Earth Pressure 3433032238, 9783433032237

The subject of earth pressure is one of the oldest and most extensive chapters in soil mechanics and foundation engineer

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Earth Pressure
 3433032238, 9783433032237

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Earth Pressure

Earth Pressure Achim Hettler Karl-Eugen Kurrer

Author Achim Hettler

Johann-Sebastian-Bach-Str. 9 76437 Rastatt Germany Karl-Eugen Kurrer

Gleimstr. 20a 10437 Berlin Germany

All books published by Ernst & Sohn are carefully produced. Nevertheless, authors, editors, and publisher do not warrant the information contained in these books, including this book, to be free of errors. Readers are advised to keep in mind that statements, data, illustrations, procedural details or other items may inadvertently be inaccurate. Library of Congress Card No.:

applied for Cover

Marie-Hélène H.-Desrue

British Library Cataloguing-in-Publication Data

A catalogue record for this book is available from the British Library. Bibliographic information published by the Deutsche Nationalbibliothek

The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at . © 2020 Wilhelm Ernst & Sohn, Verlag für Architektur und technische Wissenschaften GmbH & Co. KG, Rotherstraße 21, 10245 Berlin, Germany All rights reserved (including those of translation into other languages). No part of this book may be reproduced in any form – by photoprinting, microfilm, or any other means – nor transmitted or translated into a machine language without written permission from the publishers. Registered names, trademarks, etc. used in this book, even when not specifically marked as such, are not to be considered unprotected by law. Print ISBN: 978-3-433-03223-7 ePDF ISBN: 978-3-433-60898-2 ePub ISBN: 978-3-433-60896-8 oBook ISBN: 978-3-433-60895-1 Typesetting: SPi Global, Chennai, India Printing

Printed in the Federal Republic of Germany Printed on acid-free paper

For Marie-Hélène Achim Hettler and Claudia Karl-Eugen Kurrer

vii

Preface “You only have a future if you understand the past” Wilhelm von Humboldt, 1767–1835 Decades have passed since the publication of an entire book on the subject of earth pressure. For this purpose, refer for example to “Erddrucktheorien” by Árpád Kézdi from 1962 or to part II of the series on excavations by Anton Weißenbach from 1975, which essentially includes earth pressure issues although in this case mainly concerned with its application for excavation walls. In the meantime, the topic has been treated repeatedly as part of works in the fields of soil mechanics and foundation engineering, see for example “Bodenmechanik” by Gerd Gudehus from 1981 or