectral acceleration) be expected to change, but there may also be significant permanent settlement and rotation of the structures, which could be as damaging as the structural motions. The ability to capture the permanent behaviour is particularly important considering that many of the earthquakes mentioned previously were associated with strong aftershocks, the most notable recent example being the Darfield (2010) and Christchurch (2011) earthquakes, which occurred less than 6 months apart, that is, before a substantial amount of repair could be conducted on damaged structures following the first earthquake. Permanent soil deformations will change the behaviour of the underlying soil. Settlement may lead to soil stiffening as a result of densification, making it potentially better able to transmit ground motions into the structure (and thereby potentially increasing the structural response). A bias in permanent rotation may be amplified in subsequent earthquakes owing to P–˜ effects. Current linear elastic approaches cannot incorporate such effects. This paper, therefore, aims to study SSSI utilising methods that can incorporate non-linear elasto-plastic soil behaviour. First, dynamic centrifuge modelling is used to create a database of physical test data of pairs of adjacent low-rise simple structures having shallow foundations on sand, considering situations (a) where the structures have similar properties and (b) where the properties are dissimilar. Tests of the structures in isolation are also performed for comparison. In all cases, series of strong motions are applied to the models as an idealised representation of a sequence of

INTRODUCTION Current seismic design of building structures considers the response of a building in isolation from its neighbours. Many of the most damaging earthquakes over the last 20 years, however, have struck heavily populated and highly urbanised areas, including those in Kobe (1995), Kocaeli (1999), Athens (1999), Wenchuan (2008) and the Christchurch series (2010–2011). Although damage may be expected to be high in these areas as there are many more ‘targets’ for the earthquake, the close spacing of the building stock will result in interaction between adjacent structures through the ground, a phenomenon which is here termed structure–soil–structure interaction (SSSI). There have been few previous attempts to study SSSI and these have often been highly simplified; however, it may be expected that, depending on the layout of adjacent buildings and their relative dynamic properties, the effects of SSSI may have either a beneficial or detrimental effect on the overall response of the structures, through changes in the local soil– structure interaction. Previous studies of SSSI have generally focused on understanding changes in dynamic characteristics of adjacent rigid blocks (e.g. Tsogka & Wirgin, 2003) or simple oscillators (Alexander et al., 2013), in each case on a linear elastic medium as a representation of the soil. Both shallow (Betti, 1997) and deep foundations (Padroń et al., 2009) have previously been considered. Although these studies have provided useful information in terms of changes in fundamental periods of vibration and elastic spectral response, Manuscript received 4 April 2014; revised manuscript accepted 10 February 2015. Published online ahead of print 13 April 2015. Discussion on this paper closes on 1 October 2015, for further details see p. ii. University of Dundee, Dundee, UK. † Arup, Edinburgh, UK; formerly University of Dundee, Dundee, UK.

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KNAPPETT, MADDEN AND CAUCIS 100

strong aftershocks. Non-linear finite-element modelling is then used to simulate these tests and is validated against the centrifuge test data. This will utilise the constitutive modelling approach outlined in Al-Defae et al. (2013), which was shown to closely simulate the dynamic constitutive behaviour of soil identical to that used herein at similar relative density in sequences of strong earthquakes for slope models atop a level bearing layer. The influence of modelling assumptions will also be discussed. Finally, the finite-element approach is used to gain insight into the influence of building arrangement and aftershocks on the seismic response of pairs of adjacent structures.

24 6

Aluminium alloy

Steel 3

Steel

6

12

12 40

CENTRIFUGE MODELLING All of the tests were conducted using a model scale of 1:50 and tested at 50g using the 3 .5 m radius beam centrifuge at University of Dundee, UK. All subsequent parameters are given at prototype scale, unless otherwise stated. Scaling factors for centrifuge modelling can be found in Muir Wood (2004).

.

40 (a)

40

100

48

Model structures Two different structural models were produced that were designed to represent some of the key characteristics of lowrise buildings. Such buildings will generally form a much larger proportion of the building stock within an urban area compared to high-value, high-rise structures and are also less likely to have undergone detailed seismic design. They may therefore be more vulnerable and contribute more significantly to the overall cost of seismic damage following a major earthquake. Each structure was modelled as a single-bay, singledegree-of-freedom (SDOF) sway frame on separated strip foundations, 100 mm long and 40 mm wide at model scale (5 m 3 B ¼ 2 m at prototype scale). Steel mass plates were used to represent the dynamic mass of the structure, and vertical aluminium alloy plates, the same length as the foundations, were used to represent the sway stiffness of the structures (that would be provided by the columns). The centre-to-centre spacing between the foundations (s) was kept the same between the models, but the height was varied to change the position of the centre of mass above the soil surface and therefore alter the proportion of rocking to sway deformation between the two structures. The model structures are shown in elevation with model scale dimensions (in mm) in Fig. 1. The fundamental natural period (Tn0) of a building structure is typically related to its height, with taller buildings being laterally more flexible per unit mass and hence having higher Tn0 (e.g. Goel & Chopra, 1997; BSI, 2005). To design the model structures, the relationship given in Eurocode 8 (BSI, 2005) was used for building structures less than 40 m high T n0 ¼ C t H 0 75

60

60

6

140

6

138

6 12

12 40

40 (b)

40

Fig. 1. Single-degree-of-freedom model structures: (a) ‘short’ period structure; (b) ‘long’ period structure. All dimensions at model scale in mm

founding plane). From equation (1), Tn0 ¼ 0 .33 s and 0 .65 s for the ‘short’ and ‘long’ period structures, respectively. The characteristics of the models are compared to the measured properties of a variety of real structures (after Goel & Chopra (1997) and Stewart et al. (1999)) in Fig. 2. The supported mass and stiffness of the plates representing the columns were then selected to match these values of Tn0. The mass at the top of each structure (Meq) was selected first, which fixed the value of the bearing pressure (q) for a given static factor of safety (FSv). The thickness of the vertical plates was then selected to provide a bending stiffness, EI (and therefore lateral sway stiffness, Keq) such that the required Tn0 was achieved for the mass selected, using sffiffiffiffiffiffiffiffi M eq (2) T n0 ¼ 2 K eq

(1)

where H is the overall height of the building and Ct ¼ 0 .085 (as an approximation for a steel moment resisting frame (MRF)). In each case, the structures are assumed to be SDOF idealisations of multi-storey structures that have a uniform distribution of mass and stiffness with height. The centre of mass of such structures would be approximately at the mid-height, and so in order to ensure the correct amount of overturning moment for a given value of H, the structure with the shorter period (hereafter termed ‘short structure’) represents a prototype structure H ¼ 6 m high (centre of mass 3 m above founding plane). The structure with the longer period (hereafter termed ‘long structure’) represents a structure H ¼ 15 m high (centre of mass 7 .5 m above the

A summary of the properties of the two structures at prototype scale is provided in Table 1. This table includes an estimate of FSv when surface bearing on a uniform

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SEISMIC STRUCTURE–SOIL–STRUCTURE INTERACTION BETWEEN BUILDING STRUCTURES 3

in the following section). Tests PM005 and PM006 tested the same adjacent pair of two long structures, but with different edge-to-edge (inter-building) spacing between the structures; PM008 was comparable to PM006 in terms of spacing, but tested the two short structures as an adjacent pair. Finally, tests PM009 and PM011 tested pairs of one short and one long structure; by reversing the order of the structures in the box, the effect of earthquake direction relative to the arrangement of structures could be considered, as the ground motions were always applied in the same direction. Spacing between structures in the adjacent cases was kept to a minimum to consider the condition where SSSI effects will likely be most significant (assuming that these reduce as spacing increases). A summary of the test configurations is given in Table 2.

Field data (Stewart et al., 1999)

Fundamental natural period, Tn0: s

Short structure Long structure Eurocode 8 (Ct 0·085)

2

1

0

0

10

20 Building height, H: m

30

40

Model preparation and soil properties As the tests reported here focus on the effects of structural properties and arrangement on SSSI, a single set of soil properties was used in all of the tests. Dry HST95 Congleton silica sand was air-pluviated into an equivalent shear beam (ESB) container to a target relative density of Dr ¼ 55–60% (the range accounts for the accuracy with which this property can be replicated and measured within a model soil bed) to produce a uniform layer with a prototype thickness of 10 m. The measured relative density achieved in each test is shown in Table 2. The design and performance of the ESB container is described by Bertalot (2012) and the basic soil properties for HST95 are given in Table 3 after Lauder (2011). As the rigid base of the container lay below the layer of sand, the ground profile represents ground type E according to Eurocode 8 (BSI, 2005). During pluviation, the soil was instrumented with type ADXL78 MEMS accelerometers (� 70g range) manufactured by Analog Devices, as shown in Fig. 3 (PM003 and PM006 are shown as examples). These were used to measure the input motion (point E), free-field ground motion 1 m below the ground surface (point F) and accelerations beneath (D) and between (G) the structures at the same depth as the

Fig. 2. Fundamental period of model structures and comparison to field data Table 1. Properties of model structures (prototype scale) Parameter: units

‘Short’ structure

Total height, H: m Height to C-of-M: m Natural period, Tn0: s Bearing pressure, q: kPa Static factor of safety, FSv Meq: t Keq: MN/m Vertical plate EI: MN m2/m Footing spacing, s: m

‘Long’ structure

6 3 0 .33 161 9 .5 235 87 .2 19 .6 4

15 7 .5 0 .65 276 5 .5 469 44 .1 155 .1 4

deposit of dry sand as used in the centrifuge tests (relative density Dr ¼ 58%, unit weight ª ¼ 16 .2 kN/m3 and peak friction angle �9p ¼ 408), which was determined using 0.5ªBsª N ª FSv ¼ (3) q

Table 3. State-independent physical properties of HST95 silica sand (after Lauder, 2011)

In determining FSv, Nª and sª were calculated for �9p ¼ 408 following Salgado (2008) and Lyamin et al. (2007), respectively, although it should be noted that almost identical values are found using the relationships provided in Eurocode 7 (BSI, 2004). The same foundation type and size was used for both structures, which resulted in a higher value of FSv for the short structure due to its lower applied bearing pressure. These model structures were used in a total of seven centrifuge tests. Two of these, PM003 and PM004, tested the single structures in isolation (ground properties are described

Property

Value

Specific gravity, Gs D10: mm D30: mm D60: mm Cu Cz Maximum void ratio, emax Minimum void ratio, emin

2 .63 0 .09 0 .12 0 .17 1 .9 1 .06 0 .769 0 .467

Table 2. Centrifuge testing programme Test ID

Configuration

Structure type (left)

PM003 PM004 PM005 PM006 PM008 PM009 PM011

Isolated Isolated Adjacent, similar Adjacent, similar Adjacent, similar Adjacent, dissimilar Adjacent, dissimilar

Short Long Long Long Short Long Short

Structure type (right)

Long Long Short Short Long

103


Inter-building spacing: m

Relative density: %

N/A N/A 2 1 1 1 1

58 59 57 58 58 59 56

KNAPPETT, MADDEN AND CAUCIS Direction of shaking

Initial tilt: degrees 40

Accelerometer [A]

[F]

10 (200) [E]

1·0

Initial settlement: mm

1·5 1 (20)

[D]

Medium dense sand Dr 55–60%

LVDT

[C]

[B]

1 (20)

20 0·5 0 0

0·5

1·0

PM003 PM004 PM006 PM008 PM009 (short) PM009 (long) PM011 (short) PM011 (long)

1·5

20 40 60 80

1/250 1/50

100

33·5 (670) (a)

120 140

[A]

[K]

Direction of shaking

Medium dense sand Dr 55–60%

Accelerometer [C][H]

[I]

[F]

[G] 1 (20)

[E]

LVDT

was band-pass filtered between 0 .8 Hz and 8 Hz (40–400 Hz at model scale) using a zero-phase-shift digital filter to remove components of the signal that were outside the range that can be accurately controlled by the EQS. The time history of this demand motion, normalised by peak acceleration, is shown in Fig. 5(a). The Kobe earthquake was known to be particularly damaging to infrastructure, and the motion selected has a number of repetitive acceleration peaks close to the peak ground acceleration (ag), for example, between 9 and 14 s in Fig. 5(a). In each test a 0 .1g motion was initially applied to the model to characterise the dynamic behaviour of the system when the soil strains were small (negligible permanent settlement and rotation occurred during these motions). Subsequently, three nominally identical 0 .5g motions were applied, representing a strong earthquake and two strong aftershocks. The initial 0 .1g motion could also be

1 (20)

10 (200)

[B]

Fig. 4. Initial conditions of model structures (settlement and rotation) measured following centrifuge spin-up to 50g and prior to earthquake shaking

1 (20)

33·5 (670) (b)

Fig. 3. Layout of centrifuge tests: (a) isolated structure case (PM003 shown); (b) adjacent structure case (PM006 shown). All dimensions at prototype scale in m (model scale in mm)

free-field instrument, to examine near-field effects of SSSI on the ground motion. Identical accelerometers were also attached to the structures, as shown in Fig. 3. These were used to measure the dynamic motion of the foundations and equivalent mass. Through high-pass filtering and integration, the dynamic displacements were determined at these locations with rotational bias removed, and the difference between them represented the dynamic inter-storey drift. The container and model soil beds were loaded onto the centrifuge and the structures were placed on the surface after loading using a post-level to place them as accurately as possible. In all cases the structures occupied a maximum of the central 33% of the ESB (Fig. 3(b)) to minimise any potential boundary effects within the container. An overhead gantry was then placed above the structures allowing linear variable differential transformers (LVDTs) to be placed as shown in Fig. 3 to measure average settlement and global rotation (tilt) of the structures. During spin-up of the centrifuge the response of the LVDTs was recorded such that the initial settlement and tilt of the structures, prior to earthquake shaking, were known. These initial conditions are presented in Fig. 4.

Normalised acceleration

1

0

0

10

1

20 Time: s

30

40

(a)

4

Normalised spectral acceleration, Se/ag

0·5g

Dynamic excitation Following spin-up, a sequence of strong ground motions was applied to each of the models using the Actidyn QS67-2 servo-hydraulic earthquake simulator (EQS), the performance of which is detailed in Bertalot et al. (2012) and Brennan et al. (2014). In each case, a horizontal motion recorded at the Nishi-Akashi recording station in the Mw ¼ 6 .9 Kobe earthquake (1995) was used. This record was downloaded from the PEER (Pacific Earthquake Engineering Research) NGA database and, unscaled, had a peak acceleration of ag ¼ 0 .43g. For the purpose of these tests, the motion was rescaled to 0 .1g and 0 .5g nominal peak accelerations and

0·1g Eurocode 8 (ground type A)

3

2

1

0 0

1

2 Period: s (b)

3

4

Fig. 5. Input ground notion from Nishi-Akashi recording station (Kobe in 1995): (a) normalised time history; (b) response spectra for nominal 5% damping

104


Structural viscous damping ratio: %

SEISMIC STRUCTURE–SOIL–STRUCTURE INTERACTION BETWEEN BUILDING STRUCTURES 10 considered as a small pre-shock prior to the 0 .5g main Finite-element model, equation (4) shock. A final 0 .1g motion was then applied to provide a recharacterisation of the behaviour at smaller strains following Short structure 8 the substantial changes imparted to the soil fabric by the Long structure preceding motions. The use of the same underlying record to represent all ground motions applied was an idealisation of Typical steel MRF the earthquakes being generated by the same source (and 6 therefore having similar characteristics). The repeatability of the motions is demonstrated in Fig. 5(b) in terms of the nominal 5% damped response spectrum of the actual re4 corded motion at point E in Fig. 3, normalised by ag (12 0 .5g motions and four 0 .1g motions are shown). As this represents the ‘bedrock’ input motion, the design spectrum 2 from Eurocode 8 for such material (ground type A) is also shown in Fig. 5(b) for context (BSI, 2005). Range of input motion 0 0

FINITE-ELEMENT MODELLING As the centrifuge model structures and foundations were long in the direction perpendicular to shaking, finite-element simulations of all of the tests were conducted in plane strain using Plaxis 2D 2012. Compared to the centrifuge model shown in Fig. 3, the dimensions of the model domain were extended laterally to 100 m and combined with non-reflecting boundary elements controlling the dynamic stresses along the vertical boundaries (after Lysmer & Kuhlmeyer, 1969) to represent semi-infinite soil conditions; that is, boundary deformations at the location of the centrifuge container wall which are controlled by the dynamic deformation of the adjacent soil. This boundary condition can also be modelled by horizontal node-to-node ties between the two vertical boundaries of a model the width of the soil tested in the centrifuge. Compared to this alternative, the method used has a higher element requirement for the same mesh density, but allowed future extension of the model beyond the two structures considered here (although the extended results are not reported in this paper). This same approach has been used previously in modelling the behaviour of slopes during a sequence of strong aftershocks in the same soil and model container, as described by Al-Defae et al. (2013). The equivalent mass and vertical plates of the structures were modelled numerically using elastic plate elements having the same bending stiffness per metre length and mass as the centrifuge models (Table 1). The footings were modelled as an elastic continuum with Young’s modulus of 210 GPa, Poisson ratio of 0 .3 and unit weight of 76 .5 kN/m3 to match the properties of the steel footings in the centrifuge tests. Damping in each structure was modelled using Rayleigh’s approach to determine the equivalent viscous damping () 1 ¼ cm (4) þ ck ( f n ) 4 f n

2

4 6 Frequency: Hz

8

10

Fig. 6. Modelling of structural damping in finite-element simulations

incorporates non-linear elastic behaviour which is dependent on both confining stress level and induced strain, with Mohr–Coulomb plasticity having isotropic hardening. To examine the influence of soil parameter correlations, two different sets of parameters were used in simulations of the centrifuge tests. The first, by Brinkgreve et al. (2010), is non-soil specific, having been developed based on fitting to a database of historical element test data for different sands. This only requires relative density as an input parameter from which all of the constitutive parameters are obtained, potentially allowing for complete parameterisation from routine in-situ tests that can be used to estimate relative density, such as the standard penetration test (SPT) or cone penetration test (CPT) if further laboratory testing data were not available. The second set of parameters used was specifically calibrated for the HST95 sand used in the centrifuge tests through additional (routine) soil element testing, including direct shear tests and oedometric compression tests of the sand across a wide range of relative densities. A complete description of this set of parameters and the methods used to find them is given in Al-Defae et al. (2013). This second set of parameters also uses relative density as the input parameter. A summary of both sets of correlations is provided in Table 4. For the simulation of a particular centrifuge test, the actual relative density from Table 2 was used to obtain the constitutive parameters. Previous finite-element modelling of the behaviour of the test soil at a similar density in the same ESB container at 50g by Al-Defae et al. (2013) has shown that some additional Rayleigh damping was required in addition to the implicit hysteretic damping included in the constitutive model, to control higher frequency components of the deformation and replicate dynamic accelerations accurately within the soil. Therefore, the same mass- and stiffness-proportional Rayleigh parameters from this previous study were used for the soil in all simulations presented here, namely, cm ¼ 0 .0005 and ck ¼ 0 .005. Simulations using the Brinkgreve et al. (2010) correlations represent those that could be achieved without doing an extensive amount of site investigation; those using the AlDefae et al. (2013) correlations would potentially allow improved predictions at the cost of performing an additional soil-specific calibration. Their comparison later in the paper will demonstrate how important such a calibration is to the accurate prediction of the dynamic soil and structural response. In each case, the simulations were conducted in two

where fn are natural frequencies (of the structures). The mass- and stiffness-proportional coefficients were found by fitting equation (4) to measured damping of the model structures determined using the logarithmic decrement method applied to impulse test data, and simultaneously ensuring that the relationship is relatively flat across the full range of input motion frequencies. This is shown in Fig. 6, where mean values are shown, along with error bars representing the maximum and minimum range of the experimental impulse test data. From this figure, cm ¼ 0 .4 and ck ¼ 0 .001 were selected for use in both structures. It should also be noted that both models represent the damping of typical steel structures (,2%) well, in addition to the natural period matching shown in Fig. 2. The soil was modelled using the ‘Hardening soil model with small-strain stiffness’ (Benz, 2006). This soil model

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KNAPPETT, MADDEN AND CAUCIS Table 4. Constitutive model parameters for finite-element modelling Parameter

Brinkgreve et al. (2010)

HST 95 (Al-Defae et al., 2013)

Units

�9p c9 ł9 ref Eoed ref E50 Eurref �ur G0ref �s,0 .7 Rf m ª

12.5Dr þ 28 0 12.5Dr � 2 60Dr Eref oed 3Eref oed 0.2 68Dr þ 60.00 2 � Dr (310�4 ) 1 � 0.13Dr 0.7 � 0.31Dr 4Dr þ 15.0

20Dr þ 29 0 25Dr � 4 25Dr þ 20.22 1.25Eref oed 3Eref oed 0.2 50Dr þ 88.80 1.7Dr þ 0.67(310�4 ) 0.9 0.6 � 0.1Dr 3Dr þ 14.5

degrees kPa degrees MPa MPa MPa – MPa – – – kN/m3

Note: All reference stiffness parameters (indicated by superscript ‘ref’) are defined at p9 ¼ 100 kPa; the hardening soil model subsequently adjusts the stiffness as the confining stress changes within the continuum.

An example of the sensitivity of the rotation of isolated structures to initial conditions is shown in Fig. 8, where the three large earthquakes for test PM004 are shown, along with the input and free-field surface time histories of ground acceleration. It can be seen that if the initial rotation is matched, the finite-element model produces a very good prediction of the development of rotation throughout the first large shock and the subsequent large aftershocks. The predicted rotations are completely different using the idealised initial rotation condition. This sensitivity is exacerbated when adjacent structures are considered. Fig. 9 shows a comparison similar to Fig. 8 but for test PM006. In this particular case, the rotation of the right-hand structure is matched well in the first motion, but the left-hand structure is not (likely due to a local heterogeneity in this particular centrifuge test). As soon as the rotations begin to deviate from the centrifuge result, the match in subsequent strong aftershocks cannot be good, as the initial conditions in the subsequent earthquakes will be different. This does not imply that the finite-element model is wrong per se, just that the assumption of a uniform deposit of soil cannot model the highly subtle variations in the real soil of the centrifuge test, and that the rotation behaviour is highly sensitive to this. This is discussed further in the next section.

stages: Following initial stress generation to K0 conditions (where K0 ¼ 1 � sin �9p ), the structure is ‘built’ during a static stage; that is, the weight of the structure(s) is used to load the soil, obtain the initial stress and deformation fields within the model and achieve static equilibrium. Once this is complete the second stage applies a dynamic input motion to the bottom of the model, in each case matching that recorded in the corresponding centrifuge test. The motions were input as ground displacement histories, determined by high-pass filtering and integration of the accelerometer records; filtering before integration to obtain velocity, and again, before integrating velocity to obtain displacement ensured that there was no permanent ‘wander’ due to any offset in the accelerometer recordings or integration of random noise within the signal. Dynamic displacement data were then obtained at points in the finite-element model which matched those in the centrifuge tests, as shown in Fig. 7 (compare to Fig. 3). Where acceleration data were required, these were obtained by numerically double differentiating the appropriate displacement–time history. For each of the simulations using the different parameter sets, the initial conditions were first determined direct from the initial static stage – this represents ‘ideal’ soil conditions. These were not the same as those in the centrifuge tests (Fig. 4), as it was impossible to achieve perfectly level placement of the structures and avoid small variations in soil properties in preparing the real soil. As the rotational behaviour is likely to be highly influenced by any initial bias in the system, a third set of simulations was also conducted, using the HST95 parameter set, but with additional vertical point loads applied to the foundation (points B, C, H and I) during the initial static step to generate a couple which forces the structure to have the initial rotations shown in Fig. 4. Such a couple, superimposed on the initially equally divided vertical loads, simulates the difference in vertical loads between footings induced by the resultant static moment on the structure that is consistent with the measured structural rotation. The magnitude and direction of the couple in each case were determined by trial and error within the initial static step of the corresponding finiteelement model, so the resulting static moment and rotation are consistent with the non-linear behaviour of the foundation soil. By adding a couple, the average bearing pressure across the whole structure is unchanged (it has consistent mass), despite the loads on the different footings being distributed differently. This third set of simulations will demonstrate the importance of knowing the initial conditions of a structure prior to an earthquake (such as could be measured by surveying).

VALIDATION OF FINITE-ELEMENT MODEL AGAINST CENTRIFUGE TEST DATABASE As there is a substantial amount of test data (seven tests, each having five earthquakes’ worth of data), the performance of the finite-element model and effects of both material properties and initial conditions are here summarised in terms of the following key performance indicators: (a) peak ground acceleration near the soil surface in the free-field (point F); (b) peak ground acceleration at point G (in the case of adjacent structure models); (c) peak cyclic drift across the superstructure (between the foundation and the mass plates) in each earthquake; (d ) post-earthquake settlement and (e) post-earthquake structural tilt (global rotation). It should be noted that (c) is a measure of super-structural demand, whereas (d ) and (e) are measures of foundation performance which may affect the post-earthquake serviceability of the structure. Figure 10(a) shows the soil amplification factor in the free field (SFF) from the centrifuge tests and Fig. 10(b) presents the performance of the numerical simulations in replicating this parameter using the different sets of material parameters. The factor SFF is the ratio of the peak ground acceleration at point F divided by that at point E, and the

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SEISMIC STRUCTURE–SOIL–STRUCTURE INTERACTION BETWEEN BUILDING STRUCTURES 1m 7m Centre of mass, A

Positive rotation

Footings: B (left) and C (right) G

D

Free field, F 1m

Soil base, E 1m

(a) Centre of mass: A (left) and K (right) Positive rotation G – 1 m deep in soil – equidistant between structures

Footings: H (left) and I (right)

Footings: B (left) and C (right)

20 m

1m D (left) and J (right)

Free field, F 1m

Soil base, E 1m (b)

Fig. 7. Example layouts and finite-element mesh for numerical simulations: (a) PM003; (b) PM006. All dimensions at prototype scale

value of 1 .4 suggested by Eurocode 8 for ground type E is also shown for context (BSI, 2005). Examples of the amplification in the time histories of motion can be seen in Fig. 8(b) and Fig. 9(c). Only the cases with ideal initial conditions are shown for the finite-element model data, as the initial conditions were not found to affect this parameter. Generally, both sets of material parameter correlations produce similar predictions of the centrifuge data, including the observation from the centrifuge that the amplification is generally larger in the smaller earthquakes. There is also a noticeable increase in SFF in the smaller earthquake following the strong aftershock sequence in Fig. 10(a), presumably as a result of soil densification and a resulting stiffening of the soil response. The amplification factors are generally lower in the higher strength earthquakes as there is increasing soil inelasticity, which limits the transfer of cyclic shear stress. (Ultimately shear decoupling may occur if the ratio (S 3 ag /g) becomes equal to tan 9p ; that is, the cyclic shear stresses become equal to the shear strength of the soil. This does not happen here as the peak friction angle of the soil would require S ¼ 1 .68 for ag ¼ 0 .5g in the free field and none of the measured values for this strength of input motion is this high in Fig. 10(a).) Figure 11 shows the changes to the soil amplification in the near field of the structures (only data for the adjacent structure models are shown). SNF is the ratio of the peak ground acceleration at point G divided by that at point E. The centrifuge data in Fig. 11(a) show, in general, a slight

attenuation of ground motion close to (between) the adjacent structures. In the two finite-element cases using the ideal initial conditions (Fig. 11(b)), this behaviour is not well represented, showing instead a predominance towards amplification in the near field. When the initial conditions are correctly replicated, this tendency is reduced and a better match to the centrifuge data is obtained. This may suggest that the non-symmetrical changes to the stress distribution beneath the structures induced by the non-uniform load distribution between the footings increases the asymmetry in the interactions between the incident and reflected waves beneath the structures, encouraging destructive interference as these waves are superimposed. Figure 12 compares the magnitude of peak drift recorded for each structure in each earthquake. These are clustered into a group of smaller values, representing the response of the short structures, and a larger set for the long structures. The variation in the magnitude of the drift is partially associated with the actual achieved strength of the input motion, variations in soil amplification in each test (e.g. Fig. 10) and SSSI. Fig. 12 suggests that the dynamic response is best simulated when the initial conditions are correctly simulated, with the idealised cases leading to an underprediction of super-structural response. Figure 13 shows the post-earthquake structural settlements. These are over-predicted by the Brinkgreve et al. (2010) set of material parameters, and under-predicted using the soil-specific HST95 parameters. The over-prediction

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KNAPPETT, MADDEN AND CAUCIS 1·5

2·5

HST95 ideal initial conditions

2·0

0·5

0 0

50

100

Centrifuge test HST95 ideal initial conditions HST95 measured initial conditions

HST95 measured initial conditions

Structural tilt: degrees

Structural tilt: deg

1·0

Centrifuge test

150

Time: s 0·5

1·5 1·0 0·5 0 0

50

0·5

100

150

1·0

1·0

1·5 1·5

(a)

(a) 1·5

1·0

Centrifuge test

Free field, F

HST95 ideal initial conditions 1·0

Structural tilt: degrees

Input, E

Acceleration: g

Time: s

0·5

0 0

50

Time: s

100

150

HST95 measured initial conditions

0·5 0 0

50

Time: s

100

150

0·5

1·0

0·5

1·5

(b)

Fig. 8. Effect of initial conditions on simulated rotational response of an isolated structure during a sequence of strong earthquakes (data for PM004 shown): (a) rotations; (b) ground motions at bedrock (E) and free field (F) (as recorded in the centrifuge)

(b)

1·0

Free field, F

Acceleration: g

Input, E

within the former is likely to be attributable to the lower soil strength (peak friction angle – see Table 4) meaning that soil yield occurs earlier and greater settlements are accrued. The Brinkgreve et al. (2010) set of parameters was similarly found by Al-Defae et al. (2013) to over-predict permanent seismic slope deformations in the same sand (at a similar relative density). These observations demonstrate the benefits of investing additional effort and resources in performing a soil-specific model calibration on accurate prediction of permanent deformations. The importance of correct simulation of the initial conditions is also highlighted, as for peak drift. Using the idealised initial conditions resulted in a starting position with comparative amounts of settlement of each foundation (low initial rotation). Applying the couple to generate the measured initial conditions generally resulted in an increase in the load on one of the foundations, while reducing it on the other. As the soil response is non-linear, the foundation under greater compressive loading will be pushed into a more inelastic part of the load–settlement curve, resulting in greater settlements. This would be expected in the centrifuge too, assuming that the stress distribution is similarly altered as a result of the measured initial conditions. As settlement of the structure is the average of the settlements of the two foundations, this may explain the lower settlements for the idealised initial conditions compared to both the case with measured initial conditions and the centrifuge data, which match well. Figure 14 compares the post-earthquake permanent rotations (tilts) of the structures. Fig. 14(a) includes all of the

0·5

0

0

50

100

150

Time: s

0·5

(c)

Fig. 9. Effect of initial conditions on simulated rotational response of adjacent identical structures during a sequence of strong earthquakes (data for PM006 shown): (a) rotation of left structure; (b) rotation of right structure; (c) ground motions at bedrock (E) and free field (F) (as recorded in the centrifuge)

data, and shows a number of significant outliers, particularly along the x-axis (i.e. the finite-element model is predicting large rotations). These points are associated with the later earthquake motions of the adjacent structure tests when substantial permanent rotations had accrued in the finiteelement model owing to the successive strong shaking, and are to be expected given the example results from Fig. 9. In Fig. 14(b), only the data from the first 0 .1g and first 0 .5g motions are plotted, which results in a strong positive correlation, but only when the initial conditions are correctly modelled. This also serves to highlight the data for the two sets of simulations which used idealised conditions, where it

108


SEISMIC STRUCTURE–SOIL–STRUCTURE INTERACTION BETWEEN BUILDING STRUCTURES 2·5

Centrifuge Eurocode 8, ground type E

2·0

Centrifuge

2·0

1·5

1·5

SNF/SFF

Free-field amplification, SFF

2·5

1·0

1·0

0·5

0·1g EQ

0·5g EQs

0·1g EQ

1

2 3 4 Earthquake event no. (a)

5

0·5 0·1g

0 0

0·5g

0·1g

0

6

0

2·5

1

2 3 4 Earthquake event no. (a)

5

6

1·5

SNF/SFF, centrifuge test

SFF, centrifuge test

2·0

1·5

1·0 Brinkgreve et al. (2010)

1·0

0·5

Brinkgreve et al. (2010)


0·5


Parity (1:1)

HST95 measured initial conditions Parity (1:1)

Eurocode 8, ground type E 0 0

0·5

1·0 1·5 SFF, finite-element model (b)

2·0

0

2·5

0

Fig. 10. Effect of soil material parameter correlations on prediction of site effect in the free field through a sequence of strong earthquakes: (a) centrifuge data, adjacent structure tests; (b) comparison of finite-element method predicted values to centrifuge data

0·5 1·0 SNF/SFF, finite-element model (b)

1·5

Fig. 11. Effect of soil material parameter correlations and structural initial conditions on amplification/attenuation of ground motion in the near field (point (G), adjacent structure cases): (a) centrifuge data; (b) comparison of finite-element method predicted values to centrifuge data

can be seen that neither set of material properties provides any correlation with the centrifuge results.

|Peak drift|, centrifuge test: mm

100

Validation summary Linear relationships were fitted to the individual data sets in Figs 10(b), 11(b), 12, 13 and 14(b) (and also for data of SNF plotted as centrifuge against finite-element model), using a least-squares fitting procedure. The gradient of these relationships demonstrates, on average across the full dataset, the degree of over- or under-prediction. The inverse of these gradients, plotted as percentages, are summarised in Fig. 15 for the performance indicators (a)–(e). This shows that in order to achieve the best simulation of soil structure interaction and SSSI (at least for pairs of structures), it is necessary to both obtain a soil-specific set of model parameters (as also concluded by Al-Defae et al. (2013)) and to model the actual initial (rotation) conditions of the structure(s). When applied in practice to field structures, this could be measured based on structural surveying of the building stock, and would need to be updated if this varied with time since construction. When both material properties and initial conditions are correctly modelled, all five of the performance indicators can generally be predicted within 10% averaged error across the 35 different earthquake and structure combinations considered in this paper, although

80

60

40 Brinkgreve et al. (2010) HST95 ideal initial conditions

20

HST95 measured initial conditions Parity (1:1) 0 0

20 40 60 80 |Peak drift|, finite-element model: mm

100

Fig. 12. Effect of finite-element method modelling assumptions on estimation of peak drift

there are some outlying points, which are perhaps to be expected given the extensive amount of earthquake shaking applied to each model, and therefore the potential for small differences to become amplified by the end of the earthquake sequence.

109


KNAPPETT, MADDEN AND CAUCIS 200

Brinkgreve et al. (2010) HST95 ideal initial conditions

Finite-element model/centrifuge: %

Post-EQ settlement, centrifuge: mm

300

HST95 measured initial conditions Parity (1:1)

200

100

0

150 Finite-element method over-prediction 10%

100

10% Finite-element method under-prediction 50

0

0 100 200 300 Post-EQ settlement, finite-element model: mm

SFF

Fig. 13. Effect of finite-element method modelling assumptions on estimation of post-earthquake settlement

Post EQ tilt, centrifuge: degrees

|Peak Post-EQ Post-EQ drift| settlement tilt* *Tilt value for initial earthquakes only

Brinkgreve et al. (2010) HST95 ideal initial conditions HST95 measured initial conditions

3·0

All other values include aftershock data

Fig. 15. Summary of finite-element method validation against centrifuge test data

2·0 1·0

3·0 2·0 1·0

0

0

1·0

2·0

for perfectly ideal and identical ground. In this section, finite-element analyses using the HST95 material model and ideal initial conditions are therefore compared to demonstrate the effects of the presence of an adjacent structure on the resulting SSSI and structural response, all other conditions being equal, using the same Kobe earthquake motion and order of consecutive motions as described previously. The results are summarised in Figs 16–18 and will be discussed together at the end of the section. Fig. 16 shows the peak drifts normalised by those of the isolated structures. The data are separated by earthquake strength and by structure type, and each data point represents an average across the different motions; for example, for the case of similar tall structures, the 0 .5g point in Fig. 16 is the average of the EQ2, EQ3 and EQ4 responses for both structures. Fig. 17(a) shows a comparison of the permanent movements (settlement and tilt) of a long structure when it is either on its own (‘isolated’), adjacent to an identical structure (‘Similar’, with ‘L’ and ‘R’ denoting left and right

3·0

Brinkgreve et al. (2010) HST95 ideal initial conditions HST95 measured initial conditions Parity (1:1) Tilt 1/50

1·0 2·0 3·0

Post-EQ tilt, finite-element model: degrees (a) 1·0

Post EQ tilt, centrifuge: degrees

SNF

0·5

0 1·0

0·5 0·5

1·0

0

0·5

1·0

Brinkgreve et al. (2010) HST95 ideal initial conditions Series 1 Parity (1:1)

130

Change in drift due to SSSI, compared to an isolated structure: %

Post-EQ tilt, finite-element model: degrees (b)

Fig. 14. Effect of finite-element method modelling assumptions on estimation of post-earthquake permanent rotation (structural tilt): (a) all earthquakes; (b) earthquakes EQ1 (first 0 .1g motion) and EQ2 (first 0 .5g motion) only

INSIGHTS INTO SSSI OF PAIRS OF ADJACENT STRUCTURES Although the centrifuge test data are valuable as a means of validating the finite-element model and understanding the importance of the modelling assumptions (material properties and initial conditions), direct comparison across tests, particularly in terms of settlement and tilt, is not ideal, owing to the different initial conditions in the tests (Fig. 4). The rotation behaviour has also been shown to be highly sensitive to the exact ground conditions which match well, but not perfectly across the different centrifuge models (relative density in Table 2). The finite-element method, however, presents an opportunity to compare model behaviour

SSSI detrimental

120

110

100

90

Long structures (0·5g) Long structures (0·1g)

80

Short structures (0·5g) Short structures (0·1g)

70

Similar (s 2 m)

Similar (s 1 m)

SSSI beneficial Dissimilar

Overall

Fig. 16. Summary of effect of adjacent structure properties and earthquake strength on drift amplification/attenuation due to SSSI (ideal initial conditions)

110


SEISMIC STRUCTURE–SOIL–STRUCTURE INTERACTION BETWEEN BUILDING STRUCTURES Static

EQ1

EQ2

EQ3

EQ4

EQ5

Static

0

EQ1

EQ2

EQ3

EQ4

EQ5

0

Final settlement: mm

50

100

Similar, R

Dissimilar, R

1·00

1·00

0

0·50

Dissimilar, L

100

150

0·50

Similar, L

50

150

Final rotation: degrees


Final settlement: mm

Isolated

Isolated

0·50

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Similar, R Dissimilar, L

0·50

Dissimilar, R 1·00

1·00 Static

EQ1

EQ2

EQ3

EQ4

Static

EQ5

(a)

EQ1

EQ2

EQ3 EQ4 (b)

EQ5

Fig. 17. Effect of adjacent structure properties, earthquake strength and strong aftershocks on development of settlement and tilt due to SSSI (ideal initial conditions): (a) long structures; (b) short structures

1·0

of the pair relative to the direction of the earthquake and the geometry shown in Fig. 7), or next to a smaller structure (‘Dissimilar’). Fig. 17(b) shows similar data for a short structure when it is isolated, next to an identical structure, or next to a larger structure. As the rotation may be different depending on whether the structure is on the left or right of the arrangement (this essentially represents the earthquake motions being in opposite directions), Fig. 18 shows the average of the absolute rotations of the ‘L’ and ‘R’ cases from Fig. 17. Based on Figs 16–18, the following insights can be drawn.

Isolated


Similar Dissimilar

0·5

(a) Insights for the case where a structure is situated next to an identical neighbour. (i) The drift may or may not be increased, depending on the natural period of the structure in question and the strength of the earthquake. In this study, in small earthquakes inducing a smaller strain soil response, SSSI increased drift for long period structures and reduced it for short period structures; in larger earthquakes with a strong elasto-plastic near-field response, SSSI increased drift for both types of structure by between 2 and 10% (Fig. 16). (ii) The settlement is either unaffected, or slightly reduced due to the adjacent structure providing additional confinement to the soil beneath the foundations (Fig. 17). (iii) The magnitude of the rotation of the structure increases (Fig. 18), and it rotates away from its neighbour (compare hollow circle and hollow square markers in Fig. 17(a) or Fig. 17(b); the sign convention is shown in Fig. 7). This outward ratcheting is thought to occur as plastic soil deformation is confined by the adjacent structure while the structure is rotating towards its neighbour, whereas soil deformation while rotating outwards (away from its neighbour) is not. This means that

0 Static

EQ1

EQ2

EQ3

EQ4

EQ5

EQ3

EQ4

EQ5

(a)

1·0 Isolated


Similar Dissimilar

0·5

0 Static

EQ1

EQ2 (b)

Fig. 18. Magnitude of structural tilt (average of both earthquake directions): (a) long structures; (b) short structures

111


KNAPPETT, MADDEN AND CAUCIS either similar or dissimilar type. This demonstrated that the structural drift and co-seismic settlement could be reduced or increased as a result of SSSI, depending chiefly on the properties of the adjacent structure (building height was considered here, in terms of changes to the fundamental period and foundation bearing pressure). This suggests that, through further study, it may be possible in the future to prescribe dynamic properties in seismic design to exploit beneficial effects of SSSI with the surrounding urban environment. However, in all cases, permanent rotation (tilt) of the structure was observed to increase compared to the isolated case as a result of SSSI, and so consideration must also be given to effective ways of remediating this.

there would be a net rotation outwards in a given cycle of deformation, which would then progressively accrue in subsequent cycles due to P–˜ effects. (b) Insights for the case where a structure is situated next to a shorter neighbour (having lower natural period and bearing pressure). (i) SSSI increases drift compared to the isolated case, but by less than when the neighbouring structure is identical, and in both small and large earthquakes (Fig. 16). (ii) The settlement is increased (Fig. 17(a)). (iii) The magnitude of rotation of the structure increases, by more than when the neighbouring structure is identical (Fig. 18(a)). It rotates towards its neighbour (positive rotation if on the left of the pair and negative rotation if on the right, Fig. 17(a)). (c) Insights for the case where a structure is situated next to a taller neighbour (having higher natural period and bearing pressure). (i) SSSI appears to increase drift in larger earthquakes but reduce it in smaller earthquakes compared to the isolated case. Irrespective of earthquake strength, however, drift is larger than the case when the neighbouring structure is identical (Fig. 16). (ii) Settlement is reduced (Fig. 17(b)). (iii) The magnitude of rotation of the structure increases, by less than when the neighbouring structure is identical (Fig. 18(b)). It rotates away from its neighbour (negative rotation if on the left of the pair and positive rotation if on the right, Fig. 17(b)).

ACKNOWLEDGEMENTS This work was supported by the Engineering and Physical Sciences Research Council (EPSRC) under grant no. EP/ H039716/1, which the authors acknowledge with thanks. The authors would additionally like to express their sincere gratitude to Mark Truswell and Colin Stark at the University of Dundee for their assistance in performing the centrifuge tests.

NOTATION ag B Ct Cu Cz ck cm c9 Dr D10 D30 D60 Eoed Eur E50

These conclusions should not be considered to be general, as there is a need to perform further simulations with different types of structure (particularly to investigate the effect of different building widths and foundation types) and on different types of ground to demonstrate generality. However, they do demonstrate that the presence nearby of even a single adjacent structure can have a dramatic effect on a structure’s seismic response compared to a consideration of the same structure and underlying ground in isolation. These effects appear to be either beneficial or detrimental, depending on the relative dynamic properties of the adjacent structures and strength of the earthquake (i.e. soil response). This suggests that through further study it may be possible to exploit the beneficial effects of SSSI and avoid the detrimental ones to improve the seismic performance of the built environment.

EI emax emin FSv fn Gs G(0) g H Keq K0 Meq Mw m Nª p9 q Rf Se

CONCLUSIONS This paper has examined how the performance of a simple structure is altered when it is situated close to an adjacent structure, as a first step towards a better understanding of the seismic response of densely packed urban areas. Dynamic centrifuge modelling was conducted such that the full non-linear behaviour of the soil could be incorporated into the SSSI. This generated a database of performance data against which non-linear finite-element models were validated. The importance of both generalised or soil-specific material properties and the initial geometric configuration of the structure (initial conditions) was investigated, and it was demonstrated that accurate simulations could be achieved so long as soil-specific material properties can be determined and the initial conditions are known. (This would require building surveys for field application and laboratory testing of soils to generate site-specific soil property calibrations.) The finite-element approach was subsequently used to investigate the effects of the presence of an adjacent structure of

S(FF,NF) s sª Tn0 ª �s,0 .7 �ur � �9p ł9

112


peak ground acceleration at bedrock/input footing width (in plane of shaking) empirical period determination factor coefficient of uniformity coefficient of curvature stiffness-proportional Rayleigh damping coefficient mass-proportional Rayleigh damping coefficient cohesion intercept relative density particle diameter at which 10% is smaller particle diameter at which 30% is smaller particle diameter at which 60% is smaller oedometric tangent stiffness (in compression) unloading–reloading stiffness triaxial secant stiffness (at 50% of deviatoric failure stress in drained triaxial compression) elastic bending stiffness maximum void ratio minimum void ratio static vertical factor of safety natural frequency specific gravity of soil grains (small strain) shear modulus acceleration due to gravity (¼ 9 .81 m/s2) height equivalent lateral sway stiffness coefficient of lateral earth pressure at rest equivalent mass moment magnitude power-law index for stress-dependency of stiffness footing bearing capacity factor mean effective stress bearing pressure deviatoric failure ratio spectral acceleration Eurocode 8 equivalent soil factor (free-field, near-field) footing spacing (centre-to-centre) footing shape factor fundamental natural period soil unit weight (dry) shear strain at G/G0 ¼ 0 .7 Poisson ratio (unload–reload) equivalent viscous damping (secant) peak angle of friction dilation angle

SEISMIC STRUCTURE–SOIL–STRUCTURE INTERACTION BETWEEN BUILDING STRUCTURES REFERENCES

BSI (2004). BS EN 1997-1:2004: Eurocode 7: Geotechnical design – Part 1: General rules. British Standards Institution, London, UK. BSI (2005). BS EN 1998-1:2005: Eurocode 8: Design of structures for earthquake resistance – Part 1: General rules, seismic actions and rules for buildings. British Standards Institution, London, UK. Goel, R. K. & Chopra, A. K. (1997). Period formulas for momentresisting frame buildings. J. Structl Engng 123, No. 11, 1454– 1461. Lauder, K. (2011). The performance of pipeline ploughs. PhD thesis, University of Dundee, UK. Lyamin, A. V., Salgado, R., Sloan, S. W. & Prezzi, M. (2007). Twoand three-dimensional bearing capacity of footings in sand. Geótechnique 57, No. 8, 647–662, http://dx.doi.org/10.1680/ geot.2007.57.8.647. Lysmer, J. & Kuhlmeyer, R. L. (1969). Finite dynamic model for infinite media. ASCE J. Engng Mech. Div. 95, No. 4, 859–887. Muir Wood, D. (2004). Geotechnical modelling. Abingdon, Oxfordshire, UK: Spon. Padroń, L. A., Azna´rez, J. J. & Maeso, O. (2009). Dynamic structure–soil–structure interaction between nearby piled buildings under seismic excitation by BEM–FEM model. Soil Dynamics Earthquake Engng 29, No. 6, 1084–1096. Salgado, R. (2008). The engineering of foundations. New York, NY, USA: McGraw-Hill. Stewart, J. P., Seed, R. B. & Fenves, G. L. (1999). Seismic soilstructure interaction in buildings, Part II: Empirical findings. J. Geotech. Geoenviron. Engng 125, No. 1, 38–48. Tsogka, C. & Wirgin, A. (2003). Simulation of seismic response in an idealized city. Soil Dynam. Earthquake Engng 23, No. 5, 391–402.

Al-Defae, A. H., Caucis, K. & Knappett, J. A. (2013). Aftershocks and the whole-life seismic performance of granular slopes. Geótechnique 63, No. 14, 1230–1244, http://dx.doi.org/10.1680/ geot.12.P.149. Alexander, N. A., Ibraim, E. & Aldaikh, H. (2013). A simple discrete model for interaction of adjacent buildings during earthquakes. Comput. Structs 124, 1–10. Benz, T. (2006). Small-strain stiffness of soils and its numerical consequences. PhD thesis, University of Stuttgart, Germany. Bertalot, D. (2012). Behaviour of shallow foundations on layered soil deposits containing loose saturated sands during earthquakes. PhD thesis, University of Dundee, UK. Bertalot, D., Brennan, A. J., Knappett, J. A., Muir Wood, D. & Villalobos, F. A. (2012). Use of centrifuge modelling to improve lessons learned from earthquake case histories. Proceedings of the 2nd European conference on physical modelling in geotechnics, Eurofuge 2012, Delft, the Netherlands. Betti, R. (1997). Effects of the dynamic cross-interaction in the seismic analysis of multiple embedded foundations. Earthquake Engng Structl Dynam. 26, No. 10, 1005–1019. Brennan, A. J., Knappett, J. A., Bertalot, D., Loli, M., Anastasopoulos, I. & Brown, M. J. (2014). Dynamic centrifuge modelling facilities at the University of Dundee and their application to studying seismic case histories. In Proceedings of the 8th international conference on physical modelling in geotechnics (eds C. Gaudin and D. J. White), pp. 227–233. London, UK: Taylor & Francis Group. Brinkgreve, R. B. J., Engin, E. & Engin, H. K. (2010). Validation of empirical formulas to derive model parameters for sands. In Numerical methods in geotechnical engineering (eds T. Benz and S. Nordal). Rotterdam, the Netherlands: CRC Press/Balkema.

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Heron, C. M. et al. (2015). Geótechnique 65, No. 5, 442–451 [http://dx.doi.org/10.1680/geot.SIP.15.P.020]

A new macro-element model encapsulating the dynamic moment–rotation behaviour of raft foundations C . M . H E RO N , S . K . H A I G H † a n d S . P. G . M A DA B H U S H I †

The interaction of shallow foundations with the underlying soil during dynamic loading can have both positive and negative effects on the behaviour of the superstructure. Although the negative impacts are generally considered within design codes, seldom is design performed in such a way as to maximise the potential beneficial characteristics. This is, in part, due to the complexity of modelling the soil– structure interaction. Using the data from dynamic centrifuge testing of raft foundations on dry sand, a simple moment–rotation macro-element model has been developed, which has been calibrated and validated against the experimental data. For the prototype tested, the model is capable of accurately predicting the underlying moment–rotation backbone shape and energy dissipation during cyclic loading. Utilising this model within a finite-element model of the structure could potentially allow a coupled analysis of the full soil–foundation–structure system’s seismic response in a simplified manner compared to other methods proposed in literature. This permits the beneficial soil–structure interaction characteristics, such as the dissipation of seismic energy, to be reliably included in the design process, resulting in more efficient, cost-effective and safe designs. In this paper the derivation of the model is presented, including details of the calibration process. In addition, an appraisal of the likely resultant error of the model prediction is presented and visual examples of how well the model mimics the experimental data are provided. KEYWORDS: centrifuge modelling; dynamics; earthquakes; footings/foundations; soil/structure interaction

protection, a comprehensive model of the soil–foundation interaction behaviour is required. Previous researchers have investigated different methods of incorporating these beneficial characteristics into the design process. Ultimately all methods have to be incorporated into a numerical model of the overall soil–foundation– structure system. This can be done either by detailed numerical simulations of the soil behaviour (Abate et al., 2010; Gelagoti et al., 2012), by equivalent simplified springs (Raychowdhury & Hutchinson, 2009; Wotherspoon & Pender, 2010; Anastasopoulos & Kontoroupi, 2014) or by using macro-element models (Paolucci et al., 2007; Grange et al., 2009; Chatzigogos et al., 2011). The latter will form the focus of this paper. The use of a macro-element simplifies the modelling process by reducing the large number of elements and non-linear relationships (such as the elastoplastic behaviour of the soil and non-elastic soil–structure interface movements) down to a single element which combines these non-linearities into one constitutive law. In order for such a model to be accepted for use in design, it needs to be fully validated against physical modelling and/or field data. Despite soil–foundation interaction being capable of providing seismic protection to a variety of foundation types, this research focuses on shallow raft foundations located on dry sand beds. Paolucci et al. (2007) made use of data from 1g pseudodynamic tests and tests conducted on a large 1g shaking table to compare against predictions from a macro-element model. The variations between the physical model and numerical analysis results are similar in both cases in that the numerical analysis over-predicts permanent rotations and under-predicts settlements compared to the physical model results. The authors note that the error in the numerical analysis increases with the magnitude of excitation. The results presented in their paper do, however, have one of the best correlations between experimental and numerical data presented in the literature.

INTRODUCTION Performance-based seismic design, whereby a system is designed based on deformation limits rather than load limits, offers the potential for safe economical design in seismic engineering. Permitting a certain amount of ductility in a system minimises the cost of the structure when designing for extreme but infrequent load cases such as earthquakes. This in turn allows for a more efficient design and construction procedure. With the ability to accurately model the behaviour of manufactured structural elements, this design philosophy has to date been utilised extensively in structural designs. Examples of such a method are the introduction of ductility in the design of beams and columns of a steelframe structure or the use of unbonded tendons to give ductile behaviour of beam–column joints (Holden et al., 2003; Pampanin, 2005; Ou et al., 2010; Smith et al., 2011). With designs being led by structural engineers combined with a perceived or real lack of ability to characterise the seismic response of the soil, ductility has not been widely included into the design of the soil–foundation system. However, recent research (Gajan & Kutter, 2008; Anastasopoulos et al., 2010; Pender, 2010; Gelagoti et al., 2012) has increasingly shown the potential merits of doing so. Utilising ductility within the underlying ground can potentially reduce the cost of the overall system and can provide significant levels of seismic protection. In order to optimise designs and provide quantitative assessments of the level of seismic

Manuscript received 4 April 2014; revised manuscript accepted 17 February 2015. Published online ahead of print 15 April 2015. Discussion on this paper closes on 1 October 2015, for further details see p. ii. Department of Civil Engineering, University of Nottingham, Nottingham, UK. † Department of Engineering, University of Cambridge, Cambridge, UK.

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HERON, HAIGH AND MADABHUSHI of allowing the response to simple motions to be analysed and understood prior to examining the behaviour when more complex earthquake traces are used. This does, however, limit the extent to which the model developed in this paper can be validated for use with real earthquake motions. The sand and foundations were contained within a rigid container with a Perspex window, allowing the movements of the foundation and soil to be imaged at a high rate during the test. To reduce the effect of the rigid boundary conditions, a plastic material, Duxseal (Steedman & Madabhushi, 1991), was used to limit the amount of energy which was reflected from the container walls. In addition to high-speed photography, an extensive instrumentation array, consisting of miniature piezoelectric accelerometers and micro-electromechanical system (MEMS) accelerometers, was used to monitor the response of the ground, foundation and superstructure.

Similarly, Grange et al. (2009) present a macro-element model which is validated against data from a large-scale 1g shaking table test. In this case, the model predicted the rotations of the structure relatively accurately; however, it over-predicted settlement by approximately 50%. This is contrary to the results presented by Paolucci et al. (2007) where the model under-predicted the settlements. The model presented by Gajan & Kutter (2009) tracks the geometry of the soil–foundation interface and determines the loading on the foundation, thus the moment, rotation and settlement are evaluated simultaneously. Six user-defined parameters are required, as well as nine non-user-defined parameters, these having been back-calibrated against centrifuge data. Gajan & Kutter (2009) present several comparisons between the predictions made by the contact interface model and experimental data obtained from pseudo-dynamic centrifuge tests – with moments, shear forces and settlements all being predicted accurately. However, despite performing true-dynamic centrifuge testing, the authors did not present any comparisons between the model predictions and these data; such a validation is vital in order for the model to be adopted. In addition, the large number of user and non-user-defined parameters adds to the complexity of implementing this model for alternative prototype scenarios and reduces confidence that the prediction can be extrapolated beyond the precise situation studied – 15 independent parameters allowing almost any behaviour to be replicated with appropriate values chosen. Although there have been numerous other macro-elements presented in literature, with varying degrees of validation and capabilities, the three outlined above are prominent in the field and exemplify the challenges of developing such models. The majority of published models have to be questioned because they have been calibrated using data from tests in which the soil stress-state and loading were not accurately replicated. Ideally, a simple model, rigorously calibrated and validated using data from tests which represent the prototype stress-state, would be available to practising engineers to use in the seismic design of foundations. In this paper a simplified macro-element model developed based on fundamental geotechnical principles will be presented. The calibration and validation of the model against a collection of centrifuge data, collected from true-dynamic testing of a raft foundation located on dry sand, will also be shown, concluding with a realistic appraisal of the new model, its capabilities and limitations. The model consists of two separate components; a backbone curve which forms the overall shape of the moment–rotation cycles and an energy dissipation component which converts the model backbone into a fully developed moment–rotation cycle. Prior to presentation of the model, the experimental programme used to derive and validate the model will be described.

Test details The structure tested was a single-degree-of-freedom (when the base is fixed) stiff structure with a high centre of gravity and was located on a shallow raft foundation. The foundation was located on dry Hostun HN31 sand (Flavigny et al., 1990) prepared to loose and dense states using an automated sand pourer (Zhao et al., 2006). A typical model layout is shown in Fig. 1. Table 1 summarises details of the structure used and Table 2 summarises the tests conducted. Included in Table 1 is the vertical static factor of safety for each test, which was calculated assuming a Coulomb soil, a rough foundation–soil interface (Davis & Booker, 1971) and a shape correction factor as detailed in Eurocode 7 (BSI, 2004). The pseudo-dynamic factor of safety for each test is

Model structure

Model container

High-speed camera

EXPERIMENTAL PROGRAMME Data from two dynamic centrifuge tests, incorporating numerous excitations, were used to develop the model described in this paper. The tests were conducted using the 10-m diameter Turner beam centrifuge (Schofield, 1980), which was operated such that the g-level in the region of the foundation was 44g. A stored angular momentum (SAM) actuator (Madabhushi et al., 1998) was used to subject the models either to constant frequency, constant amplitude sinusoidal ground motions or to subject them to a constant displacement and decreasing frequency ground motion (referred to as a sine-sweep). The amplitude, frequency and duration of each shake can be adjusted in-flight, allowing a range of motions to be tested. Subjecting the models to constant frequency and amplitude motions has the advantage

500 mm

240 mm

(a)

Accelerometer (b)

Fig. 1. Model layout: (a) photograph of model set-up on centrifuge; (b) schematic diagram of instrument and structure layout

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DYNAMIC MOMENT–ROTATION BEHAVIOUR OF RAFT FOUNDATIONS by the difference in displacement between the two vertical accelerometers on either side of the foundation divided by the distance between the instruments. The moment was taken as the overturning force provided by the inertial acceleration of the structure mass multiplied by the distance between the base of the foundation and the centre of gravity of the structure. The moment contribution from the rotational inertial component was found to be negligible compared to the overturning moment and was therefore not included in the production of the plots presented in this paper.

Table 1. Details of model structure (model scale values in brackets) Bearing pressure: kPa Fixed base natural frequency: Hz Height of centre of gravity: m Base width: m Overall height: m Length: m Construction material Static vertical factor of safety (loose soil) Static vertical factor of safety (dense soil)

82 (82) Stiff: . 9 (. 400) 2 .4 (0 .054) 2 .2 (0 .050) 4 .0 (0 .092) 9 .4 (0 .214) – (Steel) 7 .2 7 .8

DEVELOPMENT OF THE BACKBONE MODEL Fundamentally, the soil beneath the foundation is experiencing a load–unload cycle as the foundation rocks during seismic loading, with the load being provided by the moment loading transmitted to the ground by the foundation. It is intuitive therefore initially to consider an established model for the stress–strain behaviour of soil, such as that developed by Oztoprak & Bolton (2013) (equation (1)). Utilising a mobilisable strength design framework, as proposed by Osman & Bolton (2005), the applied moment from the foundation can be considered to induce a representative shear stress in the deformation mechanism within the soil, �rep. The rotation can also be shown to be compatible with a certain magnitude of shear strain in the mechanism, ªrep. Equation (1) can then be used to link these representative stresses and strains and hence to link moments to rotations. If the stresses and strains are assumed to be proportional to the moments and rotations, respectively, as shown in equations (2) and (3), combining equations (1), (2) and (3) allows the formulation of an overall relationship between moment and rotation, as shown in equation (4). The choice of values for the R and S proportionality constants will be the main focus of this section.

detailed in Table 2 and was calculated using the approach presented by Butterfield & Gottardi (1994), as previously implemented for a similar purpose by Loli et al. (2014). Data processing The results presented in this paper are derived using data collected from the MEMS accelerometers located on the structure, with quoted base input accelerations being determined from the piezoelectric accelerometer attached to the outside base of the model container. The data were initially processed by filtering out high-frequency noise using an eighth-order Butterworth filter with a cut of frequency of 400 Hz. The MEMS accelerometers used have inbuilt filters at 400 Hz, hence no data were eliminated. Although the fixed base natural frequency of the stiff structure exceeded 400 Hz, negligible internal deformation would have occurred during these tests due to the excitation frequency (50 Hz) being significantly below the natural frequency and therefore no information is lost by not examining frequencies above 400 Hz. To obtain the experimental moment–rotation behaviour, the acceleration data were double integrated and then high-pass filtered above 10 Hz in order to remove the accumulation of displacement error created by the integration process. These calculated displacements were validated against displacements obtained from imaging (particle image velocimetry (PIV)) techniques. The rotation was evaluated

G ¼ G0

1 ª � ªe a 1þ ªr

(1)

�rep ¼ R 3 M

(2)

Table 2. Details of tests conducted (model scale values in brackets) Test name

Relative density

CH10

50%

CH11

80%

Earthquake details EQ number

Input acceleration: g

Frequency: Hz

Duration: s

Pseudo-dynamic factor of safety

EQ01 EQ02 EQ03 EQ04 EQ05 EQ06 EQ07 EQ08 EQ01 EQ02 EQ03 EQ04 EQ05 EQ06 EQ07 EQ08 EQ09 EQ10

0 .14 (6 .0) 0 .25 (11 .0) 0 .18 (8 .0) 0 .16 (7 .0) 0 .18 (8 .0) 0 .14 (6 .0) 0 .09 (4 .0) 0 .09 (4 .0) 0 .11 (5 .0) 0 .23 (10 .0) 0 .16 (7 .0) 0 .14 (6 .0) 0 .14 (6 .0) 0 .11 (5 .0) 0 .11 (5 .0) 0 .11 (5 .0) 0 .11 (5 .0) 0 .36 (16 .0) �

1 .1 (50) 1 .1 (50) 1 .1 (50) 1 .1 (50) 1 .1 (50) 1 .1 (50) 0 .9 (40) 0 .9 (40) 1 .1 (50) 1 .1 (50) 1 .1 (50) 1 .1 (50) 1 .1 (50) 1 .1 (50) 1 .1 (50) 1 .1 (50) 0 .9 (40) 1 .1 (60) �

22 (0 .5) 22 (0 .5) 22 (0 .5) 22 (0 .5) 22 (0 .5) 22 (0 .5) 22 (0 .5) 22 (0 .5) 22 (0 .5) 22 (0 .5) 22 (0 .5) 22 (0 .5) 22 (0 .5) 22 (0 .5) 22 (0 .5) 22 (0 .5) 22 (0 .5) 440 (�10) �

2 .2 0 .8 1 .3 1 .7 1 .6 1 .9 2 .8 2 .6 2 .2 0 .9 0 .9 0 .9 0 .9 1 .4 1 .8 2 .4 1 .9 0 .9

� Indicates sine-sweep motion.

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HERON, HAIGH AND MADABHUSHI ªrep ¼ S 3 Ł

(3)

G0 (S=R)Ł � � M¼ S jŁj � ªe a 1þ ªr

The vertical stress can be split into two components; a constant component across the entire base and a linear component, which varies from zero at one end to a maximum at the other end of the foundation. It is this linear component that provides the restoring moment to the foundation. If the maximum difference in stress between the two edges of the foundation is 2rŁb, where r is a scalar to convert displacement into vertical stress (i.e. the subgrade reaction modulus), the moment about the centre of the foundation being provided by the stress distribution can be evaluated using equation (5). A further conversion factor, f, is used to convert the peak gradient of the vertical stress into a representative shear stress, as shown by equation (6). Combining equations (2), (5) and (6) allows the parameter R to be evaluated as shown by equation (7). This indicates that, at low rotation, before uplift has occurred, R is independent of the rotation and peak normal stress, only being a function of the factor converting normal stress to shear stress, f. Further comments on the choice of parameter f will be made later.

(4)

Concern may arise from the lack of cut-off in the adopted hyperbolic model; however, the peak shear strains within the soil were approximately 5% (as determined from the image analysis) and therefore it is being assumed that no cut-off is required prior to this level of induced strain. It is worthwhile to comment on the intrinsic parameters of the hyperbolic model proposed by Oztoprak & Bolton (2013). Although average values for the ªe and ªr parameters are proposed, relationships linking their value to the confining stress are also presented. Similarly, parameter a is related to the coefficient of uniformity of the sand. Values for parameters G0, a, ªe and ªr were hence calculated as presented in Table 3. The confining stress was calculated at a depth of a quarter of the foundation width which, from image analysis of the deformation mechanism, appeared to be a sensible representative depth. The small change in confining stress with changing density did not vary the values calculated for ªe and ªr, as shown in Table 3, and hence a single value can be used in the subsequent analysis. It could be postulated that the strain in the mechanism would be directly proportional to foundation rotation and hence S might be constant. Conversely, the shear stress conversion parameter, R, would be anticipated to be a function of the rotation magnitude Ł, as the mechanism changes size and shape. R will also depend on whether the foundation is in full contact with the ground (non-uplift – ‘NUL’) or has rotated sufficiently to form a gap between its bottom surface and the underlying sand (uplift – ‘UL’). In order to understand the relationship between shear stress and moment, the bearing pressure distribution beneath the foundation (and how it changes as rotation increases) needs to be considered. Turning attention first to the nonuplift case, a simple linear distribution of bearing pressures is adopted, as shown in Fig. 2. It may be postulated that a more complex pressure distribution, similar to that found under a static, rigid, shallow foundation, should be considered. Despite the static case being well documented, there is little information on the stress distribution under a foundation sited on sand being subjected to true dynamic loading, with the inertia of the foundation and soil both likely to affect the true distribution. Therefore, in the lack of a proven alternative, a linear distribution is the most sensible choice.

Constant component Gradient component: rθ2b

Vertical stress

(5)

�rep

(6)

� 3f R(Ł) ¼ ¼ 2 (7) M b With regard to the uplift case, a linear variation in the bearing pressure is again assumed in order to maintain simplicity in the overall model, as shown in Fig. 3. In this case an extra parameter, �, defining the distance to the uplift point from the centre of the foundation, has been added. The value of � can be determined by resolving forces vertically, as shown in equation (8), which results in � being a function of the foundation width (b), the nominal bearing pressure (�n), the rotation (Ł) and the subgrade reaction modulus (r). By calculating the restoring moment about the centre of the foundation, the restoring moment can be evaluated as shown in equation (9). Similarly to the NUL case, the representative shear stress is taken as a factor, f, times the variation in normal stress across the foundation (equation (10)). Combining equations (2), (9) and (10) results in the formulation for parameter R shown in equation (11). Unlike the NUL case, R is a function of r, Ł and f. � �0.5 4b� n ?:�¼ �b (8) rŁ 2b� n M¼ (2b � �) (9) 3 (10) �rep ¼ f 3 [rŁ(b þ �)] � �0.5 4b� n frjŁj � rjŁj " (11) R(Ł) ¼ ¼ � �. # M 2b� n 4b� n 0 5 3b � 3 rjŁj

2b

θ

2rŁb3 3 ¼ f 3 (2rŁb)

M¼

Fig. 2. Assumed bearing pressure distribution under foundation for non-uplift case

�

Table 3. Values taken for intrinsic model parameters Relative density p9: kPa ªe: % ªr: % G0: MPa

50%

80%

60 .0 6 .5 3 10�4 3 .7 3 10�2 75

60 .6 6 .5 3 10�4 3 .7 3 10�2 99

Vertical stress

b

Gradient component: rθ(b �)

Fig. 3. Assumed bearing pressure distribution under foundation for uplift case

118


DYNAMIC MOMENT–ROTATION BEHAVIOUR OF RAFT FOUNDATIONS rotation stiffness percentage errors. These two tests have different relative densities but the same magnitude of earthquake, with the good comparison being for dense sand. It is interesting to note that if the small-strain shear modulus (G0) calculated for the dense test was used for the loose test model then the model would predict the data more accurately. It could therefore be the case that the density was not achieved precisely for the loose test, or the previously induced earthquake had caused sufficient densification as to increase the shear modulus towards the dense sand magnitude. Fig. 5 shows the error distribution curves resulting from implementing the generalised parameters to model the tests used to obtain the parameters. Based on these distributions, with a 95% confidence level, the model predicts the peak moments to within 30% and the small-rotation stiffness to within 50%. The large stiffness error is likely to be a result of inaccurate values for the small-strain shear modulus. The model could be used to back-calculate appropriate G0 values by using the generalised model parameters and determining the best choice of G0 to best fit the model to the data. With a formulated, calibrated and validated model for the moment–rotation backbone curve it is now possible to examine how best to include energy dissipation.

The modified hyperbolic model parameter R can therefore be evaluated for different rotation magnitudes if values of r and f are known. With reference to the f parameter, it was found that when different values were implemented in the optimisation code, the other model parameters, r and S, countered the change, resulting in approximately the same overall quality of fit between the model and the data. Thus parameter f has been set to be equal to one in the following analysis, as this was found to result in the best fit after optimisation of the r and S parameters. This leaves the parameters r and S to be determined by calibration of the model against the experimental data.

Choosing optimal model parameters Having established the appropriate equations it is now possible to calibrate the model against the centrifuge data in order to determine the two unknown parameters, r and S. Owing to the non-linear response to changes in the r and S parameters, it is not possible simply to determine the ideal parameter choices for each dataset and then take an average of the obtained values. Instead a least-squares analysis based on the error between the model prediction and the experimental moment backbone curve was performed and the best choice for the r and S parameters was determined. The model assumes some uplift and hence only tests in which uplift was observed were taken for the back-calculation of the r and S parameters. It worth noting that the tests with evident uplift were only the ones with pseudo-dynamic factor of safety values of less than one (Table 2). Generalised parameter values for r and S of 4 .3 3 107 and 0 .21, respectively, were obtained from the back-calculation analysis. Figure 4 compares the data and model moment–rotation curves for two datasets: one with the greatest and one with the least error when using the generalised model parameters. Also quoted in the figure are the peak moment and small-

Stiffness error: 47% Moment error: 30%

0·1

Test: CH10 EQ2

Moment: MN m/m

0·2

DEVELOPMENT OF ENERGY DISSIPATION MODEL Damping in soils is known to be largely independent of loading frequency (Pyke, 1979) and therefore a hysteretic damping model is most suitable for including damping in the moment–rotation cycles. Hysteretic damping could be included within the model through a mathematical construct which includes a damping coefficient and an imaginary unit. The imaginary unit is required to synchronise the damping with the rotational velocity as opposed to the rotation. Alternatively, the hysteretic damping can be included through a purely analytical method as proposed by Masing (1926) and Pyke (1979). Masing proposed a set of rules which use the initial backbone load–unload curve to develop a representation of the fully damped cyclic response. The rules proposed by Masing are outlined below and are shown diagrammatically in Fig. 6 for a simple stress–strain cycle.

0

3·5 3·0

0·1

2·5 0·02

0·04

Stiffness error: 13% Moment error: 3·8%

2·0 1·5

0·1 1·0

Test: CH10 EQ2

Moment: MN m/m

0·2

0 (a)

0·02

Probability

0·2 0·04

0

0·1

0·5 0 60

Data Model

0·2 0·04

0·02

0 Rotation: rad (b)

0·02

40

20

0 Error: %

20

40

60

Moment error (normal distribution)

0·04

Stiffness error (normal distribution)

Fig. 4. Data–model moment–rotation backbone comparisons: (a) worst case; (b) best case

Fig. 5. Error distributions resulting from use of generalised model parameters

119


HERON, HAIGH AND MADABHUSHI 1·0

0·4

Step 1: Loading backbone 0·2

Moment: MN m/m

0·5

τ/τy

Step 3: Loading backbone scaled by 2 0

Step 2: Unloading backbone scaled by 2

0

0·2 0·5

0·4 0·04 1·0

5

0 γ/γy

5

0·02

0 (a)

0·02

0·04

0·02

0 Rotation: rad (b)

0·02

0·04

0·4

Fig. 6. Application of original Masing rules 0·2

2.

The shear modulus on each loading reversal assumes a value equal to the initial modulus for the initial loading curve. The shape of the unloading or reloading curve is the same as that of the initial loading curve, except that the scale is enlarged by a factor of two in both the x- and y-directions.

Moment: MN m/m

1.

The second of the original Masing rules detailed above fails to deal with asymmetrical loading, for example when there is a smaller load–unload cycle within a larger overall load–unload cycle. Such a loading pattern results in accumulation of shear stress beyond the yield shear stress. Pyke (1979) proposed modifications to the original Masing rules in order to deal with such scenarios. These modifications would need to be considered when real earthquake motions are modelled; however, for the constant amplitude ground motions used during the experimental testing presented in this paper, the original Masing rules are sufficient. Direct implementation of the Masing rules to the moment–rotation backbone results in a cycle as shown in Fig. 7(a). Although significant damping has been included, as shown by the area enclosed within the loops, the cycles no longer follow the typical shape of the moment–rotation curves. The Masing rules were developed for a simple load–unload cycle applied to a soil column, hence at all times during the loading and unloading cycle there will be a geometrically identical mechanism at work. As discussed previously, the moment– rotation loading consists of two mechanisms; pre-uplift and post-uplift. Hence the simple application of the Masing rules to the entire moment–rotation backbone curve does not result in a correct representation of the energy dissipation. As Masing rules apply to a scenario in which a consistent mechanism acts during loading and, given the two mechanisms (non-uplift and uplift) at work during moment–rotation loading, modifications to the rules are required to allow them to accurately model the moment–rotation cycles. The initially proposed modifications involve splitting the backbone curves at the uplift point and thus having four separate sections instead of two; non-uplift loading and unloading, and uplift loading and unloading. The moment–rotation path would now follow the listed sequence of sections: loading uplift – unloading uplift – double unloading non-uplift – unload uplift – loading uplift – double loading non-uplift (Fig. 7(b)). This set of modified rules still results in the same change in moment and rotation as the original rules;

0

0·2

0·4 0·04

Fig. 7. Application of (a) original and (b) modified Masing rules to moment–rotation backbone

however, the cycle still does not follow precisely the characteristic moment–rotation shape, as shown in Fig. 7(b). The scaling of the sections that follow a load reversal is key in defining the cycle shape. Although scaling these legs by a factor of two for simple stress–strain cycles, as originally proposed by Masing, is applicable for those situations, the complex moment–rotation behaviour being modelled and the requirement to divide the backbone into two sections necessitates further modifications to the original rules. In order to obtain a moment–rotation model more representative of the true moment–rotation behaviour, the magnitude of the scaling applied to the section after a load reversal point requires further examination. It should be noted that there is some asymmetry in the experimental data, but owing to the inherent symmetry of the model, a symmetric rotation profile is input. This results in some difference in the backbone curves but this approach is required so that when energy dissipation is added to the model, drift of the cycles along the moment or rotation axis does not occur. A peak rotation, which is the mean of the positive and negative experimental peak rotations, was used for the analysis. After thorough investigation it was found that bespoke scaling was required post load-reversal in order to model the true moment–rotation cycles with different scalars being applied in the rotation (RS) and moment (MS) directions, as shown in Fig. 8. The magnitude of moment reduction post

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DYNAMIC MOMENT–ROTATION BEHAVIOUR OF RAFT FOUNDATIONS RS values, the error between the predicted cycle and an average experimental cycle was calculated. A least-squares method was applied to determine the optimal parameters. Values of 0 .84 and 1 .07 for RS and MS, respectively, were obtained for the dense sand tests, while for loose sand, values of 0 .84 and 1 .50 were obtained. This indicates that indeed there is an effect of relative density on the overall cycles beyond its influence on the backbone curve. A larger value for MS implies a larger area within the moment– rotation cycle, indicating more energy dissipation – an expected characteristic for loose sand. Fig. 9 shows the damping errors (between the model and the data) obtained for ‘optimal choice’ and generalised parameters with the dense sand tests. Further improvements to the accuracy of the model could be made with further experimental testing, providing more data against which the model parameters can be calibrated.

δM δθ

MS δM

Moment

RS δθ

DISCUSSION AND VALIDATION OF MODEL Although Figs 5 and 9 give some indication of the success of the model, it is worthwhile to examine more closely the accuracy of the predictions made by the model. Using the generalised parameters, as summarised in Table 4, different measures of error between the model and the experimental data can be obtained as shown in Fig. 10. It should be noted that the presented generalised parameters are currently only valid for prototype scenarios similar to what was modelled during this testing programme – a rigid, shallow, raft foundation sited on dry sand and subjected to horizontal sinusoidal excitations, up to 0 .25g in magnitude, which propagate

Rotation Section

Scaling Load UL

(RS, MS)

Unload UL

(2–RS, 2–MS)

NUL

2

1·0

Fig. 8. Proposed modifications to Masing rules involving situationdependent scaling Normalised probability

0·8

load reversal is the moment scalar, MS, times the moment change across the uplift backbone section, with the same rule being applied to the rotation magnitude. Following this initial scaled uplift reversal section a double non-uplift (NUL) section follows, as this was found to follow the pattern of the data successfully. To ensure moment or rotation drift does not occur, the end point of the reversal path must coincide with the opposite end of the backbone curve. Therefore, the uplift section prior to the next reversal point is scaled by 2-RS and 2-MS. In this way, the same change in moment and rotation occurs as if the original Masing rules were followed. As can be observed in Fig. 8, the shape formed following these rules now appears much more like the expected form of a moment–rotation cycle. Therefore this set of modified Masing rules has been adopted and the optimal RS and MS values can be determined by calibration against the experimental dataset. In a similar manner to the determination of the r and S parameters for the backbone curve, the optimum values for the moment and rotation scalars can be determined by finding the combination of values that best fits the experimental data. The ability to include damping accurately through this modified Masing method relies heavily upon an accurate backbone curve being available. In order to calibrate the moment and rotation scalars reliably, the optimum values for r and S were used for each test instead of the generalised parameters, thus ensuring the highest possible accuracy of the backbone shape. Soil density will affect the amount of damping obtained and despite the backbone model taking account of density, it is diligent to examine different relative densities separately in the analysis. For a range of MS and

0·6

0·4

0·2

0

40

20

0 Error: %

20

40

Best RS, MS, r and S per test All generalised parameters

Fig. 9. Damping error distributions resulting from use of optimum and generalised backbone model parameters with obtained RS and MS values for dense tests

Table 4. Summary of generalised model parameters Parameter Subgrade reaction modulus, r Rotation-shear strain scalar, S Dense rotation scalar, RS Dense moment scalar, MS Loose rotation scalar, RS Loose moment scalar, MS

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Value 4 .3 3 107 0 .21 0 .84 1 .07 0 .84 1 .50

HERON, HAIGH AND MADABHUSHI 4

be expected to reduce as anomaly tests would not bias the overall error statistics to the same degree. In fact, further testing was performed which included a further four centrifuge tests and 45 dynamic excitations. Unfortunately during these tests an issue with the experimental set-up resulted in some restriction being applied to the free-movement of the foundation, hence these data have been excluded from the previous discussion. If an identical analysis procedure, as was described previously, is used with the data from these additional tests, the model is still shown to be effective at modelling the response, albeit with different r and S parameter values (due to the experimental issue) and an increase in the standard deviation of the errors obtained (due to the inconsistency of the experimental issue). This does add further reassurance as to the validity of the proposed model. However, further testing is still advisable, not only for the reasons outlined above but also because testing a more diverse range of prototype scenarios would allow the model to be refined and validated for use in a wider range of design cases. It is also worthwhile to examine some comparisons between the model and experimental moment–rotation cycles. Fig. 11 shows the results from the test with the minimum damping error, with an error of 3 .5%. Fig. 12, on the other hand, shows the case resulting in the largest damping error (from within the subset of tests used to calibrate the model) between the model and experimental data – with an error of 23%. This is again the test shown in Fig. 4, which showed the largest error with the generalised backbone parameters. It was, however, found that modifying the small-strain shear modulus to the value used for the dense tests corrected the error in the backbone. As shown in Fig. 12, when the smallstrain shear modulus is again increased to the same level as

Probability

3

2

1

0

100

50

0 Error: % (a)

50

100

100

50

0 Error: % (b)

50

100

Probability

1·5

1·0

0·5

0

Moment error (at peak rotation) Small-rotation stiffness error Damping error

Fig. 10. Error distributions according to different measures for: (a) a subset of tests; (b) all tests

0·15

upwards through the sand layer. The subset of six tests which exhibited uplift behaviour was used in the calibration of the final model and the error distributions using the generalised parameters from Table 4 are shown in Fig. 10(a), with the mean and standard deviation values obtained being summarised in Table 5. Therefore, the maximum expected error with a 95% confidence level would be 33%, 51% and 34% for the moment, stiffness and damping, respectively. Fig. 10(b) on the other hand shows the distribution of error resulting from applying the generalised parameters to a larger subset of tests, in which sufficiently large rotations occurred such that the model was able to be applied (ten tests), with the mean and standard deviation values being given in Table 5. As observed from the figure and the values quoted in Table 5, the errors increase by approximately three times when the entire set of tests is examined. However, it must be remembered that several of the earthquakes fired during the tests were very small and hence uplift was minimal, resulting in difficulty determining accurate values for the damping within the cycles. With further testing and an expansion of the useable dataset against which the model can be calibrated, the errors would

0·05

CH11F3E3

Moment: MN m/m

0·10

0

0·05

0·10

0·04

0·02

0 Rotation: rad

0·02

0·04

Experiment Model

Fig. 11. Comparison between experimental data and model – smallest error

Table 5. Mean and standard deviation of errors Error type

Subset of six tests Mean: %

Moment Small-rotation stiffness Damping

7 .4 7 .9 9 .2

Standard deviation: % 12 .7 21 .4 12 .5

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Subset of ten tests Mean: % 20 .8 25 .3 33 .7

Standard deviation: % 26 .7 35 .5 41 .2

DYNAMIC MOMENT–ROTATION BEHAVIOUR OF RAFT FOUNDATIONS 0·15

remembered that they were obtained from applying a model developed to capture uplift behaviour to experimental situations in which significant amounts of uplift did not always occur and hence increased comparative errors are inevitable. It has been shown, however, that further validation and calibration of the model, especially regarding the selection of small-strain shear modulus, could lead to a rigorous and easily implementable model. The ability of the model presented in this paper to model the changing mechanism under the foundation in such a simple manner, and yet still to accurately replicate experimental data acquired from true dynamic centrifuge testing, thus presents a novel contribution to this field. It should be noted that many proposed models, such as those proposed by Gajan & Kutter (2009) and Chatzigogos et al. (2011), are capable of predicting the full moment– rotation–settlement behaviour, whereas the model presented here predicts the moment–rotation behaviour only. However, this new model is comparatively simpler, with significantly fewer user-defined input parameters required. Naturally there are limitations that require further exploration in the proposed model, given that one specific prototype scenario was tested; these include, for example, how the rigidity of the foundation affects the accuracy of the model. In addition, experimental data on the stress distribution under a rocking foundation would be useful to further refine the relationship between applied moment and representative shear stress.

0·05

CH10F2E2

Moment: MN m/m

0·10

0

0·05

0·10

0·04

0·02

0 Rotation: rad

0·02

0·04

Experiment Model Model with higher G0

Fig. 12. Comparison between experimental data and model – largest error

that used for the dense sand tests, the model very accurately predicts the peak moment and energy dissipation. This highlights the importance of accurate evaluation of the smallstrain shear modulus when implementing the model described in this paper. As discussed in the introduction to this paper, there have been numerous other macro-element models proposed by researchers striving to encapsulate the moment–rotation response of shallow foundations. The majority of these models, including the three presented earlier (Paolucci et al., 2007; Gajan & Kutter, 2009; Grange et al., 2009), are validated from data acquired from tests performed at an incorrect stress level and/or without true dynamic loading being applied. For example, Paolucci et al. (2007) present moment–rotation cycles obtained from small-scale tests performed at 1g with pseudo-dynamic loading being applied to a square shallow pad foundation. The peak magnitude of rotation applied to the foundation was 3 mrad, compared to a rotation magnitude of around 20 mrad recorded during the experimental programme described in this paper. As Paolucci et al. (2007) did not subject the foundations to substantial rotation, uplift did not occur and therefore only the almost linear non-uplift behaviour is observed. Although the numerical model prediction successfully mimics the experimental data, it is unclear what would happen if the model were to be used for situations with larger magnitudes of rotation when uplift does occur. Similarly, Chatzigogos et al. (2011) develop a theoretically rigorous model and provide comparisons against numerical simulations, other proposed macroelement models and two different sets of data, with generally favourable comparisons being presented. However, the datasets used for validation are again from testing in which the complicated nature of true dynamic loading was not considered and hence the validation is thus far limited. A particular strength of this new model arises from the fact that it was calibrated using true dynamic centrifuge data and thus questions regarding differences between the model and the real design scenario are avoided. The data used for the comparisons presented in this paper have not been specially selected to show particularly favourable correlations; instead an open appraisal of the model has been presented, at least for the prototype examined. Errors such as those presented in Fig. 10 might initially cause concern; however, it must be

CONCLUSIONS In this paper a model for the moment–rotation behaviour of shallow raft foundations located on dry sand beds and subjected to medium-sized seismic excitations has been developed and validated. This is intended to be included as a macro-element within an overall numerical model of the entire soil–foundation–structure system. Appropriate simplifying assumptions have been made such that the final model did not become overly complex, a key element in the novelty of this model. Even with these simplifying assumptions the model was found to be able to reliably predict the observed experimental behaviour obtained from centrifuge testing, provided the small-strain shear modulus could be accurately determined. The peak moments and energy dissipated were replicated reliably with a maximum damping error of around 20%. The ability of such a simplified model to perform reliably potentially paves the way, following validation against a wider range of prototypes, for it to be included within an appropriate model of an overall soil– structure–foundation system.

ACKNOWLEDGEMENTS The authors would like to acknowledge the collaborative and financial support received through the European Community’s Seventh Framework programme (FP7/2007–2013) under grant agreement number 227887 (SERIES – Seismic Engineering Research Infrastructures for European Synergies). The support from the staff at the Schofield Centre, University of Cambridge is also gratefully acknowledged.

NOTATION a b f G G0 g

dimensionless parameter from Oztoprak & Bolton (2013) half footing width model parameter – vertical to shear stress ratio shear modulus small strain shear modulus acceleration due to gravity (taken to be 9 .81 ms2)

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HERON, HAIGH AND MADABHUSHI Holden, T., Restrepo, J. & Mander, J. B. (2003). Seismic performance of precast reinforced and prestressed concrete walls. J. Structl Engng 129, No. 3, 286–296. Loli, M., Knappett, J. A., Brown, M. J., Anastasopoulos, I. & Gazetas, G. (2014). Centrifuge modeling of rocking-isolated inelastic RC bridge piers. Earthquake Engng Structl Dynam. 43, No. 15, 2341–2359. Madabhushi, S. P. G., Schofield, A. N. & Lesley, S. (1998). A new stored angular momentum (SAM) based earthquake actuator. In Centrifuge ’98: proceedings of the international conference, ISTokyo ’98 (eds T. Kimura, O. Kusakabe and J. Takemura), pp. 111–116. Rotterdam, the Netherlands: Balkema. Masing, G. (1926). Eigenspannungen und verfestigung beim messing. In Proceedings of the 2nd international congress of applied mechanics (ed. E. Meissner), pp. 332–335. Zu¨rich, Switzerland: Oren Fu¨ssli. Osman, A. & Bolton, M. (2005). Simple plasticity-based prediction of the undrained settlement of shallow circular foundations on clay. Geótechnique 55, No. 6, 435–447, http://dx.doi.org/ 10.1680/geot.2005.55.6.435. Ou, Y.-C., Wang, P.-H., Tsai, M.-S., Chang, K.-C., & Lee, G. C. (2010). Large-scale experimental study of precast segmental unbonded posttensioned concrete bridge columns for seismic regions. J. Structl Engng 136, No. 3, 255–264. Oztoprak, S. & Bolton, M. D. (2013). Stiffness of sands through a laboratory test database. Geótechnique 63, No. 1, 54–70, http:// dx.doi.org/10.1680/geot.10.P.078. Pampanin, S. (2005). Emerging solutions for high seismic performance of precast/prestressed concrete buildings. J. Advd Concrete Technol. 3, No. 2, 207–223. Paolucci, R., Prisco, C. & Vecchiotti, M. (2007). Seismic behaviour of shallow foundations: large scale experiments vs. numerical modelling and implications for performance based design. Proceedings of the 1st US–Italy seismic bridge workshop, Pavia, Italy. Pender, M. J. (2010). Integrated earthquake resistant design of structure–foundation systems. Proceedings of the 5th international conference on recent advances in geotechnical and earthquake engineering and soil dynamics, San Diego, CA, USA, paper no. SOAP 7. Pyke, R. M. (1979). Nonlinear soil models for irregular cyclic loadings. J. Geotech. Engng Div., ASCE 105, No. 6, 715–726. Raychowdhury, P. & Hutchinson, T. (2009). Performance evaluation of a nonlinear Winkler-based shallow foundation model using centrifuge test results. Earthquake Engng Structl Dynam. 38, No. March, 679–698. Schofield, A. N. (1980). Cambridge geotechnical centrifuge operations. Geótechnique 30, No. 3, 227–268, http://dx.doi.org/ 10.1680/geot.1980.30.3.227. Smith, B. J., Kurama, Y. C. & McGinnis, M. J. (2011). Design and measured behavior of a hybrid precast concrete wall specimen for seismic regions. J. Structl Engng 137, No. 10, 1052–1062. Steedman, R. S. & Madabhushi, S. P. G. (1991). Wave propagation in sand medium. Proceedings of the 4th international conference on seismic zonation, Stanford, CA. Oakland, CA, USA: Earthquake Engineering Research Institute. Wotherspoon, L. & Pender, M. (2010). Effect of uplift modelling on the seismic response of shallow foundations. Proceedings of the 5th international conference on recent advances in geotechnical and earthquake engineering and soil dynamics, San Diego, CA, USA, paper no. 5 .12a. Zhao, Y., Gafar, K., Elshafie, M. Z. E. B., Deeks, A. D., Knappett, J. A. & Madabhushi, S. P. G. (2006). Calibration and use of a new automatic sand pourer. In Physical modelling in geotechnics: proceedings of the 6th international conference on physical modelling in geotechnics (eds C. W. W. Ng, Y. H. Wang and L. M. Zhang), pp. 265–270. Boca Raton, FL, USA: CRC Press.

moment model parameter – moment scalar effective mean normal stress model parameter – shear stress scalar model parameter – rotation scalar subgrade reaction modulus model parameter – shear strain scalar distance from centre of footing to uplift point shear strain strain parameter from Oztoprak & Bolton (2013) strain parameter from Oztoprak & Bolton (2013) representative shear strain level within the soil deformation mechanism ªy yield shear strain �Ł change in rotation �M change in moment Ł rotation �n nominal bearing pressure � shear stress �rep representative shear stress level within the soil deformation mechanism �y yield shear stress

M MS p9 R RS r S � ª ªe ªr ªrep

REFERENCES Abate, G., Massimino, M. R., Maugeri, M. & Muir Wood, D. (2010). Numerical modelling of a shaking table test for soil– foundation–superstructure interaction by means of a soil constitutive model implemented in a FEM code. Geotech. Geol. Engng 28, No. 1, 37–59. Anastasopoulos, I. & Kontoroupi, T. (2014). Simplified approximate method for analysis of rocking systems accounting for soil inelasticity and foundation uplifting. Soil Dynam. Earthquake Engng 56, 28–43. Anastasopoulos, I., Gazetas, G., Loli, M., Apostolou, M. & Gerolymos, N. (2010). Soil failure can be used for seismic protection of structures. Bull. Earthquake Engng 8, No. 2, 309–326. BSI (British Standards Institution) (2004). BS EN 1997-1:2004: Eurocode 7. Geotechnical Design. General Rules. London, UK: British Standards Institution. Butterfield, R. & Gottardi, G. (1994). A complete three-dimensional failure envelope for shallow footings on sand. Geótechnique 44, No. 1, 181–184, http://dx.doi.org/10.1680/geot.1994.44.1.181. Chatzigogos, C., Figini, R., Pecker, A. & Salencon, J. (2011). A macroelement formulation for shallow foundations on cohesive and frictional soils. Int. J. Numer. Analyt. Methods Geomech. 35, May, 902–931. Davis, E. H. & Booker, J. R. (1971). The bearing capacity of strip footings from the standpoint of plasticity theory. Proceedings of the 1st Australian–New Zealand conference in geomechanics, Melbourne, pp. 276–282. Sudney, Australia: Institution of Engineers Australia. Flavigny, E., Desrues, J. & Palayer, B. (1990). Le sable d’Hostun. Rev. Française Geótechnique 53, 67–70 (in French). Gajan, S. & Kutter, B. L. (2008). Capacity, settlement, and energy dissipation of shallow footings subjected to rocking. J. Geotech. Geoenviron. Engng 134, No. 8, 1129–1141. Gajan, S. & Kutter, B. L. (2009). Contact interface model for shallow foundations subjected to combined cyclic loading. J. Geotech. Geoenviron. Engng 135, No. 3, 407–419. Gelagoti, F., Kourkoulis, R., Anastasopoulos, I. & Gazetas, G. (2012). Rocking isolation of low-rise frame structures founded on isolated footings. Earthquake Engng Structl Dynam. 41, No. 7, 1177–1197. Grange, S., Kotronis, P. & Mazars, J. (2009). A macro-element to simulate dynamic soil–structure interaction. Engng Structs 31, No. 12, 3034–3046.

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Bilotta, E. et al. (2015). Geótechnique 65, No. 5, 391–400 [http://dx.doi.org/10.1680/geot.SIP.15.P.016]

Importance of seismic site response and soil– structure interaction in the dynamic behaviour of a tall building founded on piles E . B I L OT TA � , L . D E S A N C T I S † , R . D I L AO R A † , A . D ’ O N O F R I O � a n d F. S I LV E S T R I �

A tall public building in Naples (Italy) has recently undergone a seismic vulnerability assessment, following the new Italian code requirements. The building is about 100 m high and is founded on a piled raft floating in a thick layer of soft pyroclastic and alluvial soils. On the basis of a conventional subsoil classification, the inertial seismic actions on the building would lead to expensive measures for seismic retrofitting. By contrast, if site effects and soil–structure interaction are adequately addressed the picture is completely different. First, free-field seismic response analyses highlighted the beneficial effects of a peat layer, acting as a natural damper on the propagation of shear waves. Finiteelement analyses of pile–soil kinematic interaction were then carried out to define the foundation input motion, which was found not to be significantly affected. The effects of inertial interaction were evaluated accounting for soil–foundation compliance; they resulted in an increase of the structural period of vibration, while the overall damping did not change compared to that of the fixed-base structure. The increased structural period led to further reduction of spectral acceleration. The results could lead to significant impacts on the seismic assessment of slender buildings founded on piles embedded in deformable soils. KEYWORDS: dynamics; footings/foundations; piles; soil/structure interaction

(a) assessment of the seismic input on the foundation accounting for kinematic interaction (FIM ¼ foundation input motion) (b) calculation of the dynamic impedance functions associated with vertical and horizontal translation, as well as with torsional rotation and rocking (c) analysis of the inertial interaction of the building subjected to the FIM and supported by visco-elastic springs, characterised by the impedance functions determined above.

INTRODUCTION Soil–structure interaction (SSI) may be an important issue in the assessment of the seismic vulnerability of a building. Depending on the relative stiffness between the structure and the soil–foundation system, it is generally expected that the dynamic SSI induces a significant increase of the fundamental period of the structure and an increase of damping, thus reducing the seismic demand on the structure (e.g. Veletsos & Meek, 1974). Recent studies (e.g. Han, 2002) have shown that the seismic response of a tall building supported on a pile foundation may be difficult to predict correctly, if the complex dynamic interaction problem is not handled with care. Neglecting such interaction, for instance by modelling the tall building as having a fixed base, cannot represent the actual seismic response, since the overall stiffness of the system is overestimated and the damping is underestimated. Equally, simplifying the problem by modelling a real pile foundation as a fictitious equivalent footing leads to no better prediction. Particularly in the case of large-diameter piles, the important contribution of the foundation system to the rocking stiffness of the building would be neglected. As a consequence, too low natural frequencies and too large displacements would be calculated. In such a case, the assessment of the seismic vulnerability of the building would be inaccurate, hence expensive and likely useless retrofitting could be undertaken to meet the seismic safety requirements. An adequate procedure to consider the soil–foundation– building interaction is based on the substructures method (Gazetas, 1984; Makris et al., 1996; Mylonakis et al., 1997) and it is implemented by subdividing the analysis into three different stages

In this paper stages (a) and (b) are described in detail, with reference to a case study of a tall building on a pile foundation floating in a deformable subsoil (section entitled ‘Case study’). The seismic actions on the building are calculated in terms of response spectra by free-field seismic response (section entitled ‘Seismic site response’) and kinematic interaction (section entitled ‘Foundation input motion’) analyses. The numerical calculation procedure of the (six-components) impedence matrix and the relevant modifications of the spectral ordinates due to the changes of the natural frequencies and the overall damping are finally assessed (section entitled ‘Influence of pile–foundation compliance’). CASE STUDY The analysed building, located in the eastern area of Naples (Italy), is a 29-storey reinforced concrete tower, with a height of 107 .4 m, built in the early 1980s. The tower, with a stiffening core, is rigidly connected to a pile foundation by a reinforced concrete box structure, made up by a lower raft of thickness up to 1 m and an upper 40 cm slab, joined by vertical reinforced concrete walls 6 m high. The 82 piles are unevenly distributed on a large area of 3300 m2 (Fig. 1); they were drilled in alluvial and volcanic soils with a length of 42 m and a diameter varying between 1800 mm and 2200 mm (Viggiani & Vinale, 1983; Mancuso et al., 1999). The reconstruction of the subsoil layering was based on the results of boreholes and cone penetration tests (CPTs)

Manuscript received 1 April 2014; revised manuscript accepted 11 March 2015. Discussion on this paper closes on 1 October 2015, for further details see p. ii. � University of Napoli Federico II, Naples, Italy. † University of Napoli Parthenope, Naples, Italy.

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BILOTTA, DE SANCTIS, DI LAORA, D’ONOFRIO AND SILVESTRI actions on the building was hence evident; therefore, the seismic site response (SSR) in ‘free-field’ conditions was analysed. Also, the complex SSI involving the building, the ground and the pile foundation was recognised as deserving of greater insight. The former (SSR) aspect will be summarised briefly in the next section, whereas the latter (SSI) will be investigated in greater detail in the later sections entitled ‘Foundation input motion’ and ‘Influence of pile– foundation compliance’.

d 1·8 m

N

d 2·0 m d 2·2 m

SEISMIC SITE RESPONSE On the basis of the available data from geotechnical investigation, a regular layering, characterised as shown in Fig. 2, was adopted to carry out one-dimensional SSR analyses with the linear equivalent approach in the frequency domain, by using the code EERA (Bardet et al., 2000). The decay of normalised shear modulus, G/G0, and the variation of the damping ratio, D, with the shear strain, ª, were defined (Fig. 3) by resonant column tests carried out on undisturbed specimens of pozzolana (Vinale, 1988) or based on data from literature for the other soils. Peat behaviour was characterised by using experimental data reported by Wehling et al. (2003). A visco-elastic bedrock was assumed at 60 m depth, with shear wave velocity Vsb ¼ 800 m/s and damping ratio D ¼ 0 .5% assigned to the stiff ‘A’ formation, based on preliminary analyses (Bilotta et al., 2013b) showing that the amplification function of the subsoil was largely independent of any reasonable assumption about the variability of the VS below 60 m. Seven natural accelerograms were extracted from the European strong motion database (ESD), through the software Rexel 3 .5 (Iervolino et al., 2009), compatible with the spectrum specified by the code for the life safety limit state criteria. Fig. 4 shows with thin lines the response spectra associated with each accelerogram, scaled to amax ¼ 0 .19g. It is worth noting that at structural periods higher than 1 s the average spectrum (thick grey line) is fully compatible with the code-specified reference spectrum for a ground type A (black line). A comparison between the profiles of the initial as opposed to the mean mobilised stiffness resulting from the one-dimensional SSR analyses is shown in Fig. 5. The response spectra calculated at the surface for each input signal (thin grey lines) and their average (thick black line) are plotted in Fig. 6. The average input spectrum (dotted line) and the code-defined spectra for any possible ground type are also shown in the figure. It may be observed that the type C and D spectral ordinates at periods higher than 2 s overestimate the mean values predicted by the SSR analyses.

Y

X 70 m

Fig. 1. Piled raft – plan view

carried out in that area, during the design and construction of the building (Vinale, 1988). The schematic east–west stratigraphic section (Fig. 2(a)) shows that the foundation subsoil profile consists of made ground (R), laying above volcanic ash (C), and pyroclastic silty sand (cohesionless pozzolana, Ps), alternating with alluvial materials (peat, T, and sand, S). Underneath, the Neapolitan yellow tuff (NYT) is replaced in some zones by weakly cemented pozzolana (Pc), which can be viewed as a weathered and weaker kind of the same soft rock. This formation rests on stiff alternating layers of ash, sand and pozzolana (A) with uncertain depth. The shear wave velocity profile, VS, shown in Fig. 2(b), is based on interpretation of cross-hole and down-hole tests, carried out in the same area down to 60 m (Vinale, 1988). Below such a depth, no direct measurements of VS were available and the profile was extrapolated to about 100 m on the basis of the results of deep CPT by means of regional correlations between qc and VS (Rippa & Vinale, 1983). According to the national technical code (NTC, 2008), adopting similar soil classification criteria as Eurocode 8 (CEN, 2003), the ground type might be classifiable between C and D, because the equivalent velocity VS,30 is about 180 m/s. In a preliminary study carried out by Bilotta et al. (2013a), linear pseudo-static finite-element method analysis of the building was performed by assuming a fixed base and by simplifying the complex structure into main seismoresistant elements modelled with one- and two-dimensional elements (‘frame’ and ‘shell’, respectively). The standard code seismic actions relevant to the lifesafety limit state (SLV) were first used; in this case they corresponded to a return period as high as 712 years and a peak reference acceleration of 0 .19g at this site. A parametric study of the seismic response of the building was carried out by changing the input spectra according to the different ground types. These first analyses showed that the seismic performance of the tall building was unsatisfactory when the code-specified demand spectra corresponded to the most unfavourable ground conditions, namely subsoil class C or D. Nevertheless, it is worth remembering that the standard classification criteria may be reliable only when VS continuously increases with depth: this is not the case, since a clear inversion in the velocity profile can be observed in Fig. 2(b), due to the presence of a relatively shallow layer of peat between 10 and 12 m. The need for the definition of more realistic seismic

FOUNDATION INPUT MOTION The effect of kinematic interaction between the foundation and the surrounding soil is generally of reducing the motion transmitted to the superstructure. The amount of such a reduction depends mainly on: excitation frequency, pile diameter and soil stiffness. Increasing the values of the first two parameters or decreasing the soil stiffness, will lead to higher mismatch between free field and foundation input motion. The reduction of spectral acceleration due to pile–soil kinematic interaction was expressed by Di Laora & de Sanctis (2013) with simplified formulations applicable to a single pile embedded in homogeneous and two-layer subsoil models. For the layered subsoil at hand, such simple formulations cannot provide sufficiently accurate results. There-

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IMPORTANCE OF SEISMIC SITE RESPONSE AND SOIL–STRUCTURE INTERACTION

Building

H 107 m

E

Made ground (R)

W

Volcanic ash (C) 15 30 45

Peat (T)

Sand (S)

Cohensionless pozzolana (Ps)

Neapolitan Yellow tuff (NYT)

60

Cemented pozzolana (Pc)

75

Alternating layers of ash, sand and pozzolana (A)

90 (a) 0

200

VS: m/s 400

600

800

Made ground (R) Volcanic ash (C) Peat (T) 20

Sand (S)


Depth: m


40

60

Alternating layers of ash, sand and pozzolana (A)

80

100

(b)

Fig. 2. (a) Ground conditions and (b) shear wave velocity profile

between displacements and forces at the heads of a couple of piles is expressed using flexibility coefficients, by assuming that the presence of a second pile does not affect the deformation of the loaded pile. For horizontal modes of vibration, it is assumed that the deflections of both piles are identical. Finally, the raft is assumed to be rigid and clear to the soil. The mobilised stiffness profile from SSR analyses (bold line in Fig. 5) has been adopted in the analyses. Figure 7 shows the ratio between the peak accelerations of the pile head and the free-field ground motion as a function of the loading frequency. It can be noted that, for low frequency, the foundation and the free-field motions are

fore the kinematic interaction problem has been analysed by way of numerical simulations in the frequency domain, by assuming a linear visco-elastic model for the soil and including the fixed head single pile or the pile group. To this aim, the code Dynapile 2.0 (Ensoft, 1999) has been used, based on the ‘consistent boundary matrix’ method (Kausel, 1974; Blaney et al., 1976). This is a hybrid procedure that models the soil–pile interaction through the finite-element method in the vertical direction and applies closed-form solutions along the horizontal direction. Group effects are taken into account by means of frequency-dependent interaction factors, according to the superposition approach. For vertical and rocking behaviour, the relationship

127


BILOTTA, DE SANCTIS, DI LAORA, D’ONOFRIO AND SILVESTRI 1·0

1·0

Rock input signals Average

0·8

0·8

Code spectrum, subsoil type A (NTC) 0·6

G/G0

Sa : g

0·6

0·4

0·4

0·2

0·2

0 105

104

103

102 γ: % (a)

100

101

0

101

0

25

1

2 Structural period, T: s

3

4

Fig. 4. Spectra associated with the input signals Made ground and volcanic ashes

20

Peat

0 .5 Hz, the kinematic interaction is certainly negligible. The results shown in Fig. 7 demonstrate that piles are unable to modify the free-field seismic motion, which is instead strongly affected by the presence of the peat layer. In the following the foundation input motion has been therefore assumed coincident with the free-field ground motion.

Sand Pozzolana (PS, PC)

15

D: %

Alternating layers of ash, sand and pozzolana 10

INFLUENCE OF PILE–FOUNDATION COMPLIANCE Simplified SSI model The dynamic response of a building founded on piles embedded in a deformable soil may be different from that of a similarly excited, identical structure resting on a rigid ground. The factors responsible for such a different behaviour are: (a) the flexibility of the pile–foundation system; (b) the vibrational energy dissipated by the wave radiation and by the internal soil damping. The above factors were both addressed for the case under examination. Fig. 8 shows a simple oscillator on a flexible foundation, whose dynamic compliance is modelled by two springs (K and KR) associated to translational and rotational oscillations and a pair of dashpots (c and cR) attached in

5

0 105

104

103

102 γ: % (b)

100

101

101

Fig. 3. (a) Normalised shear modulus and (b) damping ratio plotted against shear strain

coincident, whereas only at high frequencies the discrepancy is noticeable; in addition, minor group effects are observed. As a matter of fact, considering that for this specific case study the structural fundamental frequency is of the order of amax: g 0

0·1

0·2

0·3

0

G: MPa 100

200

Made ground (R) Volcanic ash (C) 10

Peat (T)

10

Mobilised stiffness from EERA Initial stiffness

20

Sand (S)


Depth: m

Back-analysis of pile load test

Depth: m


20

30

30

40

40

50

50

60

60

Fig. 5. Average profiles of initial and mobilised stiffness in SSR analyses and profile from pile load test back-analysis

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IMPORTANCE OF SEISMIC SITE RESPONSE AND SOIL–STRUCTURE INTERACTION 1·4

From the dynamic equilibrium of the replacement oscillator, the fundamental period along the i-axis, T~ i , of a building modelled as a single-degree-of-freedom (SDOF) system on a compliant base, and the associated apparent damping, ~ �i , can be expressed in the form (Veletsos & Meek, 1974; BSSC, 2004) vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! u u ki K i h2 t ~ Ti ¼ Ti 1 þ (1a) 1þ Ki K Łj !�3 ~i T ~ �i ¼ �0,i þ �ci (1b) Ti

Surface signals 1·2

Average surface signals Average rock signals

Sa: g

1·0 0·8

type D C

0·6

B A

0·4 0·2 0

0

1

2 Structural period, T: s

3

where Ti is the fundamental period of the fixed-base structure; ki is the stiffness of the fixed-base structure; h is the height of the centre of mass of the building computed from the top of the foundation plate; Ki is the horizontal stiffness of the foundation; KŁj is the rotational stiffness around the jaxis perpendicular to i; �ci is the structural damping associated with oscillations along the i-axis; �0,i is the foundation damping factor, that is, the contribution due to both radiation and hysteretic damping of the foundation system. Considering that Ti is lower than T~ i , the interaction reduces the effectiveness of the structural damping. When Ti is small compared to T~ i , the contribution of the structural damping may be significantly reduced. However, this reduction is usually compensated by the increase in the apparent damping due to the foundation. Depending on the ratio of T~ i over Ti, the apparent damping might be larger or smaller than the structural damping. The expression for �0,i supplied by Veletsos & Meek (1974) is only applicable to the case of a shallow foundation resting on an elastic halfspace and, hence, it is not suitable for the case of a piled foundation. A novel exact formulation for structures resting on generic springs and dashpots has been proposed by Maravas et al. (2007, 2014). According to this method, the damping of the overall system for vibration mode along the x-axis, for example, can be expressed by the following equations " # �X �Ł �CX ~ �X ¼ S X 2 þ þ øX (1 þ 4�2X ) ø2ŁY (1 þ 4�2ŁY ) ø2CX (1 þ 4�2CX )

4

Fig. 6. Acceleration response spectra computed at surface plotted against mean input spectrum and code-defined seismic actions for different ground types 1.25

Acceleration ratio, ap /aff

1.00

0.75

0.50 Single pile

0.25

Pile group 0

0

1

2 3 Frequency, f: Hz

4

5

Fig. 7. Ratio between foundation and free-field accelerations as a function of loading frequency

parallel to the springs. The overall system is commonly referred to as a ‘replacement oscillator’. In the same figure, the ratio between the mass acceleration, ast, and that of the free-field motion, aff, is plotted against frequency. The foundation compliance acts as a low-pass filtering device; as a result, the fundamental frequency of the replacement oscillator is shifted far apart from the natural frequency of the fixed-base structure. In addition, the energy dissipated by the piled foundation might lead to an increase of the damping ratio of the replacement oscillator, which is referred to as ‘apparent damping’; thus, a reduction of the peak acceleration corresponding to the natural frequency is usually expected. U X UR

(2a) ~ 2X ø with "

1 1 1 þ 2 þ 2 SX ¼ 2 2 2 øX (1 þ 4�X ) øŁY (1 þ 4�ŁY ) øCX (1 þ 4�2CX )

#�1 (3)

ast/aff Fixed base

Compliant base

h

KR

(2b)

U mg

K c

2 ¼ S X =(1 þ 4~ �X )

f

cR

Fig. 8. Replacement oscillator (left) and transfer functions of the fixed as opposed to the compliant base structure (right)

129


BILOTTA, DE SANCTIS, DI LAORA, D’ONOFRIO AND SILVESTRI by a total of 76 piles (Fig. 9), all having a diameter of 2 m. Such idealisation is a reasonable approximation for engineering purposes. The Dynapile analyses have been performed by assuming the same profile of the mobilised soil stiffness as for the kinematic interaction analysis (Fig. 5). The code allowed first the determination of the impedance functions of a single pile and then those of the whole pile group. The elastic properties of the soil representative of the response of single piles under vertical loads, and hence the rotational impedance, are also affected by the pile installation technique. In order to evalute the modification of soil properties induced by pile installation, the procedure suggested by Mandolini & Viggiani (1997) was adopted. Hence the small-strain stiffness of each layer was first evaluated from cross-hole and down-hole investigations (Vinale, 1988). The stiffness profiles were then normalised by the stiffness of the first layer to obtain a non-dimensional profile. Finally, the mobilised stiffness was calculated by performing iteratively an elastic analysis of the single pile, until matching the initial slope of the experimental load–settlement curve. The initial vertical stiffness of the single pile as deduced by averaging four load tests was KZ0 ¼ 2381 MN/m (Mandolini & Viggiani, 1997). Interestingly, the back-figured stiffness profile, shown in Fig. 5, is very similar to the mobilised stiffness profile deduced from free-field SSR analyses. This similarity led to the choice of the mobilised stiffness as the equivalent stiffness for the subsequent linear elastic analyses. Figure 10 illustrates the real and imaginary part of the impedance functions, (Kh( f ), Ch( f )) and (Kv( f ), Cv( f )), for a single pile for horizontal (Fig. 10(a)) and vertical (Fig. 10(b)) modes of vibration. The two components of the impedance function are plotted against frequency up to a value of 5 Hz. The vertical static stiffness, obtained by extrapolating the dynamic stiffness function to zero frequency, meets exactly the experimental initial stiffness KZ0 ¼ 2381 MN/m backfigured from loading test. The static horizontal stiffness of the fixed-head single pile is KH0 ¼ 683 MN/m. Figure 11 shows the real and imaginary parts of the impedance functions of the pile group associated with the horizontal modes of vibration along the x-axis (KX) and y-axis (KY) and with rocking modes of vibration around the y-axis (KŁY) and x-axis (KŁX). It is worth noting that the

where

rffiffiffiffiffiffiffi KX øX ¼ m rffiffiffiffiffiffiffiffi K ŁY øŁY ¼ mh2

(4) (5)

are fictitious uncoupled natural frequencies of the system under rocking and swaying oscillation of the base, and rffiffiffiffiffiffi kX (6) øCX ¼ m is the natural oscillation frequency of the undamped fixedbase structure. In equations (2) and (3), �X and �ŁY are the damping terms of the foundation in the vibrational modes along the x-axis and around the y-axis, respectively, while �CX is the damping ratio related to the horizontal motion of the structure along the x-axis. By approximating to unity the terms expressed as (1 þ 4�2 ), with � being any foundation or structural damping term, it is possible to rearrange equations (2) and (3) into !�1 � � 1 1 1 �1 1 h2 1 2 ~ ~X ¼ ø þ þ ) kX ¼ þ þ K X K ŁY k X ø2X ø2ŁY ø2CX ~�X ¼

!

k~X � þ KX X

~ kX h K ŁY

2

!

�ŁY þ

!

~ kX � k X CX

(7a)

(7b)

¼ ÆX �X þ ÆŁY �ŁY þ ÆCX �CX where ~k X is the translational stiffness along the x-axis of the overall system. The same procedure may be applied to the motion along the y-direction. Rewriting the expressions by Maravas et al. (2014) as in equations (7a) and (7b) has the advantage of offering an insight into the physics of the interaction phenomenon. The apparent damping is a linear combination of the damping ratios pertaining to the fixed-base structure, the swaying oscillation and the rocking oscillation of the foundation, weighted for the three coefficients ÆX, ÆŁY and ÆCX. It can be verified that the sum of above coefficients is 1. As a result, if damping ratios �X and �ŁY are equal to �CX, the apparent damping must be equal to �CX. For very stiff foundation systems, ÆX and ÆŁY are negligible, and the apparent damping of the replacement oscillator coincides with that of the fixed-base structure. Since stiffness and damping terms pertaining to the foundation are frequency-dependent, an iterative procedure is necessary to obtain the apparent damping of overall system. To this aim, frequencies and damping ratios corresponding to the natural circular frequency of the structure øci can be used as starting values, thereby calculating by means of ~ 2i : Thereafter, equation (7a) a new value for the frequency ø ~ 2i , until new estimation of impedances may be obtained for ø convergence. Generally, two or three iterations are sufficient to get accurate results. Finally, the value of the apparent damping may be calculated from equation (7b).

d 2·0 m

Y

X

Evaluation of translational and rotational impedance The rotational and horizontal stiffness components of the dynamic compliance have been evaluated by means of the code Dynapile 2.0 (Ensoft, 1999). The analyses have been performed by referring to a symmetric layout characterised

56·8 m

Fig. 9. Simplified pile layout considered in the analyses carried out with Dynapile

130


IMPORTANCE OF SEISMIC SITE RESPONSE AND SOIL–STRUCTURE INTERACTION 1000

The exact solution supplied by Maravas et al. (2014) has been then applied to evaluate both the modified period and the overall damping of the complete system, in order to assess the role of the radiation and hysteretic damping associated with the foundation motion. The frequency-dependent terms �i and �Łj, associated with swaying and rotational modes of vibration of the pile foundation, can be easily obtained from the impedance functions by the following expression

Single pile impedance, Kh, Ch: MN/m

Real part Imaginary part 750

500

�i ¼

250

0

C Łj (ø) �Łj ¼ 2K Łj (ø) 0

1

2 3 Frequency, f: Hz (a)

4

5

0

1

2 3 Frequency, f: Hz (b)

4

5

Single pile impedance, Kv, Cv: MN/m

(8)

where Ci(ø) and Ki(ø) are the imaginary and real parts of the dynamic stiffness associated with swaying along the i-axis, while CŁj and KŁj are those associated with rocking around the j-axis. For the case at hand, the method by Maravas et al. (2014) provides the results shown in Table 1. It would be straightforward to show that the increase in oscillation period along both directions is coincident with that evaluated by the approach proposed by Veletsos & Meek (1974); that is, by taking the static stiffness of the foundation, in agreement with the findings by Maravas et al. (2014). This implies that the adopted procedure can be applied without any iteration; in other words, simply by computing with equation (1a) the natural frequency of the replacement oscillator. Also, the contribution of the foundation motion to the overall damping is very small in comparison with that associated with the structure. For example, ÆY and ÆŁX are only about 5% and 10%, respectively, while ÆCY is 85%. Taking into account that �X and �ŁY are very close to �CX, the overall damping is practically coincident with that of the structure (i.e. ~ �X ¼ ~ �Y ¼ 5%). This is mainly due to the fact that the stiffness of the structure, kX, is small in comparison to the stiffness of the foundation.

3000

2000

1000

0

C i (ø) 2K i (ø)

Fig. 10. Impedance functions associated with (a) horizontal and (b) vertical modes of vibration for the single pile

horizontal impedance functions, KX and KY, are practically coincident. The same is true for the rotational components. The vertical and the torsional components of the stiffness matrix, KZ and KT, are not reported since they do not affect the increase of the oscillation period and the apparent damping. For both directions, the value of the real part at zerofrequency is the static horizontal stiffness, corresponding to the raft restrained against rotation, that must be introduced into equation (1) in order to evaluate the vibration period of the replacement oscillator. From the Dynapile analyses, the following are obtained: KŁY ¼ 1 .316 3 107 MNm (rotational static stiffness around y-axis), KŁX ¼ 1 .351 3 107 MNm (rotational static stiffness around x-axis), KX ffi KY ¼ 6215 MN/m (horizontal static stiffnesses around x- and y-axes). The natural periods of the fixed-base building along the two directions x and y were already calculated by Bilotta et al. (2013a). The stiffnesses ki of the fixed-base SDOF equivalent to the building were computed from the first vibration periods assuming a seismic mass equal to 0 .7W/g, with W being the weight of the building (BSSC, 2004). The height was assumed according to BSSC (2004) as h ¼ 0 .7H, with H being the total height of the building. From such values, the ratios between the fundamental periods of the building on compliant and fixed bases, T~ =T , were computed according to equation (1) along both directions. The resulting ratios equalled 1 .16 along the x-axis and 1 .08 along the y-axis, representing an appreciable increase of the structural period due to foundation compliance. A reduction of the seimic action on the central core of the building and on the foundation can be expected on the basis of such an increment.

Effect of soil–foundation compliance on seismic actions The reduction of the seismic actions achieved by taking into account the site response and the dynamic SSI in the problem at hand is presented in Fig. 12. The average spectrum, computed from the SSR accelerograms by assuming ~ �X ¼ ~ �Y ¼ 5%, is compared to the spectrum for ground type D. By considering the effects of SSR on the fixed-base structure, at the first vibration mode (TY ¼ 2 .28 s) the spectral acceleration is reduced by 47 .2% compared to that predicted by the code. An additional reduction (9 .5%) of the inertial action can be achieved by accounting for the increase in the structural period (T~ Y ¼ 2 .47 s). For the second vibration mode (TX ¼ 1 .62 s), the reduction due to SSR is equal to 28 .3%; a further significant reduction of 15 .1% is due to the deformability of the pile group. Summarising, the overall reduction of inertial action due to both seismic response and SSI is 56 .8% along the y-axis and 43 .5% along the x-axis. CONCLUSIONS A tall public building in Naples (Italy) recently underwent a seismic vulnerability assessment, following the new Italian code requirements. After a preliminary unsatisfactory evaluation based on code-specified spectra, the seismic actions were re-evaluated by a more sophisticated approach, giving credit to seismic site effects and dynamic SSI. Seismic site response was evaluated by one-dimensional equivalent linear analyses, which highlighted the beneficial

131


BILOTTA, DE SANCTIS, DI LAORA, D’ONOFRIO AND SILVESTRI 3 10

4

3 104 Real part Imaginary part

2 104

KY, CY: MN/m

KX, CX: MN/m

2 104 1 104

0 1 104

0 1 104

0

1

2 3 Frequency, f: Hz (a)

4

2 104

5

3 107

3 107

2 107

2 107

KθX, CθX: MN/m

KθY, CθY: MN/m

2 104

1 104

1 107

0

1 107

1 107 1

2 3 Frequency, f: Hz (c)

4

1

2 3 Frequency, f: Hz (b)

4

5

0

1

2 3 Frequency, f: Hz (d)

4

5

1 107

0

0

0

5

Fig. 11. Real and imaginary parts of the impedance functions associated with horizontal swaying along the (a) x-axis and (b) yaxis, and rocking around the (c) y-axis and (d) x-axis Table 1. Results of calculations according to Maravas et al. (2014) TX: s

T~ X: s

~f X: Hz

KX (~f X): MN/m

CX ( ~f X): MN/m

�X (~f X)

KŁY (~f X): MN m

CŁY (~f X): MN m

�ŁY (~f X)

1 .62

1 .89

0 .528

5 .311 3 103

5 .461 3 102

5 .14%

1 .312 3 107

1 .185 3 106

4 .52%

TY: s

T~ Y: s

~f Y: Hz

KY (~f Y): MN/m

CY (~f Y): MN/m

�Y (~f Y)

KŁX (~f Y): MN m

CŁX (~f Y): MN m

�ŁX (~f Y)

2 .28

2 .47

0 .404

5 .697 3 103

5 .308 3 102

4 .66%

1 .372 3 107

1 .221 3 106

4 .45%

In order to evaluate the contribution of the combined radiation and hysteretic damping, an exact solution recently proposed in the literature was adopted, requiring the evaluation of the frequency-dependent impedance components associated with swaying and rocking oscillation of the foundation. The overall damping of the compliant base system was found to be practically coincident with that of the fixed-base structure. Such a result can be attributed to the larger foundation stiffness compared to that of the structure. Owing to SSR effects and dynamic SSI, the inertial actions were overall reduced by 57% and 43% for the first and second vibration modes. It is therefore inferred that the assessment of the above factors is mandatory for reliable and sustainable predictions of the seismic performance of buildings like the one considered in this study.

effects of a peat layer, acting as a natural damper on the propagation of seismic waves. As a consequence, the spectral ordinates at the surface were reduced by as much as 47% and 28%, for the first two vibration modes of the building. The attention was then focused on the role of SSI. Pile– soil kinematic interaction analyses were first carried out to assess the so-called foundation input motion (FIM): the results showed that, in this specific case, the filtering action exerted by the piles did not affect the FIM. The pile foundation compliance relevant to inertial interaction was then computed by the numerical code Dynapile, referring to mobilised stiffness profile. The rotational and translational dynamic stiffnesses were found to be poorly affected by the frequency in the range of periods corresponding to the first vibration modes of the building. The swaying and rocking components of the foundation impedance made it possible to evaluate the increase of the structural period of the compliant base system, which was found to be 1 .08 and 1 .16 for the first and second modes, respectively.

ACKNOWLEDGEMENTS The activity was carried out as part of WorkPackage 5 ‘Soil–foundation–structure interaction’ of the sub-project on

132


IMPORTANCE OF SEISMIC SITE RESPONSE AND SOIL–STRUCTURE INTERACTION 1·0

3

4

Kv( f ), Cv( f ) real and imaginary parts of the single pile vertical impedance KZ0, KH0 static axial and horizontal stiffness of single pile KŁj rotational stiffness of the foundation around j-axis ki stiffness of the fixed-base structure along i-axis m mass qc CPT cone resistance Sa spectral acceleration SX model parameter Ti fundamental period of the fixed-base structure T~ i fundamental period of the compliant-base structure UX, UR, U displacement components of the SDOF system VS shear wave velocity in the soil VS,30 equivalent shear wave velocity defined by Eurocodes Vsb bedrock shear wave velocity ÆCX,ÆŁY,ÆX weighting coefficients ª shear strain ~ �i apparent damping ratio associated to oscillations along i-axis �ci structural damping ratio associated to oscillations along i-axis �i foundation damping ratio associated to oscillations along i-axis �Łj foundation damping ratio associated to oscillations around j-axis �0,i foundation damping factor associated to oscillations along i-axis øCX natural circular frequency of the fixed-base structure along x-axis øX,øŁY fictitious frequencies ~ i natural circular frequency of the replacement ø oscillator

3

4

REFERENCES

Subsoil type D (NTC)

0·8

Seismic response analysis

Sa : g

0·6

0·4

0·2

0

TY 0

1

~ TY

2 Structural period, T: s (a)

1·0

0·8

Sa : g

0·6

0·4

0·2

0

TX 0

1

~ TX

2 Structural period, T: s (b)

Bardet, J. P., Ichii, K. & Lin, C. H. (2000). EERA a computer program for equivalent-linear earthquake site response analyses of layered soil deposits. Los Angeles, CA, USA: University of Southern California, Department of Civil Engineering. Bilotta, A., Sannino, D., Fretta, A., Nigro, E. & Manfredi, G. (2013a). Influenza della categoria di sottosuolo sulla vulnerabilita` sismica di edifici alti. Proceedings ANIDIS conference atti del convegno ANIDIS 2013, Padova. Padova, Italy: Padova University Press (in Italian). Bilotta, E., Bilotta, A., Del Prete, I., d’Onofrio, A., Nigro, E. & Silvestri, F. (2013b). Influenza delle condizioni locali di sottosuolo sulla risposta sismica di un edificio pubblico di notevole altezza. Proceedings ANIDIS conference 2013, Padova. Padova, Italy: Padova University Press (in Italian). Blaney, G. W., Kausel, E. & Roesset, J. M. (1976). Dynamic stiffness of piles. Proceedings of the 2nd international conference on numerical methods in geomechanics, Blaksburg, Virginia. BSSC (2004). NEHRP recommended provisions for seismic regulations for new buildings and other structures, FEMA 450. Washington D.C., USA: Building Seismic Safety Council, National Institute of Building Sciences. CEN (2003). (pr)EN 1998-1:2003: Eurocode 8: Design of structures for earthquake resistance – Part 1: General rules, seismic actions and rules for buildings. Brussels, Belgium: CEN European Committee for Standardization. Di Laora, R. & de Sanctis, L. (2013). Piles-induced filtering effect on the foundation input motion. Soil Dynam. Earthquake Engng 46, 52–63. Ensoft (1999). Dynapile 2.0: A program for the analysis of piles and drilled shafts under dynamic loads. Austin, TX, USA: Ensoft Inc. Gazetas, G. (1984). Seismic response of end-bearing single piles. Int. J. Soil Dynam. Earthquake Engng 3, No. 2, 82–93. Han, Y. (2002). Seismic response of tall building considering soil– pile–structure interaction. Earthquake Engng Engng Vibration 1, No. 1, 57–64. Iervolino, I., Galasso, C. & Cosenza, E. (2009). REXEL: computer aided record selection for code-based seismic structural analysis. Bull. Earthquake Engng 8, No. 2, 339–362, http://dx.doi.org/ 10.1007/s10518-009-9146-1.

Fig. 12. Reduction of the inertial actions for (a) the first and (b) the second building vibration modes

‘Earthquake geotechnical engineering’, in the framework of the research programme funded by Italian Department for Civil Protection through the ReLUIS (University Network of Seismic Engineering Laboratories) consortium.

NOTATION free field acceleration maximum (absolute) acceleration of the time history foundation acceleration mass acceleration of the replacement oscillator damping coefficient associated to the oscillation of foundation along i-axis, with i ¼ X, Y CŁj damping coefficient associated to the oscillation of foundation around j-axis, with j ¼ Y, X c viscous dashpot coefficient associated to the translational oscillation of the foundation cR viscous dashpot coefficient associated to the rotational oscillation of the foundation D soil damping ratio fi natural frequency of the fixed-base structure along iaxis ~f i natural frequency of the compliant-base structure along i-axis G shear modulus G0 initial shear modulus h height of the centre of mass of the building K horizontal stiffness of the foundation Kh( f ), Ch( f ) real and imaginary parts of the single pile horizontal impedance Ki horizontal stiffness of the foundation along i-axis KR rotational stiffness of the foundation aff amax ap ast Ci

133


BILOTTA, DE SANCTIS, DI LAORA, D’ONOFRIO AND SILVESTRI Mylonakis, G. E., Nikolaou, A. & Gazetas, G. (1997). Soil-pile-bridge seismic interaction: kinematic and inertial effects. Part I: soft soil. Earthquake Engng Structural Dynam. 27, No. 3, 337–359. NTC (2008). D. M. 14 Gennaio 2008, ‘Norme tecniche per le costruzioni’. Gazzetta Ufficiale della Repubblica Italiana no. 29, 4 February 2008 (in Italian). Rippa, F. & Vinale, F. (1983). Experiences with CPT in Eastern Naples area. Proceedings of the 2nd European symposium on penetration testing (ESOPT), Amsterdam. London, UK: CRC Press. Veletsos, A. S. & Meek, J. W. (1974). Dynamic behaviour of building foundation systems. Earthquake Engng Structural Dynam. 3, No. 2, 121–138. Viggiani, C. & Vinale, F. (1983). Comportamento di pali trivellati di grande diametro in terreni piroclastici. Rivista Italiana di Geotecnica 17, No. 2, 59–84 (in Italian). Vinale, F. (1988). Caratterizzazione del sottosuolo di un’area campione di Napoli ai fini di una microzonazione sismica. Rivista Italiana di Geotecnica 22, No. 2, 77–100 (in Italian). Wehling, T. M., Boulanger, R. W., Arulnathan, R., Harder, L. F. & Jr Driller, M. W. (2003). Nonlinear dynamic properties of a fibrous organic soil. J. Geotech. Geoenviron. Engng 129, No. 10, 929–939.

Kausel, E. (1974). Forced vibration of circular foundations. ScD thesis, Massachusetts Institute of Technology, Cambridge, MA, USA. Makris, N., Gazetas, G. & Delis, E. (1996). Dynamic pile–soil– foundation–structure interaction: records and predictions. Geótechnique 46, No. 1, 33–50, http://dx.doi.org/10.1680/geot.1996. 46.1.33. Mancuso, C., Viggiani, C., Mandolini, A. & Silvestri, F. (1999). Prediction and performance of axially loaded piles under working loads. In Pre-failure deformation characteristics of geomaterials (eds M. Jamiolkowski, R. Lancellotta and D. Lo Presti), pp. 801–808. Rotterdam, the Netherlands: Balkema. Mandolini, A. & Viggiani, C. (1997). Settlement of piled foundations. Geótechnique 47, No. 4, 791–816, http://dx.doi.org/ 10.1680/geot.1997.47.4.791. Maravas, A., Mylonakis, G. & Karabalis, D. L. (2007). Dynamic characteristics of structures on piles and footings. Proceedings of the 4th international conference on earthquake geotechnical engineering, Thessaloniki, Paper 1672. Dordrecht, the Netherlands: Springer. Maravas, A., Mylonakis, G. & Karabalis, D. L. (2014). Simplified discrete systems for dynamic analysis of structures on footings and piles. Soil Dynam. Earthquake Engng 61, No. 62, 29–39.

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Conti, R. et al. (2014). Geótechnique 64, No. 1, 40–50 [http://dx.doi.org/10.1680/geot.13.P.031]

Some remarks on the seismic behaviour of embedded cantilevered retaining walls R . C O N T I � , G . M . B. V I G G I A N I † a n d F. B U R A L I D ’ A R E Z Z O ‡

This paper is a numerical investigation of the physical phenomena that control the dynamic behaviour of embedded cantilevered retaining walls. Recent experimental observations obtained from centrifuge tests have shown that embedded cantilevered retaining walls experience permanent displacements even before the acceleration reaches its critical value, corresponding to full mobilisation of the soil strength. The motivation for this work stems from the need to incorporate these observations in simplified design procedures. A parametric study was carried out on a pair of embedded cantilevered walls in dry sand, subjected to real earthquakes scaled at different values of the maximum acceleration. The results of these analyses indicate that, for the geotechnical design of the wall, the equivalent acceleration to be used in pseudo-static calculations can be related to the maximum displacement that the structure can sustain, and can be larger than the maximum acceleration expected at the site. For the structural design of the wall, it is suggested that the maximum bending moments of the wall can be computed using a realistic distribution of contact stress and a conservative value of the pseudostatic acceleration, taking into account two-dimensional amplification effects near the walls. KEYWORDS: design; earth pressure; earthquakes; numerical modelling; retaining walls; shear strength

either provided by the technical codes or obtained by onedimensional seismic response analyses. If the permanent displacement at the end of the earthquake is taken as a performance indicator, the choice of the equivalent acceleration should be related to the maximum displacements that the structure can sustain, with respect to different levels of design earthquake motion. The reliability of the choice of an equivalent acceleration depends crucially on the ability to predict the displacements experienced by the wall during the earthquake. Permanent displacements of retaining walls are usually computed through Newmark (1965) rigid-block analysis (Richards & Elms, 1979). According to this method

INTRODUCTION In the recent literature several cases are reported of damage or failure of gravity and cantilevered retaining walls during earthquakes (Fang et al., 2003; Madabhushi & Zeng, 2007; Koseki et al., 2012). Following the pioneering works by Okabe (1926) and Mononobe & Matsuo (1929), several studies have tackled the problem of computing dynamic earth pressures on retaining structures with a theoretical (Steedman & Zeng, 1990; Lancellotta, 2007; Mylonakis et al., 2007; Kim et al., 2010), experimental (Atik & Sitar, 2010) or numerical approach (Gazetas et al., 2004; Evangelista et al., 2010). In the last decade, following the seminal works by Newmark (1965) and Richards & Elms (1979), more and more works have been devoted to the computation of wall displacements, in the light of a performance-based design (Ling, 2001; Huang et al., 2009; Basha & Babu, 2010). In recent years, new performance-based strategies have been proposed in the literature and included in current codes of practice (PIANC, 2001; CEN, 2003; NTC, 2008) for the seismic design of retaining structures. Although characterised by different levels of complexity, all these methods rely on the idea that the structure may experience permanent displacements during the earthquake, provided the behaviour of the system is ductile. The simplest way to embody the performance-based philosophy in the seismic design of retaining structures is by an appropriate choice of the equivalent acceleration to be used in pseudo-static calculations, which has to be proportional to the maximum acceleration expected at the ground surface,

(a) the critical (yield) acceleration of the wall, ac – that is, the acceleration corresponding to which the strength of the soil is fully mobilised – is computed with respect to an assumed collapse mechanism, assuming rigidperfectly plastic behaviour for both the soil and the wall (b) for accelerations a(t ) < ac , no relative displacements occur between the soil and the wall, and both the inertia forces into the soil wedge–wall system and the internal forces in the structure increase with the applied accelerations (c) for accelerations a(t ) . ac , the wall experiences permanent displacements, but the internal forces remain constant and equal to the maximum value they attained for a(t ) ¼ ac (d ) the permanent displacements are computed by integrating the relative acceleration, a(t ) � ac , twice over the time intervals in which the relative velocities are non-zero. The critical acceleration is a key ingredient not only for the computation of the permanent displacements experienced by the wall, but also for its structural design, as it defines the maximum internal forces that the structure may ever experience during an earthquake. Centrifuge dynamic tests have shown that Newmark rigidblock analysis provides good results when applied to gravity retaining structures (Zeng & Steedman, 2000; Huang et al., 2009). Moreover, experimental dynamic tests carried out on

Manuscript received 19 February 2013; revised manuscript accepted 11 July 2013. Published online ahead of print 13 September 2013. Discussion on this paper closes on 1 June 2014, for further details see p. ii. � Dipartimento di Ingegneria Civile, Universita` di Roma Tor Vergata, and International School for Advanced Studies, Trieste, Italy. † Dipartimento di Ingegneria Civile, Universita` di Roma Tor Vergata, Italy. ‡ University of Cambridge, UK.

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CONTI, VIGGIANI AND BURALI D’AREZZO mobilisation of plastic strain). Based on the results of the parametric study, suggestions are provided for the seismic geotechnical and structural design of embedded cantilevered walls using simplified methods.

reduced-scale models (Neelakantan et al., 1992; Richards & Elms, 1992) and dynamic numerical analyses (Callisto & Soccodato, 2010) have shown that a Newmark-type calculation may also be adopted, at least qualitatively, to interpret the dynamic behaviour of embedded cantilevered walls or retaining walls with one level of support, where the wall can rotate when a state of limit equilibrium is attained in the adjacent soil. Results from both numerical (Callisto & Soccodato, 2007, 2010) and experimental (Zeng, 1990; Zeng & Steedman, 1993; Conti et al., 2012) work on the dynamic behaviour of embedded retaining walls, however, have shown that the Newmark approach does not describe the observed behaviour satisfactorily. In fact, if the critical acceleration of the system is computed with standard limit equilibrium methods, such as those adopted in European countries and the USA (Blum, 1931; Padfield & Mair, 1984; King, 1995; Powrie, 1996; Day, 1999; Osman & Bolton, 2004), seeking the pseudo-static coefficient corresponding to which limit conditions are attained in the system, two findings are of major concern.

ANALYSIS PROCEDURE Constitutive soil model The soil is modelled as an elastic-perfectly plastic material with Mohr–Coulomb failure criterion, in which, during the dynamic stage, non-linear and hysteretic behaviour is introduced for stress paths within the yield surface through a hysteretic model available in the library of FLAC 5.0 (Itasca, 2005). This strategy makes it possible to take into account both the cyclic soil behaviour and the possibility of full mobilisation of soil strength close to the excavation, at the same time avoiding the cumbersome calibration process and the high computational costs of more advanced constitutive models (Kontoe et al., 2012). The hysteretic model, which is used to update the tangent shear modulus of the constitutive law for the soil at each calculation step, consists in an extension to general strain conditions of the one-dimensional non-linear models that make use of the Masing (1926) rules to describe the unloading–reloading behaviour of soil during cyclic loading. If the simplified assumption is made that the stress state does not depend on the number of cycles, the relationship between shear stress, �, and shear strain, ª, can be written as

(a) Embedded walls may accumulate significant rigid permanent displacements concurrently with an increase of the internal forces in the structural members: that is, permanent displacements occur even before the critical acceleration is attained. (b) Internal forces in cantilevered walls may be substantially larger than those computed with conventional limit equilibrium methods in critical conditions.

� ¼ GS (ª) � ª

It follows that, at least for cantilevered walls, standard pseudo-static approaches do not provide reliable or conservative values of the yield acceleration, neither for a displacement-based analysis nor for a pseudo-static calculation. On the basis of centrifuge dynamic tests carried out on pairs of embedded propped and cantilevered walls in dry sand, Conti et al. (2012) have shown that a Newmark analysis carried out using the limit equilibrium value of the critical acceleration would yield displacements that are much smaller than observed, as the analysis would overlook the displacements experienced by the wall before the acceleration reaches the limit equilibrium critical value. According to the authors, the observed behaviour may be justified by a stress redistribution and a progressive mobilisation of the soil strength on the passive side of the wall produced by the earthquake. Recent numerical studies of the dynamic behaviour of embedded retaining walls, both cantilevered (Madabhushi & Zeng, 2006, 2007) and with one level of support (Iai & Kameoka, 1993; Callisto et al., 2008; Cilingir et al., 2011) have shown interesting aspects related to the soil–structure interaction and the constitutive modelling of the mechanical behaviour of the soil under cyclic loading. Useful guidelines for the seismic design of cantilevered retaining walls may be found in Callisto & Soccodato (2010), but these are still not exhaustive: the simplified procedure proposed by the authors does not seem conservative, as it is based on a standard pseudo-static calculation of the critical acceleration. This work is a numerical investigation of the physical phenomena that control the dynamic behaviour of embedded cantilevered retaining walls, aimed at developing suitable simplified procedures to be incorporated in recommendations and codes of practice. A parametric study was carried out on a pair of embedded cantilevered walls in dry sand, subjected to real earthquakes scaled at different values of the maximum acceleration. The earthquakes were chosen to represent a significant range of dominant frequency (governing local amplification and resonance phenomena) and peak acceleration (governing non-linearity of soil behaviour and

¼ G0 M S (ª) � ª

(1)

where GS (ª) is the secant shear modulus, G0 is the smallstrain shear modulus, and MS (ª) is the normalised secant shear modulus, defined as a (2) MS ¼ 1 þ exp �(log10 ª � x0 )=b

where a, b and x0 are model parameters that can be determined from the best fit of a specific modulus degradation curve. The tangent shear modulus, Mt , can be evaluated by differentiating equation (1) with respect to ª. Strain reversals during cyclic loading are detected by a change of the sign of the scalar product between the current strain increment and the direction of the strain path at the previous time instant. At each strain reversal, the Masing rule is invoked, and stress and strain axes are scaled by a factor of 0.5, resulting in hysteresis loops in the stress–strain curves with associated energy dissipation. As already outlined by Callisto & Soccodato (2010), an advantage of using a truly non-linear soil model for dynamic numerical simulations is that energy dissipation emerges from the hysteretic behaviour of the soil, and is not introduced artificially by including a frequency-dependent viscosity in the equilibrium equations. Seismic input Three different acceleration time histories were used in the analyses, all registered on rock outcrop during real earthquakes: Tolmezzo (T) from the Friuli earthquake of 1976, Assisi (A) from the Umbria-Marche earthquake of 1997, and Arcelik (N) from the Kocaeli earthquake of 1999. The choice of these three earthquakes is motivated by the fact that they are characterised by substantially different frequency contents. Table 1 shows the maximum values of acceleration, amax,r , duration, T5–95 , mean period, Tm (Rathje et al., 1998), and Arias intensity, Ia : Fig. 1 shows the acceleration time histories and the Fourier spectra of the

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THE SEISMIC BEHAVIOUR OF EMBEDDED CANTILEVERED RETAINING WALLS 0:5 Table 1. Ground motion parameters of input earthquakes G0 p9 ¼ KG pref pref T5–95 : s

Tm : s

Ia : m/s

0.28 0.35 0.14

4.28 4.19 7.23

0.25 0.40 1.09

0.78 0.80 0.31

where p9 is the mean effective stress, pref ¼ 100 kPa is a reference pressure, and KG is a stiffness multiplier, set equal to 1000. Soil parameters a ¼ 1.0, b ¼ �0.6 and x0 ¼ �1.5 were used for the normalised secant shear modulus in equation (2), derived from the best fit of the modulus degradation curve proposed by Vucetic & Dobry (1991) for cohesionless (PI ¼ 0) soils. Fig. 3 shows a comparison between the modulus decay curve and the equivalent damping ratio of the adopted model, and that suggested by Vucetic & Dobry (1991). The retaining walls were modelled as elastic beams connected to the grid nodes with elastic-perfectly plastic interfaces with a friction angle � ¼ 208. The bending stiffness of the walls was EI ¼ 2.7 3 105 kN m2 /m, corresponding to that of a wall consisting of 0.6 m diameter and 0.7 m spacing bored piles. The walls can be considered, for all

three signals. The recorded signals were baseline-corrected and low-pass-filtered at 15 Hz for compatibility with the dimension of the grid zones in the numerical domain; moreover, they were scaled at maximum accelerations ranging from 0.05g to 0.5g.

G/G0

Numerical model Two-dimensional, plane-strain, finite-difference analyses of a rectangular excavation of width B ¼ 16 m and depth h ¼ 4 m, in a layer of dry sand with thickness Z ¼ 16 m, were carried out. The excavation was retained by a pair of cantilevered retaining walls. Figure 2 shows the grid adopted in the numerical analyses, with an extension of 80 m, consisting of a total of 4838 elements, with a minimum size of 0.33 m near the walls. Both the refinement of the mesh and the extension of the grid were chosen after a preliminary parametric study, in order that they did not influence the numerical results during either the static or dynamic stages. The soil was modelled with a constant friction angle � ¼ 358, cohesion c9 ¼ 0, and density r ¼ 2.04 Mg/m3 : A standard non-associated flow rule was used, with angle of dilatancy ł ¼ 0. The small-strain shear modulus is given by

1·0

30

0·8

25 20

0·6 0·4

Vucetic & Dobry (1991)

15

Numerical

10

0·2

5

0 0·0001

0·001

0·01 γ: %

0·1

0·02 Tolmezzo

a: g

0·2

0·01

0 0·2

0 0·02

0·4 0·4

a: g

0·01

0

A: g

Assisi

0·2

0·2 0 0·02

0·4 0·4

a: g

0·01

0

A: g

Kocaeli

0·2

0·2 0

4

8

12

16

20 0

t: s

4

8 f : Hz

12

0 16

Fig. 1. Acceleration time histories and Fourier amplitude spectra of input earthquakes 32 m

1

0

Fig. 3. Modulus decay and damping ratio curves (from Conti & Viggiani, 2013)

0·4

0·4

(3)

D: %

Assisi Tolmezzo Arcelik

amax,r : g

A: g

Record

B

32 m h

d 16 m

Bedrock

Fig. 2. Finite difference grid (from Conti & Viggiani, 2013)

139


CONTI, VIGGIANI AND BURALI D’AREZZO A time increment of ˜t ¼ 6.25 3 10�7 s was adopted during the dynamic stage, in order to guarantee the stability of the explicit time integration scheme. Moreover, a small Rayleigh viscous damping (1%) was adopted to remove the high-frequency noise deriving from the numerical integration, but not otherwise affecting the results of the analyses. A total of 39 numerical analyses were carried out in this study, as reported in Table 2. Three different values were adopted for the embedded depth of the walls: d ¼ 3 m (Nos 1 to 9), d ¼ 4 m (Nos 10 to 21) and d ¼ 5 m (Nos 22 to 36). For the analysis with d ¼ 4 m and the Tolmezzo (T) record scaled to 0.35g, the soil stiffness multiplier was halved and doubled (Nos 37 and 38), and a thickness of Z ¼ 30 m was considered for the soil layer (No. 39).

practical purposes, as infinitely rigid (Callisto & Soccodato, 2010). The in situ stress state was prescribed in terms of the earth pressure coefficient at rest, h9 = v9 ¼ K 0 (¼ 1 � sin ). During the static stage, standard boundary conditions were applied to the model: that is, zero horizontal displacements along the lateral boundaries, and fixed nodes at the base of the grid. The excavation was carried out in four successive steps, chosen after a preliminary study in order to have the numerical results unaffected by the calculation sequence, during which the soil elements corresponding to 1 m of the excavated volume were removed. In this stage the shear modulus of the soil was set equal to 0.3G0 , corresponding to a shear strain level of about 0.1% (see Fig. 3), characteristic of the expected level of deformation during the excavation stage (Atkinson et al., 1993). After the excavation, static constraints were removed from the boundaries. The selected acceleration–time histories were applied to the bottom nodes of the grid, together with a zero velocity condition in the vertical direction, thus simulating the presence of an infinitely rigid bedrock. Standard dynamic constraints (Zienkiewicz et al., 1988) were applied to the nodes on the lateral boundaries of the grid: that is, they were free to move in both the vertical and horizontal directions, while being tied to one another in order to enforce the same displacements of the two boundaries.

DISCUSSION OF RESULTS Static stage At the end of the static stage, the soil around the excavation is approximately in limit equilibrium conditions, both in front and behind the walls. Fig. 4 shows, for the case d ¼ 4 m, the contour lines of: (a) the mobilised shear strength into the soil, defined as the ratio, /lim between the maximum shear stress and the corresponding available strength; and (b) the maximum shear strain (in %) close to the left wall. Fig. 4(a) also shows the horizontal stress distribution in the soil elements at the contact with the wall,

Table 2. Summary of numerical analyses No.

Record

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39

Assisi Assisi Assisi Tolmezzo Tolmezzo Tolmezzo Arcelik Arcelik Arcelik Assisi Assisi Assisi Assisi Tolmezzo Tolmezzo Tolmezzo Tolmezzo Arcelik Arcelik Arcelik Arcelik Assisi Assisi Assisi Assisi Assisi Tolmezzo Tolmezzo Tolmezzo Tolmezzo Tolmezzo Arcelik Arcelik Arcelik Arcelik Arcelik Tolmezzo Tolmezzo Tolmezzo

ainp : g

d: m

Z: m

0.05 0.10 0.20 0.05 0.10 0.20 0.05 0.10 0.20 0.05 0.10 0.20 0.35 0.05 0.10 0.20 0.35 0.05 0.10 0.20 0.35 0.05 0.10 0.20 0.35 0.50 0.05 0.10 0.20 0.35 0.50 0.05 0.10 0.20 0.35 0.50 0.35 0.35 0.35

3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 4 4 4

16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 30

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EI: kN m2 /m] 2.7.105 2.7.105 2.7.105 2.7.105 2.7.105 2.7.105 2.7.105 2.7.105 2.7.105 2.7.105 2.7.105 2.7.105 2.7.105 2.7.105 2.7.105 2.7.105 2.7.105 2.7.105 2.7.105 2.7.105 2.7.105 2.7.105 2.7.105 2.7.105 2.7.105 2.7.105 2.7.105 2.7.105 2.7.105 2.7.105 2.7.105 2.7.105 2.7.105 2.7.105 2.7.105 2.7.105 2.7.105 2.7.105 2.7.105

KG 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 500 2000 1000

THE SEISMIC BEHAVIOUR OF EMBEDDED CANTILEVERED RETAINING WALLS 1984; Bica & Clayton, 1989; King, 1995; Day, 1999), and a limit equilibrium method outlined herein. Fig. 5(d) shows the horizontal displacements of the wall computed at the end of each earthquake, while in Figs 5(e)–5(i) the computed contact stresses are plotted together with the dynamic active and passive limit values. The earth pressure coefficients, kAE (Okabe, 1926; Mononobe & Matsuo, 1929) and kPE (Lancellotta, 2007), are computed using, at each depth, a pseudostatic acceleration equal to the acceleration resulting from the numerical analyses (kh ¼ a/g). During the earthquakes, amplification phenomena around the excavation cause the surface accelerations behind the wall to be substantially larger than the maximum input accelerations, with amplification factors between 1.5 (No. 17) and 3.8 (No. 14). The distribution of accelerations into the soil is not uniform, owing to both amplification and phase shift between the top and the bottom of the wall; in the time instants when the acceleration reaches its maximum value, it varies almost linearly with depth behind the wall, at least in the retained part of the soil, while it is approximately constant below dredge level, and substantially lower than the maximum value at surface (Fig. 5(a)). The inertia forces in the retained soil induce an increment of the contact stresses behind the wall, the soil being in active limit conditions. As a consequence, the wall rotates, mobilising the passive resistance of the soil below dredge level progressively, until the system reaches a new equilibrium configuration. The stronger the earthquake, the greater the depth down to which the passive resistance of the soil is fully mobilised (Figs 5(b), 5(f), 5(g), 5(h) and 5(i)). Both the increment of the contact stresses behind the wall and the lower position of the resultant of the pressure distribution in front of the wall cause a significant increase of the bending moments in the wall (Fig. 5(c)). At the end of the earthquakes, the horizontal displacements of the wall correspond to an approximately rigid rotation around a pivot point between 0.8d and 0.9d (Fig. 5(d)). As far as the internal forces in the walls are concerned, the maximum bending moments may occur at instants of time when the acceleration at surface behind the wall is not maximum. As an example, Fig. 6 shows, for analysis No. 17 and for the time instants at which a ¼ amax (t ¼ 5.64 s) and M ¼ Mmax (t ¼ 5.70 s): (a) the acceleration profile, (b, c) the earth pressure distribution, (d) the bending moment distribution, and (e, f) the contour lines of the ratio /lim : As already discussed, when the acceleration behind the wall reaches its maximum value (t ¼ 5.64 s), the retained soil is in active limit state conditions, at least down to z ¼ 6 m, while all the available resistance in front of the wall is mobilised down to 2 m from the excavation bottom. At this time instant, the occurrence of a soil wedge may be observed both behind (active) and in front (passive) of the wall (Fig. 6(e)). By contrast, for t ¼ 5.70 s the soil on both sides of the wall is far from limit conditions (Fig. 6(f)): that is, the horizontal stresses are higher than the corresponding active values on the retained side, while a constant fraction of the passive resistance is mobilised below dredge level (Fig. 6(c)). This stress distribution results in higher bending moments in the wall, even if the accelerations in the retained soil are about 20–30% lower than the values computed for t ¼ 5.64 s. The behaviour exhibited by the soil–wall system at this time instant would be hardly reproduced by a simplified limit equilibrium approach.

1·00 0·95 0·90 0·85 0·80

σh: kPa

0·75

Active and passive limit states

0·70

100

50

0

50 100 150

(a)

0·28 0·24 0·20 0·16 0·12 0·08 0·04 (b)

Fig. 4. Static stage (d 4 m). Contours of (a) mobilised shear strength, /lim , and (b) maximum shear strain (%) close to left wall

together with the theoretical values of the static active and passive pressure, computed with Lancellotta’s closed-form solutions (Lancellotta, 2002). The soil behind the wall is in active limit state down to 5 m from the surface, whereas in front of the wall the passive resistance is fully mobilised only immediately below dredge level, the horizontal stresses being approximately constant in the remaining part of the embedded depth. The distribution of maximum shear strain into the soil is similar to that of the mobilised strength. Moreover, while a maximum shear strain of 0.3% is mobilised just below dredge level, an average strain level of about 0.1% is mobilised into the whole soil volume interacting with the wall during the static stage. Dynamic behaviour of retaining walls In this section, the dynamic behaviour of cantilevered retaining walls is discussed with reference to analyses Nos 14 to 17, where the walls with d ¼ 4 m are subjected to the Tolmezzo earthquake scaled to maximum input accelerations ainp ¼ 0.05g, 0.10g, 0.20g and 0.35g. Figure 5 shows, for the time instant when the acceleration behind the right wall reaches its maximum value (a) the distribution of accelerations immediately behind and in front of the wall (b) the contact horizontal stresses (c) the bending moment distribution in the wall. Figs 5(a) and 5(c) also show the critical acceleration of the wall and the corresponding bending moment distribution, computed using the Blum (1931) method, customarily adopted in the UK and other European countries, and described extensively in many works (e.g. Padfield & Mair,

Critical acceleration and limit equilibrium analysis The critical acceleration is computed with respect to an assumed failure mechanism, generally a rigid rotation about a point close to the toe for embedded cantilevered walls,

141


CONTI, VIGGIANI AND BURALI D’AREZZO a: g 0·2

0

0·2

0·4

0·6

300

200

σh: kPa 100

0

100

M: kN m/m 200

0

400

u: m 0·1

0·2

0

Static ainp 0·05g (No. 14) ainp 0·10g (No. 15) ainp 0·20g (No. 16) ainp 0·35g (No. 17)

0

2

Conti & Viggiani (2013)

0·2

ac 0·481g

ac 0·283g

4

z: m

Blum (1931)

a: g 0·2 0

6

(a)

(b)

(c)

8

(d)

0 Static

ainp 0·05g

ainp 0·10g

ainp 0·20g

ainp 0·35g

Limit cond.

Limit cond.

Limit cond.

Limit cond.

Limit cond.

4

z: m

2

6

200

0

σh: kPa (e)

200

σh: kPa (f)

0

200

0

σh: kPa (g)

200

σh: kPa (h)

0

200

σh: kPa (i)

0

8

Fig. 5. Analyses Nos 14–17 (right wall): (a) distribution of accelerations behind and in front of wall; (b) and (e) to (i) contact horizontal stresses; (c) bending moment distribution for time instant when acceleration behind wall is maximum; (d) horizontal displacements of wall at end of each earthquake

a: g 0·2 0

σh: kPa

0·2 0·4 0·6 300200100 0

σh: kPa 100 300200100 0

M: kN m/m 100

0

200

400 0

t 5.64 s t 5.64 s Limit cond.

2

t 5·64 s (a amax) 0·2 0 0·2 4

z: m

a: g

(e)

1·00 6

0·90 0·80

(a)

(b)

(d)

(c)

8

0·70

t 5·70 s (M Mmax) (f)

Fig. 6. Analysis No. 17 (right wall): (a) acceleration profile; (b), (c) earth pressure distribution; (d) bending moment distribution; (e), (f) contour lines of ratio /lim , for time instants at which a amax and M Mmax

and depends solely on the geometry of the system and the strength of the soil. The results obtained by Callisto & Soccodato (2010) show that the Blum method does not provide a reliable or conservative estimate of the critical acceleration, and hence of the maximum bending moment that the wall may ever experience during an earthquake. There are two main reasons for this.

(a) In the Blum method, the same pseudo-static acceleration is assumed for the soil in front and behind the wall, whereas, in general, the acceleration below dredge level is only a small fraction of the maximum value on the retained side, and always lower than about 0.1g (see Fig. 5(a)). (b) The pivot point is taken to be at a depth of 0.8d from dredge level, whereas it may be as deep as 0.9d during strong earthquakes (see Fig. 5(d)).

142


THE SEISMIC BEHAVIOUR OF EMBEDDED CANTILEVERED RETAINING WALLS F RHS ¼ 12ªK AE (h þ d 0 )2 þ 12ªK P (2h þ d þ d 0 )(d � d 0 )

Moreover, for pseudo-static accelerations smaller than the critical value, the Blum method assumes a constant fraction of the soil passive resistance in front of the wall, whereas during real earthquakes this is progressively mobilised from the bottom of the excavation, depending on the amplitude of the acceleration applied (see Fig. 5(b)). To take into account all these observations, a different pseudo-static distribution of contact stresses is used in this work (see Fig. 7). According to this approach (e.g. Conti & Viggiani, 2013)

F LHS ¼

þ 12ª(K P d þ K AE d 0 )(d 0 � d) "� # ! � d 0 � d K AE d 0 þ 2K P d 3 þ (d � d 0 ) 3 K AE d 0 þ K P d þ 16ªK AE (d � d 0 )2 (d þ 2d 0 )

(7)

By equating FRHS ¼ FLHS and MRHS ¼ MLHS one obtains a system of two equations in the two unknowns d and d0 : Figure 8 shows the maximum bending moment on the walls computed from all the numerical analyses, as a function of the surface acceleration behind the walls at the same time instant; analyses 37 to 39 are those carried out using different values of the small-strain stiffness of the soil, G0 , and of the total height of the wall, H. Fig. 8 also shows the limit equilibrium maximum bending moment, as a function of kh , computed according to both the Blum method and the proposed method. For completeness, the results obtained with the earth pressure distribution adopted customarily in the USA, and described in detail by Bowles (1988), King (1995) and Day (1999), are also reported in the figure. Internal forces on the walls increase as a function of the applied acceleration, until the critical acceleration is reached. As an example, for d ¼ 3 m and d ¼ 4 m the numerical maximum bending moments reach a plateau for accelerations greater than 0.3g and 0.48g respectively, their maximum values being about 185 kN m/m and 330 kN m/m respectively. In this case, critical accelerations predicted by the proposed approach, and the corresponding maximum

kh kc (d d0) h

d K Pγ d

KAEγd

KPγ(h d0)

KPγ(h d)

Fig. 7. Distribution of seismic earth pressures in proposed limit equilibrium method (from Conti & Viggiani, 2013)

Mmax: kN m/m

800 700

d: m Tolmezzo

600

Assisi Kocaeli

3

4

5

No. 37 No. 38 No. 39

Proposed method Blum method US method

500

d5m

400 300 d4m

200 100 0

d3m 0

0·2

(5)

2 M LHS ¼ 12ªK P d (d � 23d)

kh k c

KAEγd0 KAEγ(h d0)

þ K AE d 0 )(d 0 � d)

(4)

Similarly, the moment equilibrium can be established by taking the moment about the toe of the wall, generated by the forces acting on the right-hand side (MRHS ) and on the left-hand side (MLHS ), to give � � M RHS ¼ 12ªK AE (h þ d 0 )2 13(h þ d 0 ) þ (d � d 0 ) (6) þ 16ªK P (3h þ d þ 2d 0 )(d � d 0 )2

Using Fig. 7, the force equilibrium of the wall can be established by considering the forces acting on the righthand side (FRHS ) and on the left-hand side (FLHS ) of the wall, as follows.

d0

þ

1 2ª(K P d

þ 12ªK AE (d þ d 0 )(d � d 0 )

(a) the active earth pressure coefficient, KAE , is computed as a function of the assumed pseudo-static coefficient kh, while a static earth pressure coefficient, KP , is adopted for the passive resistance (b) the strength of the soil in front of the wall is progressively mobilised down to a depth d, as a function of kh (c) the position of the pivot point, d0 , depends on kh , being about 0.9d for kh ¼ kc :

d

1 2 2ªK P d

0·4

0·6

0·8

1·0

a: g

Fig. 8. Maximum bending moment as a function of surface acceleration behind walls computed at the same time instant, and limit equilibrium results

143


CONTI, VIGGIANI AND BURALI D’AREZZO 0·6

bending moments, are ac ¼ 0.29g and Mc ¼ 175 kN m/m for d ¼ 3 m, and ac ¼ 0.48g and Mc ¼ 342 kN m/m for d ¼ 4 m, showing a very good agreement with the numerical data. A more dispersed trend is observed for d ¼ 5 m, where the maximum bending moments do not seem to reach a limit value for the earthquakes applied. The fact that the pseudostatic approach does not always predict the numerical observations satisfactorily, at least for a , ac , is due mainly to the fact that the soil around the excavation is not always in limit conditions when bending moments on the walls attain their maximum value, as already discussed. However, the proposed method provides an accurate estimate of the maximum (critical) internal forces that the walls may ever attain during an earthquake, whereas the Blum and US methods always underpredict the maximum bending moments substantially. It is worth noting that the critical acceleration, and hence the maximum bending moment, increase with d – that is, the longer (and safer) the wall, the larger the bending moment it has to sustain under a strong earthquake.

ac 0·48g

a: g

0·3

ac_N 0·16g

0 0·3 0·6 (a)

u: m

0·15 0·10 Numerical Newmark analysis

0·05 0

0

4

8

t: s (b)

12

16

20

Fig. 10. Analysis No. 17: (a) acceleration time histories computed at soil surface; (b) horizontal displacements of top of right wall. Comparison between results from numerical and Newmark analysis

Newmark analysis The horizontal displacement of the top of the wall is an important parameter in performance-based seismic design, as the settlements of the ground surface behind retaining structures, and hence the potential damage to adjacent buildings, are related to the horizontal displacements of the wall (Mana & Clough, 1981; Hsieh & Ou, 1998; Kung et al., 2007; Wang et al., 2010). Fig. 9 shows the final horizontal displacements of the top of the wall, u, normalised by the total height of the wall, H, as a function of the ratio ac /amax between the critical acceleration computed with the pseudostatic approach outlined above, and the maximum acceleration behind the walls. As expected, the displacements of the walls increase as the intensity of the applied earthquake increases. However, as shown by Conti et al. (2012) on the basis of centrifuge tests, the walls can accumulate significant displacements (u/H . 1–2%), even for accelerations lower than the critical one, and hence before the available soil passive resistance is fully mobilised in front of the wall. These displacements cannot be computed by a conventional Newmark (1965) analysis, that is, assuming a yield acceleration equal to the critical value provided by the limit equilibrium analysis. Following the procedure adopted by Conti et al. (2012) for the interpretation of centrifuge dynamic tests, a Newmark calculation was carried out for each analysis, in which the yield acceleration, acN , was found by trial and error to match computed and numerical displacements at the end of each earthquake. As an example, Fig. 10 shows, for analysis No. 17, (a) the acceleration time histories computed at the soil surface, and (b) the horizontal displacements of the top of the right wall. The yield acceleration, acN , required to match the computed final displacement is only a

fraction of the critical acceleration ac that corresponds to the complete mobilisation of the soil passive strength; in other words, had the Newmark analysis been carried out using ac , the displacements of the wall would have been zero. Figure 11 shows the computed values of acN as a function of the maximum acceleration amax , both normalised by the limit equilibrium value of the critical acceleration, ac : Data from centrifuge tests (Conti et al., 2012) have been also included in the same figure for comparison. Both numerical and experimental data indicate that acN /ac increases with the maximum acceleration applied, up to about amax /ac ¼ 1.0, and it is then approximately constant, and equal to about 40%. This result is not significantly affected by the characteristics of the applied earthquakes, such as frequency content, duration or Arias intensity. GUIDELINES FOR DESIGN The seismic design of embedded cantilevered retaining walls for a given value of the maximum acceleration expected at the site, amax,1D , must address the issues both of: (a) the geotechnical design of the wall, that is, the calculation of a depth of embedment such that the permanent horizontal displacement of the top of the wall at the end of the earthquake, taken as an indicator of the performance of the retaining structure, is less than or equal to an admissible value; and (b) the structural design of the wall, that is, the definition of the structural section needed to sustain the maximum internal forces experienced by the wall during the earthquake. In the following, the authors try to provide guidelines for the seismic design of retaining structures using simplified methods, in the light of the results presented so far.

6 d: m Tolmezzo Assisi Kocaeli

5

1·0

No. 37 No. 38 No. 39

ac_N/ac

u/H: %

4

3 4 5

3

Centrifuge data

d: m Tolmezzo Assisi Kocaeli

No. 37 No. 38 No. 39

3 4 5

0·5

2 1 0

0 0

1

2

3

ac /amax

4

5

6

0

0·5

1·0

1·5

2·0

2·5

amax /ac

7

Fig. 11. Computed values of acN as a function of maximum acceleration amax , both normalised by limit equilibrium value of critical acceleration, ac

Fig. 9. Final normalised horizontal displacements of top of walls, as a function of the ratio ac /amax

144


THE SEISMIC BEHAVIOUR OF EMBEDDED CANTILEVERED RETAINING WALLS 4

amax /amax,1D

Geotechnical design Figure 12 is similar to Fig. 9, but this time the final normalised displacements of the top of the wall, u/H, are plotted as a function of the ratio ac /amax,1D , where amax,1D is the maximum acceleration at the surface (free field) computed from one-dimensional site response analyses using FLAC. Similar to Richards & Elms (1979), the data were interpolated with a power function, which is plotted in Fig. 12 together with the 98% confidence intervals. For each design earthquake motion, corresponding to which an allowable displacement is prescribed, the upperbound curve in Fig. 12 can be used to obtain the ratio ¼ ac /amax,1D and then the required critical acceleration. Clearly, the effective value of the critical acceleration for the wall must be the maximum among all the design earthquakes. Once ac is known, the depth of embedment of the wall can be computed iteratively using the proposed limit equilibrium approach, with kh ¼ ah /g ¼ ac /g, until the (critical) condition d ¼ d 0 is satisfied. Any introduction of a safety factor at this stage would ensure that the displacements experienced by the wall are less than the admissible value. Allowable displacements less than about 3% of H would result in a ratio ac /amax,1D . 1: that is, the wall should be designed to have a critical acceleration larger than the maximum acceleration expected at the site (i.e. using an equivalent acceleration that is larger than the maximum free-field acceleration). This is completely different from the performance-based design of gravity retaining walls, as they will experience permanent displacements only if their critical acceleration is lower than the maximum acceleration of the design earthquake ( , 1). As already discussed, this difference arises from the fact that embedded cantilevered walls begin to rotate for accelerations lower than the critical value.

u/H: %

5

3

2 1

0

0·1

0·2

0·3 amax,1D: g

0·4

0·5

0·6

CONCLUSIONS This paper has addressed the issue of the seismic design of embedded cantilevered retaining walls, in the light of the results obtained from an extensive set of numerical analyses of a pair of cantilevered walls in dry sand, subjected to real earthquakes scaled at different values of the maximum acceleration. The results of the analyses confirm that embedded cantilevered retaining walls experience permanent displacements even before the acceleration reaches its critical value, corresponding to full mobilisation of the shear strength of the soil. 8

3 4 5

d: m 3 4 5 Tolmezzo Assisi Kocaeli Limit equilbrium

7

No. 37 No. 38 No. 39

6

2

5

No. 37 No. 38 No. 39

4 3 2

1 0

No. 37 No. 38 No. 39

seismic response analyses, amax,1D , as a function of amax,1D : Two-dimensional phenomena clearly induce a stronger (further) amplification than that merely associated with onedimensional shear wave propagation, but, at least for the range of geometrical and mechanical factors considered in this parametric study, the ratio amax /amax,1D is always less than about 2, and not significantly affected by the ground motion parameters of the earthquakes applied. Figure 14 shows the numerical values of the maximum bending moment, Mmax , normalised by the maximum static bending moment, Mstat , as a function of amax /ac , together with the pseudo-static bending moments computed according to the proposed method. The figure shows a good agreement between numerical and pseudo-static results, for all the applied earthquakes. Moreover, limit equilibrium clearly provides an accurate estimate of the maximum (critical) internal forces that a wall with a given value of the critical acceleration may ever attain during an earthquake, which is therefore always a conservative value for the structural design of the wall. For this purpose, pseudo-static bending moments can then be computed using the earth pressure distribution outlined in this work, with a pseudo-static coefficient kh ¼ amax /g. A conservative value of amax would be amax ¼ 2amax,1D :

u/H 2·9 (ac/amax,1D)1·9 98% interval confidence

4

3 4 5

Fig. 13. Ratio amax /amax,1D between maximum acceleration computed behind walls and maximum free-field acceleration obtained by one-dimensional seismic response analyses, as a function of amax,1D

Mmax/Mstat


6

3

0

Structural design For the structural design of retaining walls, the maximum bending moment, Mmax , must be computed under a realistic distribution of contact stresses between the soil and the structure. Its value is closely related to the maximum acceleration at surface behind the walls, amax : As already observed by Callisto & Soccodato (2010) values of amax can only be computed taking into account soil–structure interaction effects, as the maximum accelerations behind retaining structures depend not only on the dynamic properties and thickness of the soil layer, but also on a number of factors, such as the geometry of the excavation, the bending stiffness of the wall and the embedded depth, which all affect the natural frequency of the soil–wall system. As an example, Fig. 13 shows the ratio amax /amax,1D between the maximum acceleration computed behind the walls, amax , and the maximum free-field acceleration obtained by one-dimensional 7


1 0

1

2

3 4 ac/amax,1D

5

6

0

7

Fig. 12. Final normalised horizontal displacements of top of walls, as a function of the ratio ac /amax,1D

0

0·2

0·4

0·6

0·8 amax /ac

1·0

1·2

1·4

1·6

Fig. 14. Maximum normalised bending moments, as a function of amax /ac : numerical and limit equilibrium results

145


CONTI, VIGGIANI AND BURALI D’AREZZO For the geotechnical design of the wall, if the permanent displacement at the end of the earthquake is taken as a performance indicator, the choice of the equivalent acceleration to be used in the pseudo-static calculations should be related to the maximum displacements that the structure can sustain, with respect to different levels of design earthquake motion. In this case, the relationship between the final displacements of the top of the wall and the ratio ac /amax,1D (Fig. 12) can be used to obtain the equivalent acceleration, for any given allowable displacement of the wall. This can be expressed as ah ¼ �amax,1D , where � (¼ ac /amax,1D) can be larger than 1. For the structural design of the wall, the maximum bending moments can be computed using a realistic distribution of contact stresses, such as that proposed in this work, and a conservative value of the pseudo-static acceleration: kh ¼ amax /g ¼ 2amax,1D : The data discussed in the paper refer only to cantilevered walls in dry sand; further research is required to clarify the role of the presence of the pore water, for either saturated or unsaturated soils.

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ACKNOWLEDGEMENTS The work presented in this paper was developed with the financial support of the Italian Department of Civil Protection within the ReLUIS research project. NOTATION a, b, x0 d d0 d GS G0 h KAE KG KP MS p9 pref ª �

constants of the hysteretic model depth of embedment depth of pivot point depth of full mobilisation of passive soil pressure secant shear modulus small-strain shear modulus excavation depth dynamic active earth pressure coefficient stiffness multiplier static passive earth pressure coefficient normalised secant shear modulus mean effective stress reference pressure shear strain; soil unit weight shear stress

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TECHNICAL NOTE

Revisiting Nigawa landslide of the 1995 Kobe earthquake H . L I N G � , H . I . L I N G † a n d T. K AWA BATA ‡

A simplified pseudo-static limit equilibrium approach is used to analyse a full-scale slope failure that occurred during the 1995 Kobe earthquake. The slope is 28 m high, having a gentle angle of 26 .58, and has a factor of safety as high as 2 .43 in the absence of an earthquake. In addition to a factor of safety, the critical failure surface obtained from pseudo-static analysis is compared with that of the actual failure surface. Contradictory to some design codes that use only a horizontal seismic coefficient, this study finds that a rational analysis must take into consideration both horizontal and vertical seismic accelerations. The analysis including vertical acceleration may give a shallower or deeper failure surface than cases where vertical acceleration is ignored. An analysis combining horizontal and upward accelerations results in a deeper failure surface than when only downward or no vertical acceleration is considered. The effects of soil properties on the slope stability are also investigated. The study suggests that for near-fault earthquakes, slopes may be subjected to vertical (upward) seismic coefficient as large as 90% that of horizontal, larger than that specified in existing design codes. KEYWORDS: earthquake; landslide; slope

drained ring shear tests to investigate possible rapid loss of strength due to liquefaction. Although ground water was observed during post-failure investigation, the writers felt that it was unlikely that the water table stayed constantly high during shaking, as was the degree of saturation in the soil mass, to allow for liquefaction at the site. Statistically, 1994 was reported to be one of the driest years. Thus, the water table was not included in the main analysis (effects of which were found to be small), and a different mechanism of sliding of the gentle slope could be involved, such as could be attributable to large horizontal and vertical accelerations. In this study, a series of pseudo-static analyses were conducted on the Nigawa slope. Both horizontal and vertical accelerations were included in the analyses. The results were compared with the observed failure surface and discussed.

INTRODUCTION The 1995 Kobe earthquake (Hyogoken Nambu earthquake, M ¼ 7 .2) resulted in many landslides along the vicinity of the fault, among which Nigawa landslide was the largest slope failure induced by an earthquake, in terms of scale and casualties. The slope was 28 m high and gentle, around 26 .58 (more details provided subsequently). The landslide covered the width and length, both of 100 m, and a maximum thickness of 15–20 m. A total volume of 120 000 m3 of soil was involved. The sliding soil mass travelled rapidly over a maximum distance of 175 m in a single event to the toe of the slope and blocked the Nigawa River. The landslide destroyed 11 houses and killed 34 residents. The geological profile for the main section of the slope (A–A9) is shown in Fig. 1. (Sassa, 1996). The base rock was of stable granite, underlain by the Osaka group layer, which was of limnic and marine deposit of granite sands and clays from the Pliocene to Middle Pleistocene age. The terrace deposits were seen at the toe of the slope. The surface layer where sliding occurred was a fill consisting of Osaka group coarse, sandy soil. Kawabe (1996) investigated the topographical maps of the site that were published by the Geospatial Information Authority of Japan in 1886, 1947 and 1990. It became apparent that the slope was a manmade high embankment, with two filling records. There has not been a rigorous failure analysis for Nigawa landslide, except by Sassa and colleagues (e.g. Sassa, 1996; Sassa et al., 1996), which focused on the mechanism of long-distance sliding. Sassa et al. (1996) conducted un-

METHODOLOGY OF ANALYSIS Limit equilibrium methods are typically used to analyse slope stability (e.g. Fredlund & Krahn, 1977; Bromhead, 1992; Duncan & Wright, 2005). By including an inertia force due to an earthquake, several mechanisms, such as planar (e.g. Seed & Goodman, 1964; Ling et al., 1999), circular and log-spiral (e.g. Leshchinsky & San, 1994; Ling et al., 1997) have been proposed to assess seismic slope stability. The earthquake inertia force is considered through a seismic coefficient, which is a percentage of earth gravity. Because of the randomness of earthquake motions, it is difficult to determine the value of the coefficient, although the peak or a reduced value may be used according to the importance of structures as stated in the design codes. The design can also be linked to slope performance expressed in terms of earthquake-induced displacements, so that different values of seismic coefficient can in principle be used for different levels of slope performance (e.g. Ling & Leshchinsky, 1995, 1997; Bray & Rathje, 1998; Stewart et al., 2003). In design, while the horizontal acceleration is considered, the vertical acceleration is usually ignored, since a factor of safety may

Manuscript received 3 December 2012; revised manuscript accepted 10 October 2013. Published online ahead of print 2 May 2014. Discussion on this paper closes on 1 October 2014, for further details see p. ii. � Columbia College, Columbia University, New York, NY, USA. † Department of Civil Engineering and Engineering Mechanics, Columbia University, New York, NY, USA. ‡ Geotechnical and Environmental Engineering, Faculty of Agriculture, Kobe University, Nada-ku, Kobe, Japan.

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LING, LING AND KAWABATA Granite Osaka group Terrace deposits Secondary deposits (fill)

B-6

110

Sliding surface B-7

Height: m

100

Initial landslide mass (before motion) Deposited landslide mass

B-8

B-9

90

B-23

B-5 B-10

80 70 60 0

50

100 Width: m

150

200

Fig. 1. Nigawa landslide: slope geometry of section A–A9 (Sassa, 1996)

portion of slope was 6 .3 m high and had an angle of 13 .138. The toe was inclined at 4 .68. The basic soil properties were obtained from a post-failure investigation (Kawasaki Chishitsu, personal communication, 2010). Fig. 2(a) shows the grain size distribution curves of two soil samples, where the mean diameter D50 was between 0 .5 and 0 .7 mm and the percentage of fines was between 25 .5 to 28 .4%. The soil was classified as silty sand and its specific gravity was 2 .645. The void ratio was 0 .558 and the natural water content was 11 .65%, which corresponded to a degree of saturation of 55 .3%. A series of direct shear tests was conducted using undisturbed soil specimens of diameter 15 cm and height 6 .2 cm at the natural water content (unsaturated condition). The specimen was sheared at a displacement rate of 1 mm/min. Four normal stresses of 19 .62 kPa, 39 .24 kPa, 58 .86 kPa and 78 .48 kPa were applied. Fig. 2(b) shows the relationships between the shear stress and horizontal displacement of

be adequate to smear the effects of vertical excitation. However, relatively large vertical components of acceleration, larger than 30% that of the horizontal, have been recorded during recent major earthquakes (e.g. Ling & Leshchinsky, 1998), such that their effects may no longer be ignored. Ling et al. (1997) indicated that effects of vertical acceleration may become prominent when combined with large horizontal component. In Eurocode 8 (CEN, 2005), both horizontal and vertical components of acceleration have to be considered in seismic design, with kv ¼ �0 .33kh (for avg /ag < 0 .6) or kv ¼ �0 .5kh (for avg /ag . 0 .6), where ag is the design value of acceleration and avg is the design value of vertical acceleration. kh and kv are the horizontal and vertical seismic coefficients, respectively. A serviceability evaluation, such as using the rigid sliding block method, is also required. Although there are several design codes in Japan (e.g. RTRI, 2007), a pseudo-static analysis of slope is also followed by a permanent displacement evaluation. The vertical component of acceleration is typically not used for slope stability analysis in Japan. Because of a large seismic coefficient used in designing against level II earthquakes, a specified factor of safety may no longer be satisfied, thus the design has to be based on serviceability. In the USA, as documented in the NCHRP (2008), seismic design of a slope does not require a vertical component of acceleration. In this study, a circular mechanism of the modified Bishop method (Bishop, 1955) is used. The analysis was conducted using the program ReSSA (Leshchinsky, 2002). Because of the length restriction, the procedure of incorporating inertia forces into the modified Bishop method is not elaborated. Note that, in this study, the vertical seismic coefficient is considered positive when vertical inertia force is acting upward. The silty sand disintegrated after failure, thus this study did not aim to evaluate the permanent displacements, which are usually obtained by assuming a rigid body motion.

100

Percent finer:%

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Sample 1 Sample 2

60 40 20 0 0·001

0·01

0·1 Grain size: mm (a)

1

10

Shear stress, τ: kPa

100

SLOPE GEOMETRY AND SOIL PROPERTIES The slope geometry was obtained from the failure section (Fig. 1) as reported by Sassa et al. (1996), but interpreted slightly differently here (see Fig. 4(a), in the results section, for the geometry used in analyses). The survey showed that the total height of the slope was 28 .1 m, and was composed of two parts. The bottom part of the slope (main slope) was 21 .8 m high and at an angle of 26 .58, whereas the top

80

σn: kPa 78·5

60 58·9 39·2

40

19·6

20 Sample 1 0

0

2

4

Sample 2

6 8 10 12 14 16 Horizontal displacement: mm (b)

18

20

Fig. 2. Soil properties: (a) grain size distribution, (b) direct shear test results

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REVISITING NIGAWA LANDSLIDE OF THE 1995 KOBE EARTHQUAKE Railway) Takarazuka Station was located closest to the Nigawa landslide site (note that this station has been relocated after the earthquake). The seismic coefficients may be selected based on the design codes as explained earlier, using reduced values. Since this study analysed a failure case, coefficients corresponding to actual peak earthquake records are considered (as will also be seen later from the results of analysis that reduced seismic coefficients based on design codes would not render reasonable results). Figure 3(b) shows the records of horizontal (east–west and north–south directions) and vertical accelerations. The seismic coefficients based on peak acceleration in various directions are: north ¼ 0 .586, south ¼ 0 .613, east ¼ 0 .365, west ¼ 0 .697. From the orientation of the major cross-section of failure (A–A9) along the north–east direction, and assuming a linear variation of coefficients between the north and east directions, the peak horizontal seismic coefficient was obtained as kho ¼ 0 .476. The seismic coefficients for the vertical inertia force (upward and downward) were kvo ¼ 0 .426 and �0 .306 (i.e. kvo /kho ¼ 0 .89 and �0 .782), respectively. In the analyses, several different combinations of seismic coefficients were used: horizontal coefficient only, horizontal and upward vertical coefficients, horizontal and downward vertical coefficients.

samples 1 and 2. Sample 1 exhibited slight softening behaviour. The cohesion (should be known as apparent cohesion in lieu of unsaturated conditions) and the angle of internal friction were obtained as c ¼ 16 .7 kPa and ¼ 38 .78, respectively. Sample 2 was of natural water content of 20 .65% or degree of saturation of 89 .1%, rendering strength parameters of c ¼ 7 .65 kPa and ¼ 36 .18. The results were of similar trend as reported by Ling & Ling (2012), where apparent cohesion reduced with increasing water content for unsaturated soil. The authors felt that direct shear tests, which allowed principal stress rotation, are more relevant than triaxial tests in representing the soil strength used in slope stability analysis. In the analyses shown subsequently, using ¼ 38 .78 (sample 1), the best fit cohesion was c ¼ 20 kPa, which is very close to that of sample 1. SEISMIC COEFFICIENTS The site was 37 .5 km from the epicenter, but merely 500 m away from the fault (Figure 3(a)). Among the acceleration records of the 1995 Kobe earthquake, JR (Japan 40

20

km

RESULTS AND DISCUSSION Figure 4 shows the slope geometry and results of stability analysis. Under static conditions, the slope had a factor of safety as high as 2 .43, which indicated a very stable slope. Considering the Kobe earthquake with peak horizontal and vertical accelerations, which acted upward and downward, respectively, the factors of safety are 0 .782 and 1 .17. For the seismic analysis without considering vertical acceleration, the yield horizontal seismic coefficient (giving a factor of safety of unity) is 0 .5. The horizontal yield seismic coefficients including vertical accelerations (upward and downward) are 0 .38 and 0 .65, respectively. Thus, the yield

km

km

6·3 m

10

N

28·1 m

A: Epicentre B: Landslide site C: Takarazuka station (a) 0·8

North–south

Potential failure surface 26·5°

kho 0·586

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21·8 m

30

km

c 20 kPa φ 38·7° γt 17·45 kN/m3

4·6°

Kobe earthquake (1995)

(a)

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3·0

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East–west

0·4

2·5

kho 0·365

Factor of safety: Fs

Acceleration: g

0·8

0 0·4 0·8 Upwards–downwards kvo 0·426

0·4 0

Fs 2·43 (static) kv /kh

2·0 1·5

0·895 (up) 0 0·782 (down)

Stable

1·0

Failure Unstable

0·5

kh 0·38 0·5

0·4 0

5

10 Time: s (b)

15

0

20

Fig. 3. (a) Location of Nigawa landslide with reference to epicentre (map data # 2010 Google, Zenrin); (b) earthquake records obtained at JR Takarazuka station

0

0·65

0·2 0·4 0·6 Horizontal seismic coefficient, kh (b)

0·8

Fig. 4. Stability analysis: (a) slope geometry, (b) factor of safety under different combinations of horizontal and vertical accelerations

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LING, LING AND KAWABATA ward vertical accelerations, when combined with the same magnitude of horizontal acceleration. The horizontal acceleration combined with upward vertical acceleration (kh ¼ 0 .38, kv /kh ¼ 0 .895) gave a failure surface closest to that observed in the field; a smaller value of kv /kh did not lead to acceptable matching of failure surface. The results implied that Eurocode 8, which has kv /kh ¼ 0 .5, is not conservative, especially in design where the failure surface is a major concern, such as deciding the reinforcement length for a reinforced slope (see Leshchinsky et al., 2014; Ling & Leshchinsky, 1998). The shape of failure surface and the yield horizontal seismic coefficient are greatly affected by the soil cohesion as shown in Fig. 5(b). Note that the angle of internal friction and the ratio of vertical to horizontal acceleration were kept the same for all cases. When cohesion is increased, a deeper failure surface and higher yield horizontal acceleration resulted. A cohesion of 20 kPa matched the failure surface of the lower part of the slope well, whereas it deviated at the upper part of the slope. Possible explanations included, but were not limited to, amplification of accelerations along the slope height, as well as the simplified two-dimensional as opposed to three-dimensional analyses. Note that in Leshchinsky et al. (1985), which used a log spiral mechanism, three-dimensional analysis gave a deeper failure surface at the top part of slope compared to that of two-dimensional analysis. The effect of the internal friction angle of soil was also investigated by varying it between 208 and 458. The cohesion (c ¼ 20 kPa) and ratio of vertical to horizontal acceleration (kv /kh ¼ 0 .895) were kept the same for all cases. The results (Fig. 5(c)) show that a small angle of internal friction rendered a lower yield acceleration, and vice versa. However, the difference in the failure surfaces was insignificant. It has to be noted that the water table was varied in a separate analysis, but showed negligible effect on the failure surface.

horizontal accelerations can be larger or smaller than the peak value of 0 .476 depending on the direction and magnitude of vertical acceleration. The inclusion of vertical acceleration in the downward direction gives a larger factor of safety compared to that when acting upward. It is clear that in order to obtain the most appropriate value of yield accelerations, the geometry of the failure surface has to be considered. Fig. 5(a) shows the different failure surfaces obtained under different combinations of horizontal and vertical accelerations. The upward vertical acceleration gives a deeper failure surface than the down-

φ 38·7° 60

c 20 kPa γ 17·45 kN/m3 kv /kh 0, khy 0·5

40

kv /kh 0·25, khy 0·46 kv /kh 0·5, khy 0·425 kv /kh 0·895, khy 0·38

kv /kh 0·25, khy 0·54 kv /kh 0·5, khy 0·59 kv /kh 0·782, khy 0·65

Observed

20

0 Unit: m 0

20

40

60

80

100

(a) φ 38·7° 60

γ 17·45 kN/m3 c 50 kPa, khy 0·493

kv /kh 0·895

c 40 kPa, khy 0·46 c 30 kPa, khy 0·425

40

c 20 kPa, khy 0·38 c 10 kPa, khy 0·315

20

CONCLUSION Nigawa landslide of the 1995 Kobe earthquake was analysed using the best available information on soil properties and earthquake characteristics. For a reasonable comparison between the results of analysis and field observation, the failure surface should be considered in addition to a factor of safety. The analysis led to the conclusion that upward vertical acceleration gave the most critical failure surface compared to cases of downward vertical acceleration or when vertical acceleration was ignored. The best-fit failure surface was obtained as kh ¼ 0 .38, which is less than the peak value of 0 .46 (0 .82 instead of 0 .5 for soil factor of 1 in Eurocode 8); and kv /kh ¼ 0 .9, which is significantly larger than 0 .33 (for kh , 0 .6) as assumed in Eurocode 8. This study indicated that vertical (upward) combined with horizontal accelerations might have contributed to the actual failure of Nigawa slope, instead of liquefaction as explained by Sassa et al. (1996). Additional failure cases should be analysed for near-fault earthquakes, so as to re-evaluate the recommended design value of kv /kh :

Observed

0 Unit: m 0

20

40

60

80

100

(b) φ 38·7° 60

40

20

γ 17·45 kN/m3 kv /kh 0·895

φ 45°, khy 0·438 φ 38·7°, khy 0·38 φ 30°, khy 0·286 φ 20°, khy 0·146 Observed

0 Unit: m 0

20

40

60

80

ACKNOWLEDGEMENTS The writers appreciated the technical information provided by Kawasaki Chishitsu Chosa, Co. The location of the failure site is shown by Google Maps in Fig. 3(a). Dr Meguro of the University of Tokyo kindly provided the earthquake records and Dr Luigi Callisto of the University of Rome La Sapienza provided information on Eurocode 8.

100

(c)

Fig. 5. (a) Effects of vertical acceleration on critical failure surface and yield acceleration; (b) effects of cohesion on critical failure surface and yield acceleration; (c) effects of angle of internal friction on critical failure surface and yield acceleration

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REVISITING NIGAWA LANDSLIDE OF THE 1995 KOBE EARTHQUAKE ReSSA is licensed by Dov Leshchinsky through Adama Engineering, Inc., Newark, DE, USA.

of slopes: design charts. J. Geotech. Engng, ASCE 120, No. 9, 1514–1532. Leshchinsky, D., Baker, R. & Silver, M. L. (1985). Three dimensional analysis of slope stability. Int. J. Numer. Analyt. Methods Geomech. 9, No. 3, 199–223. Leshchinsky, D., Kang, B., Han, J. & Ling, H. I. (2014). Framework for limit state design of geosynthetic-reinforced walls and slopes. Transportation Infrastructure Geotechnology, 10.1007/ S40515-014-0006-3. Ling, H. I. & Leshchinsky, D. (1995). Seismic performance of simple slopes. Soils Found. 35, No. 2, 85–94. Ling, H. I. & Leshchinsky, D. (1997). Seismic stability and permanent displacement of landfill cover systems. J. Geotech. Geoenviron. Engng, ASCE 123, No. 2, 113–122. Ling, H. I. & Leshchinsky, D. (1998). Effects of vertical acceleration on seismic design of geosynthetic-reinforced soil structures. Geótechnique 48, No. 3, 347–373. Ling, H. & Ling, H. I. (2012). Centrifuge model simulations of rainfall-induced slope instability. J. Geotech. Geoenviron. Engng 138, No. 9, 1151–1157. Ling, H. I., Leshchinsky, D. & Mohri, Y. (1997). Soil slopes under combined horizontal and vertical seismic accelerations. Earthquake Engng Struct. Dynamics 26, No. 12, 1231–1241. Ling, H. I., Mohri, Y. & Kawabata, T. (1999). Seismic stability of sliding wedge: extended Francais–Culmann’s analysis. Soil Dynamics Earthquake Engng 18, No. 5, 387–393. NCHRP (National Cooperative Highway Research Program) (2008). Seismic analysis and design of retaining walls, buried structures, slopes, and embankments, Report 611. Washington, D. C., USA: Transportation Research Board. RTRI (2007). Design standards for railway structures – earth structures. Japan: Maruzen Co., Ltd (in Japanese). Sassa, K. (1996). Prediction of earthquake induced landslides. In Landslides (ed. K. Senneset), pp. 115–132. Rotterdam, the Netherlands: Balkema. Sassa, K., Fukuoka, H., Scarascia-Mugnozza, G. & Evans, S. (1996). Earthquake-induced-landslides: distribution, motion and mechanisms. Soils and Found., Special Issue, 53–64. Seed, H. B. & Goodman, R. E. (1964). Earthquake stability of slopes of cohesionless soils. J. Soil Mech. Found. Div., ASCE 90, No. 6, 43–73. Stewart, J. P., Liu, A. H. & Choi, Y. (2003). Amplification factors for spectral acceleration in tectonically active regions. Bull. Seismol. Soc. Am. 93, No. 1, 332–352.

NOTATION ag avg c D50 Fs kh kho kv kvo ªt �n � �

design value of acceleration design value of vertical acceleration cohesion mean diameter factor of safety horizontal seismic coefficient horizontal seismic coefficient (peak value) vertical seismic coefficient vertical seismic coefficient (peak value) bulk unit weight normal stress shear stress angle of internal friction

REFERENCES Bishop, A. W. (1955). The use of slip circles in the stability analysis of earth slopes. Geótechnique 5, No. 1, 7–17. Bray, J. D. & Rathje, E. M. (1998). Earthquake-induced displacements of solid waste landfills. J. Geotech. Geoenviron. Engng, ASCE 124, No. 3, 242–253. Bromhead, E. N. (1992). The stability of slopes. Glasgow, UK: Blackie Academic and Professional. CEN (European Committee for Standardization) (2005). Eurocode 8: Design of structures for earthquake resistance – Part 5: foundations, retaining structures and geotechnical aspects. Brussels, Belgium: European Committee for Standardization. Duncan, J. M. & Wright, S. G. (2005). Soil strength and slope stability. New York, USA: Wiley. Fredlund, D. G. & Krahn, J. (1977). Comparison of slope stability methods of analysis. Can. Geotech. J. 14, No. 3, 429–439. Kawabe, T. (1996). Liquefaction disasters of the 1995 South Hyogo earthquake – Special reference to the origin of slope collapse at Nigawa-Yurinodai, Nishinimiya City. Tohoku J. Natural Disaster Sci. 32, 213–218 (in Japanese). Leshchinsky, D. (2002). Design software for geosynthetic-reinforced soil structures. Geotech. Fabrics Rep. 19, March/April, pp. 44–49. Leshchinsky, D. & San, K.-C. (1994). Pseudo-static seismic stability

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Kokkali, P. et al. (2014). Geótechnique 64, No. 11, 865–880 [http://dx.doi.org/10.1680/geot.14.P.064]

Static and cyclic rocking on sand: centrifuge versus reduced-scale 1g experiments P. KO K K A L I , I . A NA S TA S O P O U L O S † , T. A B D O U N a n d G . G A Z E TA S ‡

Shallow foundations supporting bridge piers, building frames, shear walls and monuments are often subjected to extreme lateral loading such as wind in offshore environments, or strong seismic shaking. Under such loading conditions, foundations may experience a host of non-linear phenomena: sliding on and uplifting from the supporting soil or even soil failure in the form of development of ultimate bearing capacity mechanisms. This type of response is accompanied by residual settlement and rotation of the supported structural system. Nevertheless, inelastic foundation performance can provide potential benefits to the overall seismic integrity of the structure. Thanks to such non-linearities, energy dissipation at or below the foundation level may eventually limit the seismic demand on structural elements. Several theoretical and experimental studies have provided encouraging evidence to this effect. This paper has a dual objective: first, to study the behaviour of shallow foundations under vertical and lateral monotonic loading and under lateral slow cyclic loading of progressively increasing amplitude; second, to explore the differences in foundation response between reduced-scale 1g and centrifuge 50g model testing. Emphasis is placed on interpreting their discrepancies by unveiling the role of scale effects. The role of soil densification due to multiple loading cycles with uplifting is also highlighted. KEYWORDS: bearing capacity; centrifuge modelling; footings/foundations; settlement

INTRODUCTION The importance of non-linear soil–foundation–structure interaction under lateral loading has been acknowledged by the engineering community, especially for offshore structures which are typically subjected to multicycle wind and wave loading. In the case of seismic shaking, with the recorded acceleration levels by far exceeding the conventional design guidelines in recent seismic events, it has become evident that inelastic foundation response is often unavoidable. Shallow foundations supporting bridge piers or building columns and shear walls may experience sliding and/or uplifting from the supporting soil, or bearing capacity failure in softer soils. Usually, such non-linear response is accompanied by permanent settlement and/or rotation. However, such mobilisation of strongly inelastic response also reduces the seismic demand and may therefore be beneficial for the seismic performance of the structural system. The potential benefits from these types of non-linearity have been indicated by several researchers (Priestley et al., 1996; Pecker & Pender, 2000; Gazetas & Apostolou, 2004; Mergos & Kawashima, 2005; Gajan & Kutter, 2008; Anastasopoulos et al., 2010a; Gelagoti et al., 2012; Kourkoulis et al., 2012). Several studies have explored the behaviour of foundations under lateral and combined loading, both theoretically (Nova & Montrasio, 1991; Butterfield & Gottardi, 1994; Paolucci, 1997; Bransby & Randolph, 1998; Martin & Houlsby, 2001; Gourvenec & Randolph, 2003; Chatzigogos et al., 2009) and experimentally (Negro et al., 2000; Gajan et al., 2005; Gajan & Kutter, 2008; Paolucci et al., 2008; Anastasopoulos et al.,

2012, 2013; Deng et al., 2012; Drosos et al., 2012). Experimental studies have significantly contributed to the understanding of the rocking response of shallow foundations. Nevertheless, many of them have been conducted at a low confining stress environment (reduced-scale 1g testing). Compared to centrifuge model testing, 1g experiments are easier and more economical to perform but cannot reproduce the actual stress field in the soil. Owing to the low prevailing confining stresses in 1g test conditions, the angle of shearing resistance and the small strain stiffness of the soil are typically much larger compared to realistic stress levels. Such issues may have a substantial effect on the measured response and therefore 1g tests should be carefully designed and the results should be interpreted accordingly. In an attempt to clarify these issues, commonly referred to as ‘scale effects’, a qualitative and quantitative comparison of the rocking response of shallow foundations obtained from centrifuge and reduced-scale 1g experiments is presented in this paper. Simple slender systems founded on dry sand are designed to be equivalent in terms of vertical factor of safety. They are then subjected to lateral monotonic loading till overturning and lateral slow cyclic loading. The response of the equivalent systems is compared in terms of moment capacity, settlement accumulation during cyclic loading, stiffness degradation and energy dissipation. The results elucidate some salient features of the behaviour of shallow foundations subjected to large deformations, offering a quantification of the role of scale effects in cyclic foundation response.

Manuscript received 10 April 2014; revised manuscript accepted 23 October 2014. Discussion on this paper closes on 1 April 2015, for further details see p. ii. Rensselaer Polytechnic Institute, Troy, NY, USA. † University of Dundee, UK; formerly National Technical University of Athens, Greece. ‡ National Technical University of Athens, Greece.

PROBLEM DEFINITION AND EXPERIMENTAL METHODOLOGY A series of centrifuge model tests were conducted in the 3 m radius, 150 g-tonne capacity centrifuge of the Centre for Earthquake Engineering Simulation at Rensselaer Polytechnic Institute (RPI). The corresponding reduced-scale 1g tests were performed at the Laboratory of Soil Mechanics of the

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KOKKALI, ANASTASOPOULOS, ABDOUN AND GAZETAS sidering the shallow nature of the rocking failure mechanism. This prototype system was appropriately scaled down according to the relevant centrifuge and 1g scaling laws (Wood, 2003) and the resulting geometries for the centrifuge and the 1g models are shown in Fig. 1. The lateral loading tests were conducted at a 50g centrifuge acceleration (1:50 scale) and a scale of 1:20 was selected for the reduced-scale 1g tests. Vertical push tests were conducted prior to the lateral push tests in order to determine the bearing capacity of the soil–foundation systems and the corresponding vertical factors of safety (FS). The experimental configurations for the centrifuge and the 1g tests are presented in the following paragraphs and details about the modelling and the critical design parameters are also provided.

National Technical University of Athens (NTUA). The experimental investigation focuses on the response of a rigid single mass slender system supported on a surface footing and subjected to monotonic and slow cyclic lateral loading. The prototype soil–structure system studied is shown in Fig. 1(a). Founded on a square surface footing of width B ¼ 3 m, the system has its centre of mass located at 6 .9 m above ground. The oscillator is rigid in order to focus on the nonlinear foundation response. The supporting soil consists of dry sand of adequate thickness D � 3 .3B (¼ 10 m), con-

6·9 m

Experimental set-up for centrifuge tests The experimental configurations for the vertical and lateral loading centrifuge tests are depicted in Fig. 2. A fourdegrees-of-freedom in-flight robot designed to perform multiple tasks while the centrifuge is spinning was used in the experimental series. The robot is capable of articulating in three linear dimensions and rotating around one axis with variable speed. It can operate in single instruction mode or follow programmatic scripts. Custom tools for the end of the robotic arm (robot end-effector) were fabricated for the vertical and lateral push tests. While the bearing capacity tests and the lateral monotonic push-over tests were conducted in manual mode, the robot was programmed for the application of the cyclic loading path assuring control and consistency of the applied displacement and velocity through all the loading cycles. The set-up for the vertical push tests is shown in detail in Fig. 2(a) (plan view) and Fig. 2(b) (side view). A model square steel foundation of width B connected to the robot custom tool was placed at the centre of a square container. Adequate distance from the lateral boundaries (5B) was assured so that boundary effects were avoided. The foundation was pushed down until bearing capacity failure was indicated. The vertical displacement was applied and recorded by the robot and the reaction load was measured by the robot load cell. A rectangular container provided two test locations for the monotonic and the slow cyclic lateral push tests. Adequate distance from the box lateral boundaries (4B) and between the two test locations (5B) was assured to minimise boundary effects and interference between the different tests (Figs 2(c) and 2(d)). The structure was a three-piece unit comprising a steel foundation, a steel column and a steel mass located at a specified height. Sandpaper was placed beneath the foundation in order to attain an adequately rough foundation–soil interface and minimise sliding. A spherical aluminium attachment on top of the structure was pushed laterally by the robot during cyclic loading. This structure– sphere assembly was properly designed so that the structure could freely move without any lateral or vertical restraints. Details of this connection are shown in Fig. 2(d) and later in Fig. 4(c). After cyclic testing, the structure was moved to the second test location for the monotonic push-over test, and lateral displacement was applied against the side face of the structure until it overturned. A biaxial load cell, connected to the robot custom tool, measured the horizontal force in both the loading (x) and the transverse (y) direction, while the vertical force was monitored by the robot load cell. On-board cameras captured the horizontal and vertical displacements and rotation of the structure. Specialised software was used to analyse the recorded videos and extract displacement–time histories.

3m Dry sand

10 m

(a)

Scale 1:50

13·8 cm

6 cm Dry Nevada sand

20 cm

(b)

Scale 1:20

34·5 cm

15 cm Dry Longstone sand

50 cm

(c)

Fig. 1. Schematic illustration of the soil–structure systems studied: (a) prototype system; (b) centrifuge model; (c) reduced-scale 1g model

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STATIC AND CYCLIC ROCKING ON SAND

5B

Square foundation BB

5B

Vertical load and displacement measured by the robot load cell

5B

B

5B

5B

5B 3·3B

Dry Nevada sand

(b)

(a) Location 1: Monotonic loading Location 2: Cyclic loading C1, C2, C3, C4: Camera views for motion tracking

3B

Location 1

C1

Location 2

C4 B

4B

B

B

5B

4B 3B

C2

C3 (c) Biaxial load cell on robot custom tool

Location of the centre of mass

Mass

Detail of attachment used in cyclic loading

Column h 2·3B Footing 4B

B

B

5B

Dry Nevada sand

4B

3·3B

(d)

Fig. 2. Centrifuge model containers and experimental set-up for vertical and lateral loading tests: (a) vertical push test: plan view; (b) vertical push test: side view; (c) lateral push test: plan view; and (d) lateral push test: side view along the loading axis

locations, the cameras focused on a square grid in order to correct for lens distortion and field of view perpendicularity. The corrected video was then loaded into the software package Tema for tracking. Fig. 3 shows a snapshot during tracking of a lateral monotonic push-over test.

Tracking targets were mounted on the structure along each axis, and high-intensity light-emitting diodes (LEDs), placed at appropriate angles to minimise glare and reflection, enhanced video quality thus facilitating motion tracking. After the cameras and the structure were placed at their final

157


KOKKALI, ANASTASOPOULOS, ABDOUN AND GAZETAS Aluminium plate attached to the foundation only to facilitate motion tracking Column 2

Foot_left

Robot custom tool connected to square footing for soil surface preparation

Column 1 Foot_ centre Foot_right

Foot_left

Foot_centre

Foot_right

[1 103 m]

T 86 500·0 ms (a)

Foot_left

0

Foot_centre

Biaxial load cell on robot custom tool

Robot–structure connection for cyclic lateral loading

SDOF system

Foot_right 5

Spirit level

[1 103 ms] 0

10

20

30

40

50

60

70

80

90

SDOF system

100

Fig. 3. Motion tracking during monotonic push-over test (snapshot from Tema software)

Camera

An important part of the set-up procedure of the lateral push tests was the surface preparation and the alignment of the structure with respect to the robot loading axis. Placing the foundation with an initial inclination in any direction could result in undesired pre-stressing of the soil or biaxial loading. Therefore, the soil surface was levelled and the structure was precisely aligned to the robot loading axis, as shown in Fig. 4. In both cases, the robot custom tools were utilised to smooth and level the soil surface without disturbing the soil density and to place the footing at the specified test location. Fig. 4(c) shows the structure placed at the cyclic loading test location and the camera view used to track the motion of the footing along the loading axis.

(b)

(c)

Fig. 4. Centrifuge lateral push tests: (a) soil surface preparation; (b) alignment and placement of structure on soil surface; (c) structure located at test location

Experimental set-up for reduced-scale 1g tests Similar experimental configurations were developed in the reduced-scale 1g test series. Fig. 5(a) shows the set-up of the bearing capacity tests and Fig. 5(b) the one used in the lateral push tests. The locations of the models with respect to the lateral boundaries of the rigid container used in the Hinged connection

Vertical slider

Screw-jack Load cell actuator h 2·3B

Load cell

5B

B

Dry Longstone sand

5B

5B 3·3B

B

Dry Longstone sand

(a)

5B 3·3B

(b) Laser displacement transducer Wire displacement transducer

Fig. 5. Model containers and experimental set-up for reduced-scale 1g tests: (a) vertical push test set-up; (b) lateral push test set-up

158


STATIC AND CYCLIC ROCKING ON SAND experimental series are also shown. A push-over apparatus, fixed to a reaction wall and consisting of a servomotor attached to a screw-jack actuator, was used to apply the vertical or horizontal displacement. Four laser displacement transducers measured the settlement of the four corners of the footing during the vertical test and a load cell connected to the edge of the actuator measured the vertical reaction force. The foundation–structure model consisted of a square aluminium footing rigidly connected to a pair of rigid steel columns that supported an aluminium slab. Steel plates were symmetrically placed above and below the slab to model the system’s mass. Sandpaper was used beneath the foundation to simulate a realistically rough soil–footing interface. In this set-up the free end of the actuator was connected to the structure model using a vertical slider and a hinged connection in series. This connection allowed the system to freely settle, slide or rotate as horizontal displacement was applied. The horizontal load was measured by a load cell inserted between the vertical slider and the hinged connection. Horizontal and vertical displacements were recorded through a system of wire and laser displacement transducers. Accurate positioning of the structure at the test location without disturbing the soil surface was achieved by a system of four mechanical jacks. Photographs of the experimental configuration are shown in Fig. 6.

Load cell

Square foundation

(a) Load cell

SDOF system

Soil properties and samples preparation Nevada 120 sand was dry pluviated to the desired density with a consistent manual technique in the centrifuge test series. Dry Longstone sand was layered in the 1g test container through an electronically controlled raining system, capable of producing sand specimens of controllable relative density. The raining system has been calibrated through a series of pluviation tests documented in Anastasopoulos et al. (2010b). Nevada 120 sand is a laboratory grade with D50 ¼ 0 .15 mm and uniformity coefficient Cu ¼ 2 .35. Longstone sand is an industrially produced fine and uniform quartz sand, also having D50 ¼ 0 .15 mm and uniformity coefficient Cu ¼ 1 .42. The grain size distribution of both sand specimens is shown in Fig. 7 and their properties are summarised in Table 1. The stress level prevailing in the 1g tests is unavoidably low; therefore, the strength characteristics of Longstone sand need to be known at a wide range of stresses. Fig. 8 shows the dependence of the angle of shearing resistance on the normal stress level, as described in Anastasopoulos et al. (2010b) based on laboratory tests performed for two relative densities (Dr ¼ 45% and 80%). The friction angle of Nevada sand at three relative densities (Dr ¼ 40%, 60% and 70%), at a reference mean effective normal stress ¼ 100 kPa, is also shown as three points in Fig. 8 (Arulmoli et al., 1992). The two stress ranges prevailing at a depth equal to one foundation width in each type of test are also depicted. The lower bound corresponds to the average geostatic stress for these densities at this reference depth. The upper bound takes into account the dead load of the superstructure. This diagram will be revisited in the discussion of the test results later on.

Wire displacement transducers (b)

Fig. 6. Photographs of the reduced-scale 1g tests: (a) vertical push test; (b) cyclic lateral push test 100

Percentage finer: %

Longstone sand 80 Nevada sand 60 40 20 0 0·01

0·1

1

10

Particle size: mm

Fig. 7. Grain size distribution of Nevada and Longstone sand Table 1. Summary of soil properties for Nevada and Longstone sand Properties

Superstructure modelling Instead of scaling the dead load of the superstructure, a different methodology was followed, namely the vertical load over capacity ratio of the compared systems was kept constant in each set of experiments. This ratio is expressed through the vertical factor of safety FS and is directly correlated to the rocking response of shallow

emax emin D50 Cu Gs

159


Nevada 120 sand

Longstone sand

0 .887 0 .511 0 .15 mm 2 .35 2 .67

0 .995 0 .614 0 .15 mm 1 .42 2 .64

KOKKALI, ANASTASOPOULOS, ABDOUN AND GAZETAS 1g tests stress range

60

vertical stiffness of the 1g soil–foundation systems. Despite these differences, some interesting common trends can be observed: while the capacity of the foundation on dense sand eventually reaches a constant value (plateau of the curve), a continuing increase of the vertical load is observed for the loose and the two-layer soil profiles. This ‘hardening’ behaviour can be attributed to the stiffening of the loose supporting soil as the footing is pushed into the ground, in combination with the contribution of the footing embedment to the bearing capacity (for all four configurations). The vertical load capacity Nu of each system is also shown in Fig. 9. These values were determined in a consistent way for all eight tests. For the tests on loose sand and on the twolayer profiles the definition of the ultimate load is not straightforward because of the hardening behaviour. In this study, Nu was defined as the load for which the rate of change of the vertical stiffness became constant. In addition, the choice of the Nu point was consistent when comparing the same profiles at 1g and high g level. A summary of the vertical load capacities as well as the ratio of the 1g to the centrifuge ultimate loads is provided in Table 2. The 1g vertical load capacities are 30–70% higher than the loads sustained in the centrifuge tests. The differences observed between the two sets of tests can be further explored in view of the scale effects affecting the 1g foundation response. Nevada and Longstone sands exhibit comparable shear resistance at large stress levels and similar densities, as evidenced in Fig. 8. Nevertheless, below 50 kPa confining pressure the shearing resistance of Longstone sand exhibits a remarkable increase. As shown in the figure, the stress level for the 1g tests falls in this range, hence the overestimation of the friction angle and thereby of the ultimate vertical loads. To further verify this response, the classical expression of Meyerhof (1951) for the bearing capacity of a surface rectangular footing was employed to back-calculate the effective friction angle of the dense and loose soil profiles. Whereas for the centrifuge tests the friction angle of the loose and the dense sand was calculated to be 33 .58 and 38 .58, respectively, the friction angles of the same profiles at 1g were 478 and 508. Interestingly, the deviation between the centrifuge and the 1g load–settlement curves becomes more considerable for the dense soil (Table 2), indicating that the overestimation of the friction angle is more prominent for a dense sand profile with distinct dilative behaviour. Regarding the two-layer profiles, the response is affected by both soil layers. Revisiting Fig. 9, it can be noticed that the behaviour resembles the response of the underlying loose sand, which is of dominant importance in this deep failure mechanism.

55

φ: deg

50

Centrifuge tests stress range

45 40 35 30 25 20 0

50

100

150 σ: kPa

Longstone sand Dr 80% peak Dr 45% peak Dr 80% post peak

200

250

300

Nevada sand Dr 70% Dr 60% Dr 40%

Fig. 8. Direct shear test results for Longstone sand: dependence of the angle of shearing resistance on stress level (Anastasopoulos et al., 2010b). Friction angle for Nevada sand: evaluation from isotropically consolidated undrained compression tests at reference mean effective normal stress 100 kPa (Arulmoli et al., 1992). The stress ranges at a depth equal to one foundation width are also depicted. The lower bound corresponds to the geostatic stress at this depth. The upper bound includes the geostatic stress and the stress induced by the structure dead weight

foundations, distinguishing the sinking from the uplifting rocking response. Additionally, the location of the centre of mass is a crucial design parameter related to the geometric non-linearity of the system through the slenderness ratio h/B, which controls the uplifting and overturning potential as well as the ultimate moment capacity of the foundation. According to the above, the mass and geometry of the structure models were designed so that the equivalent centrifuge and 1g single degree of freedom (SDOF) systems had the same FS and h/B ¼ 2 .3. In terms of the vertical factor of safety FS, the bearing capacity tests provided the ultimate vertical load of each soil–footing system and the mass of the superstructure was then adjusted to satisfy the FS criterion. As already mentioned, no attempt was made to model the flexibility of the superstructure and both models were sufficiently rigid so that the system’s response was governed by non-linear soil–foundation response. BEARING CAPACITY TESTS Two uniform soil profiles were considered: a loose sand of relative density Dr ¼ 45% and a very dense sand of Dr ¼ 90%. Two-layer soil profiles were also tested, consisting of a loose (Dr � 45%) bottom layer and a dense (Dr � 90%) upper layer depth z. In the following, the depth of the upper dense layer (z) is normalised by the foundation width: z/B. Two layered profiles were investigated: z/B ¼ 0 .25 and 0 .5. The model square foundations were subjected to vertical push tests in order to estimate the bearing capacity of four soil–foundation systems and design the centrifuge and the 1g superstructures. In the following paragraphs, a comparison between the two types of tests is presented. All results correspond to prototype units. Figure 9 compares the vertical load–settlement curves of each of the four soil profiles. A first comparison reveals discrepancies between the centrifuge and the 1g response. The 1g models sustain higher vertical loads than the corresponding centrifuge systems. The significantly steeper initial slopes of the 1g load–settlement curves indicate a larger

ROCKING RESPONSE IN VIEW OF SCALE EFFECTS As described above, an alternative methodology was followed for the superstructure design in order to maintain similitude by keeping the vertical factor of safety and the slenderness ratio of the models constant and directly comparing the centrifuge and the 1g rocking response. Thus, the loose soil profile was chosen as a reference case and the superstructure mass was adjusted and distributed so that a slenderness ratio h/B ¼ 2 .3 and a factor of safety FS ¼ 5 were achieved for each system on loose sand. Using the same superstructures, the two-layer soil–foundation systems yielded factors of safety equal to 5 .5 (for z/B ¼ 0 .25) and 7 (for z/B ¼ 0 .5). The factors of safety on dense sand were slightly different and in order to avoid misinterpretations of the test results this case is not be presented here. A summary of the vertical factors of safety is included in Table 2.

160


STATIC AND CYCLIC ROCKING ON SAND Uniform soil profiles

Two-layer soil profiles N Foundation

N

width B z

s

s

Loose or dense sand

Dense sand Loose sand

20

50 Dr 45% 13·6 MN

40

10·3 MN

30

N: MN

N: MN

15

10

38·6 MN

22·8 MN 20

5 10 0 0

0·1

0·2

0·3 s: m (a)

0·4

0·5

0

0·6

0

25

0·2

0·3 s: m (b)

0·4

0·5

0·6

25 19·5 MN

20

20

15·6 MN

14·8 MN

15 11·2 MN

N: MN

N: MN

0·1

10

5

15

10

5

0 0

0·1

0·2

0·3 s: m (c)

0·4

0·5

0

0·6

0

0·1

0·2

0·3 s: m (d)

0·4

0·5

0·6

1g tests

Centrifuge tests

Fig. 9. Load–settlement curves obtained from monotonic vertical centrifuge and 1g push tests on four soil profiles: (a) loose sand; (b) dense sand; (c) z/B 0 .25; (d) z/B 0 .5 Table 2. Summary of vertical push-down test results Soil profile

Centrifuge test, Nu: MN

1g test, Nu: MN Centrifuge model

Loose sand Dense sand z/B ¼ 0 .25 z/B ¼ 0 .5

10 .3 22 .8 11 .2 14 .8

13 .6 38 .6 15 .6 19 .5

Nu, 1g /Nu centrifuge

FS

5 11 5 .5 7

1g model 5 14 5 .5 7

1 .32 1 .69 1 .39 1 .32

systems reaching higher moment capacities and exhibiting a more pronounced uplifting behaviour than those in the centrifuge. These trends are hardly surprising

Monotonic lateral loading The soil–structure systems described above were subjected to lateral loading until they overturned. Fig. 10 depicts the moment, the settlement and the rotational stiffness of the six systems as functions of the footing rotation. The moment is calculated as the product Fh, using the horizontal force F measured by the load cell multiplied by the lever arm h. The settlement that is induced only by the lateral loading refers to the centre of the footing. Overall, the two sets of tests show several differences between them, with the 1g

(a) higher angle of shearing resistance results in higher moment capacity (b) the geometric non-linearity (uplifting) in such systems is governed by the stiffness of the soil; the stiffer soil in the 1g test leads to greater uplifting. When it comes to the inelastic response of the systems,

161


KOKKALI, ANASTASOPOULOS, ABDOUN AND GAZETAS δ

ϑ M s

2000

2000

1000

3000

1000

2000

1000

0

0 0

0·04

0·08 ϑ: rad

0·12

0

0·16

0·04

0·08 ϑ: rad

0·12

0

0·16

0

0·16

0·16

0·08

0·08

0·08

0

0·08

0 ϑ: rad

0·08

0·16

0

0·08 0·16

KR: kN m/rad

KR: kN m/rad

0 ϑ: rad

0·08

0·16

0·08 0·16

60

60

40

20

0 0·0001

0·08

0·001

0·01 ϑ: rad (a)

0·1

1

0·04

0·08 ϑ: rad

0·12

0·16

0·08

0 ϑ: rad

0·08

0·16

0·001

0·01 ϑ: rad (c)

0·1

1

0

60

KR: kN m/rad

0·08 0·16

s: kN m

0·16

s: kN m

s: kN m

z/B 0·5: FS 7

z/B 0·25: FS 5·5

M: kN m

3000

M: kN m

M: kN m

Loose sand: FS 5 3000

40

20

0 0·0001

0·001

0·01

0·1

1

ϑ: rad (b) Centrifuge tests

40

20

0 0·0001

1g tests

Fig. 10. Comparison of monotonic centrifuge and 1g push tests in terms of moment–rotation, settlement–rotation and rotational stiffness against rotation: (a) loose sand; (b) two-layer soil profile z/B 0 .25; (c) two-layer soil profile z/B 0 .5

The settlement–rotation curves of the two test systems almost coincide at small rotational amplitudes while distinct differences are noticed at larger rotations. The 1g systems uplift from the supporting soil relatively quickly (uplift is denoted by the upward change of the slope of the settlement–rotation curve), whereas the centrifuge systems keep sinking (for loose sand and z/B ¼ 0 .25) or uplift slightly and much later (for z/B ¼ 0 .5). Evidently the vertical stiffness of the 1g systems is larger, following the larger soil stiffness and friction angle. The comparison under monotonic loading is concluded with the plots of the secant rotational stiffness as a function of the angle of rotation. The accurate measurement of KR is not feasible at small rotations due to limitations in sensor accuracy. The real data are plotted for rotational amplitudes larger than 0 .008 rad and curve fitting is adopted to extrapolate the small-strain rotational stiffness (dotted lines). Invariably, all the 1g systems exhibit significantly higher rotational stiffness. The deviation becomes smaller at larger rotational

that is, the moment degradation and the subsequent overturn due to P–� effects, a very good agreement is observed in terms of the overturning angle since the systems share the same slenderness ratio. Unsurprisingly, the moment capacity reached in the 1g two-layer system of z/B ¼ 0 .5 is overestimated to a greater extent than in the other two soil profiles. Since the rocking failure only extends to a very limited depth beneath the foundation, not more than half the width, it seems that scale effects become more adverse with the presence of the dilative dense upper layer that contains the rotational failure surface. The shape of the 1g moment–rotation curve reflects this behaviour. At low rotational amplitudes and therefore relatively low confining stresses, the 1g moment–rotation curve significantly deviates from the centrifuge curve. After the peak of the curve and when the stresses induced to the soil due to lateral loading have increased (around Ł ¼ 0 .1 rad) the 1g moment–rotation curve starts approaching the centrifuge curve and they eventually converge.

162


STATIC AND CYCLIC ROCKING ON SAND 0·50

amplitudes where the systems have uplifted from the supporting soil and reduction of the effective contact area has occurred. At that point, the geometric non-linearity dominates the response and the rotational stiffness quickly degrades in all cases.

0·40 0·25

δ/δR

0·25

Cyclic lateral loading The performance of the SDOF systems when subjected to slow cyclic lateral loading is evaluated in this section. Gajan et al. (2005) have shown through a series of centrifuge tests that the response of a rocking foundation to seismic shaking can be fairly well predicted from slow cyclic tests. Fourteen cycles of increasing displacement amplitude were applied at the centre of mass of the structures (Fig. 11). The lateral displacement is normalised to the overturning displacement of the equivalent rigid block on rigid base (R ¼ B/2 ¼ 1 .5 m). Even though the chosen load sequence is not representative of a specific earthquake, it allows the systems’ performance to be compared under a wide range of displacement amplitudes, stressing them from their elastic all the way into their metaplastic regime.

0·30

0·35

0·18 0·20 0·14 0·16 0·10 0·12 0·08 0·06 0·02 0·04

0

0·25

0·50 0

2

4

6 8 Cycle number

10

12

14

Fig. 11. Normalised cyclic lateral displacement applied at the centre of mass of SDOF systems

The cyclic response of the compared systems is outlined in terms of moment–rotation and settlement–rotation in Figs 12–14. Overall, qualitative and quantitative differences are observed. The 1g systems consistently exhibit higher cyclic moment capacity and accumulate more settlement than the δ

ϑ M s Centrifuge test Loose sand: FS 5

4000

4000

2000

rM 60%

M: kN m

M: kN m

2000

0

rM 80%

0

2 000

2 000

4 000 0·12

1g test Loose sand: FS 5

0·06

0 ϑ: rad

0·06

4 000 0·12

0·12

0

0

0·1

0·1

0·2

0·06

0 ϑ: rad

0·06

0·12

s: m

s: m

0·2 0·22 m

0·3

0·3

0·4

0·4

0·5 0·12

0·5 0·12

0·43 m 0·06

0 ϑ: rad (a)

0·06

0·12

0·06

0 ϑ: rad (b)

Fig. 12. Moment–rotation and settlement–rotation curves obtained from slow cyclic tests on loose sand (Dr test; (b) 1g test

163


0·06

0·12

45%): (a) centrifuge

KOKKALI, ANASTASOPOULOS, ABDOUN AND GAZETAS δ

ϑ M s Centrifuge test z/B 0·25: FS 5·5

4000

1g test z/B 0·25: FS 5·5

4000

rM 75% 2000

rM 50%

M: kN m

M: kN m

2000

0

2 000

2 000

4 000 0·12

0

0·06

0 ϑ: rad

0·06

4 000 0·12

0·12

0

0

0·1

0·1

0·06

0 ϑ: rad

0·06

0·12

0·06

0·12

s: m

s: m

0·14 m 0·2

0·3

0·2

0·3 0·31 m

0·4 0·12

0·06

0 ϑ: rad (a)

0·06

0·4 0·12

0·12

0·06

0 ϑ: rad (b)

Fig. 13. Moment–rotation and settlement–rotation curves obtained from slow cyclic tests on two-layer soil profiles (z/B (a) centrifuge test, (b) 1g test

0 .25):

the loose sand tests but indicated by the decreasing rate of settlement accumulation in all settlement–rotation plots; (b) due to this compaction-style densification, the footing penetrates into the ground and hence it soon becomes essentially embedded; (c) on this denser soil layer the embedded footing enjoys a greater ultimate moment resistance, hence the great overstrength, especially for the 1g tests; (d) during large amplitude rotation angles (Ł . 0 .04 rad), the footing (over an already denser soil) tends to uplift, although still eventually accumulates settlement. Similar trends have been noted by Drosos et al. (2012) and Anastasopoulos et al. (2013) in 1g experiments. But even for footings on saturated clay (under undrained conditions), Panagiotidou et al. (2012) observed theoretically a cyclic overstrength, which was attributed to the beneficial role of P–� effects acting in the opposite to the loading direction. Hence, a portion of the observed overstrength is not necessarily related to densification. The most evident difference between the centrifuge and the 1g tests lies in the accumulation of settlement during cyclic loading. The settlement response of the compared systems varies upon the vertical factor of safety. The systems on loose sand (FS ¼ 5) settle at the very first loading cycles of small

corresponding centrifuge systems. Nevertheless, the qualitative comparison reveals several interesting similarities between the response observed in the centrifuge and the 1g cyclic tests. The trend established in the monotonic pushover tests, where the moment capacity of the 1g tests reached higher levels, is also observed during cyclic loading. No substantial reduction of the moment capacity with number of cycles takes place. Additionally, a considerable amount of energy is dissipated in the foundation as suggested by the wide hysteresis loops. Rotational stiffness degradation is also observed and is discussed later in more detail. The cyclic moment capacity well exceeds the monotonic backbone moment–rotation curve for all systems under comparison. This apparent moment overstrength can be quantified by an overstrength ratio rM, defined as the increase in the cyclic moment capacity divided by the monotonic moment capacity. For the loosest soil and the 1g tests it reaches 80%; for the centrifuge tests it is smaller than 60%. In both types of test rM is a function of the soil profile and the vertical factor of safety. Several interesting phenomena take place in parallel during cyclic loading: (a) densification of the soil under the footing, a phenomenon most prominent in

164


STATIC AND CYCLIC ROCKING ON SAND δ

ϑ M s Centrifuge test z/B 0·5: FS 7

4000

1g test z/B 0·5: FS 7

4000 rM 20% rM 20%

2000

M: kN m

M: kN m

2000

0

2 000

4 000 0·12

0

2 000

0·06

0 ϑ: rad

0·06

4 000 0·12

0·12

0

0·06

0 ϑ: rad

0·06

0·12

0·06

0·12

0

0·1

0·1

s: m

s: m

0·1 m

0·2

0·2 0·2 m

0·3 0·12

0·06

0 ϑ: rad (a)

0·06

0·12

0·3 0·12

0·06

0 ϑ: rad (b)

Fig. 14. Moment–rotation and settlement–rotation curves obtained from slow cyclic tests on two-layer soil profiles (z/B (a) centrifuge test; (b) 1g test

0 .5):

pronounced uplifting behaviour. This could be attributed to the nature of cyclic loading and the potential for soil stiffening after multiple loading cycles. In an attempt to understand the cyclic settlement response, the last loading cycles of the cyclic loading tests on the z/B ¼ 0 .5 profiles are isolated and the settlement–rotation curves are compared in Fig. 16. The settlement accumulated in the previous cycles has been subtracted to allow more direct comparison. The two systems follow almost the same path in the first quarter cycle (A to B) up to the point that the centrifuge system enters the unloading quarter cycle, while the 1g system is still loaded to a slightly higher rotational amplitude. The difference in the response becomes clear during the unloading branch and continues thereafter. The accumulated settlement during this half cycle is larger for the 1g system. A reduced vertical stiffness of the 1g system that affects the unloading quarter cycle and the accumulation of settlement could be implied at this point. This hypothesis for the 1g system is further illustrated in the sketch of Fig. 16(b) (not drawn to scale). During the first quarter cycle (loading from A to B), large stress concentration takes place beneath the

rotational amplitudes and tend to uplift while still accumulating settlement when larger displacement is applied. The systems on the two-layer profiles exhibit a more prominent uplifting behaviour, accumulating less settlement. This is particularly evident in the case of z/B ¼ 0 .5 (FS ¼ 7). For all soil profiles considered, the rate of settlement accumulation is larger in the 1g tests. At the end of the cyclic push test, the 1g systems have settled twice as much as the centrifuge systems. The settlement–rotation response is summarised in Fig. 15, plotting the settlement per cycle against the cycle rotation halfamplitude. The settlement is normalised to the foundation width B and the rotation is normalised to the overturning angle of the equivalent rigid block on rigid base (ŁR ¼ B/2h, where B is the block width and 2h is the block height). For all sets of systems (FS ¼ 5, 5 .5 and 7) the settlement per cycle induced during the 1g slow cyclic push tests is higher at any rotational amplitude. The divergence becomes larger as the rotation increases and the settlement obtained from the 1g tests reaches values up to double the settlement obtained from the centrifuge tests. The cyclic settlement response contradicts the monotonic settlement response where the 1g systems exhibited more

165


KOKKALI, ANASTASOPOULOS, ABDOUN AND GAZETAS ϑc

zero, and hence the geostatic stresses become prevalent and scale effects play a substantial role. And since the geostatic stresses are unrealistically low in the 1g tests, soil stiffness is much lower at this stage and until the foundation regains full contact with the soil, the potential for settlement is substantially larger. This phenomenon takes place in every unloading quarter cycle, leading to an overestimation of the settlement in the 1g tests.

sc

0·16 Loose sand FS 5

sc /B

0·12

Stiffness degradation and energy dissipation As the systems are subjected to large amplitude rotation cycles, softening occurs and rotational stiffness degradation is observed. At these amplitudes a gap may form at one side of the footing while soil yielding occurs at the other side. In the following half cycle this phenomenon reverses and the gap has to close before the opposite corner of the footing starts uplifting. This gap formation and closing that occurs at large rotational amplitudes is responsible for the rotational stiffness degradation shown in the moment–rotation plots. The significant loss of contact between the foundation and the supporting soil results in reduction of the rotational stiffness of the unloading branches and produces the characteristic S-shape hysteresis loops. Even though the rotational stiffness degradation was evident in both centrifuge and 1g tests, the comparison with respect to the shape of the hysteresis loops reveals a fundamental difference: while the centrifuge tests produce distinct S-shaped loops, a more oval shape is demonstrated in the 1g plots. S-shaped moment– rotation response has been previously observed in centrifuge and large-scale experiments for systems with relatively high factor of safety or systems on dense soil (Gajan et al., 2005; Negro et al., 2000). In the same experimental investigations the moment–rotation curves were more oval-shaped for systems with lower factor of safety or systems on lowdensity sand. This section explores the rotational stiffness degradation and energy dissipation that takes place during cyclic loading. To this end, the cyclic rotational stiffness and the damping ratio were calculated for the different loading cycles that the six systems were subjected to. Two different approaches were followed for these calculations and are illustrated in Fig. 17. According to method I, the rotational stiffness is defined as the slope of the line connecting the two tips of the moment–rotation loop. Since the maximum moment and maximum rotation do not occur simultaneously, an alternative rotational stiffness can be calculated as the ratio of the maximum moment to the maximum cycle rotation (method II). The respective elastic areas used for the calculation of the damping ratio are shown on this plot. Figure 18 focuses on the rotational stiffness degradation and Fig. 19 on the dissipated energy during cyclic loading. In order to avoid misinterpretations due to inaccuracies in test measurements, the very first and small amplitude loading cycles are not included in the plots, and only cycles 3 to 14 are considered. The top graphs of Fig. 18 show the rotational stiffness as calculated according to the methods previously described. Following the trend noticed in the monotonic push tests, the rotational stiffness measured in the 1g tests is larger at any rotational amplitude. This agrees with the more oval shape of the 1g moment–rotation loops. In the bottom graphs the cyclic rotational stiffness is normalised to the small strain rotational stiffness as defined in the monotonic push over tests. Interestingly, the normalised results from the centrifuge and 1g tests follow identical degradation trend. In both absolute and normalised rotation plots, no distinct dependence of KR on the soil profile (or alternatively the factor of safety) could be possibly extracted for the tests of the same type (centrifuge or 1g) since there

0·08

0·04

0 0

0·1

0·2 ϑc /ϑR (a)

0·3

0·4

0·2

0·3

0·4

0·3

0·4

0·12 z/B 0·25 FS 5·5

sc /B

0·09

0·06

0·03

0 0

0·1

ϑc /ϑR (b)

0·08 z/B 0·5 FS 7

sc /B

0·06

0·04

0·02

0 0

0·1

0·2 ϑc /ϑR (c)

Centrifuge tests

1g tests

Fig. 15. Evolution of settlement during cyclic push tests: normalised settlement per cycle with respect to normalised cycle rotation half amplitude: (a) loose sand; (b) two-layer soil profile z/B 0 .25; (c) two-layer soil profile z/B 0 .5. B is the foundation width and ŁR is the overturning angle of the equivalent rigid block on rigid base

foundation corner as the latter uplifts from the supporting soil. As a result, at this stage scale effects are not important, as the geostatic stresses are not prevalent. As the second quarter cycle follows (unloading from B to C), the stress field under the uplifted footing becomes practically equal to

166


STATIC AND CYCLIC ROCKING ON SAND 0·08

z/B 0·5: FS 7 14th loading cycle

D

B

Δs: m

0·04

A

0

Centrifuge curve E

ΔsAC 2·6 mm ΔsCE 6·4 mm 0·04 0·10

1g curve

C

ΔsAC 7·7 mm ΔsCE 12·7 mm

0 ϑ: rad (a)

0·05

0·05

Loading quarter cycle: A → B

0·10

Unloading quarter cycle: B → C

Initial soil surface

σ0

Soil surface at A

High stress concentration around the foundation corner and soil yielding

σ 0 during unloading quarter cycle leads to higher settlement accumulation in 1g test

(b)

Fig. 16. (a) Settlement–rotation curve for the last loading cycle on the z/B 0 .5 profiles and comparison of settlement induced in each half cycle; (b) schematic illustration of the loading–unloading sequence in the 1g test (not drawn to scale)

Method I

Method II

M

Mmax

M(ϑmax) ΔEel ϑmax

ϑmax

ϑ ΔEcycle

ΔEel ϑmax

ϑmax

ϑ ΔEcycle

M(ϑmax) KR

rotation plots. Additionally, some dependence on the soil conditions or the factor of safety is present. An increase of the damping ratio is noticed for decreasing factor of safety. Overall, the average damping values are slightly larger for the 1g tests.

M

Mmax

2M(ϑmax)

KR

2ϑmax Damping ratio

SUMMARY AND CONCLUSIONS An experimental study on dry sand was performed for the rocking response of slender systems, and for comparison between centrifuge and reduced-scale 1g tests. Equivalent (as much as possible) systems on three different soil profiles were subjected to monotonic and slow cyclic lateral loading. Before the lateral loading tests, the bearing capacity of the soil-foundation systems was measured through vertical push tests that provided additional information regarding the role of scale effects on bearing capacity. Based on the measured vertical ultimate force of each foundation, the systems were designed so as to maintain an analogy between the key dimensionless parameters of the rocking response: the vertical factor of safety FS and the slenderness ratio h/B. The key conclusions can be summarised as follows.

2Mmax 2ϑmax

ΔEcycle 4πΔEel

Fig. 17. Definition of cyclic rotational stiffness and damping ratio derived from the cyclic moment–rotation diagrams following two approaches

are no substantial differences in the rotational stiffness values of the loose and the two-layer profiles. In the centrifuge tests, the damping ratio varies from 20% to 40%, being more or less independent of the soil conditions or, alternatively, of the FS of the system. The damping ratio is slightly larger for small rotation amplitudes, but for Ł . 0 .03 rad it remains constant. These observations apply to the results obtained by either method I or II. On the other hand, different trends are noted for the 1g tests. Increasing damping ratio with rotation is seen in the 1g tests that follow method I, while damping remains more or less constant when calculated with method II. In the first case the increase might be attributed to asymmetries in the cyclic loading that are also evident in the respective moment–

•

•

The low confining pressure prevailing in the 1g tests led to overestimation of the bearing capacity, since the effective friction angle of the soil is highly dependent on the stress level. This overestimation became more prominent when a dense sand profile was considered. The comparison between equivalent centrifuge and 1g lateral loading tests provided insight in several aspects of the problem. The 1g tests exhibited qualitative

167


KOKKALI, ANASTASOPOULOS, ABDOUN AND GAZETAS 1g tests

Loose sand

Loose sand

z/B 0·25

z/B 0·25

z/B 0·5

z/B 0·5

30

30

25

25

KR: kN 104 m/rad

KR: kN 104 m/rad

Centrifuge tests

20 15 10 5

20 15 10 5

0

0 0

0·02

0·04

0·06

0·08

0·10

0

0·02

0·04

0·06

0·08

0·10

0·3

0·4

0·5

ϑ: rad (b)

1·0

1·0

0·8

0·8

0·6

0·6

KR /KRo

KR /KRo

ϑ: rad (a)

0·4

0·2

0·4

0·2

0

0 0

0·1

0·2

0·3

0·4

0·5

0

ϑc /ϑR (c)

0·1

0·2

ϑc /ϑR (d)

Fig. 18. Rotational stiffness degradation: top graphs show rotational stiffness plotted against cycle rotation half amplitude and bottom graphs show ratio of rotational stiffness KR to small-strain rotational stiffness KRo as defined in the monotonic push tests plotted against the normalised cycle rotation half amplitude. The rotational stiffness was calculated according to method I in (a) and (c), and according to method II in (b) and (d)

• •

•

even though the actual rotational stiffness was larger in the 1g tests.

similarities with the centrifuge tests capturing the highly ductile cyclic response, the stiffness degradation, the high level of energy dissipation of the rocking systems, as well as the cyclic moment overstrength. A difference was observed in the moment–rotation hysteresis loops, which were more oval-shaped in the 1g tests, offering larger energy dissipation. Densification of the sandy layers during cyclic loading clearly played a significant role in all the tests, either centrifuge or 1g. As expected, the quantitative comparison revealed discrepancies in terms of vertical and moment capacity as well as settlement accumulation. The increased moment capacity observed in the 1g tests is attributed to the overestimation of the angle of shearing resistance. The increased settlement is most probably due to the reduced vertical stiffness during the unloading phase of the loading sequence, when the effective stress exerted to the soil by the foundation is zero and the response is governed by the geostatic stresses. In terms of rotational stiffness degradation, the centrifuge and 1g systems showed the same normalised response

Summarising, the comparison presented in this paper showed that reduced-scale 1g tests can provide valuable insights to the rocking response of SDOF systems only if properly interpreted, with due consideration to the actual soil properties at very small confining pressures. This topic could be further explored with direct comparisons of true seismic shaking that could reveal potential differences related to the stress dependent dynamic soil behaviour and the characteristics of the applied ground motion. The experimental findings presented in this paper could serve as a baseline to interpret these differences. ACKNOWLEDGEMENTS The authors would like to acknowledge financial support from the EU 7th Framework research project funded through the European Research Council’s Programme ‘Ideas’, Support for Frontier Research – Advanced Grant, under contract number ERC-2008-AdG 228254-DARE.

168


STATIC AND CYCLIC ROCKING ON SAND Centrifuge tests z/B 0·25

1g tests z/B 0·25

1·0

1·0

0·8

0·8

0·6

0·4

z/B 0·25

0·6

0·4

0·2

0·2

0

0 0

0·02

0·04

ϑ: rad

0·06

0·08

0

0·10

0·02

0·04

0·06

0·08

0·10

0·06

0·08

0·10

ϑ: rad

(a) 1·0

1·0

0·8

Damping ratio

0·8

Damping ratio

z/B 0·25

Loose sand

Damping ratio

Damping ratio

Loose sand

0·6

0·4

0·6

0·4

0·2

0·2

0

0 0

0·02

0·04

0·06

0·08

0·10

ϑ: rad

0

0·02

0·04

ϑ: rad

(b)

Fig. 19. Ratio of energy dissipation during cyclic push tests with respect to cycle rotation half amplitude. Damping was calculated according to method I in (a) and method II in (b)

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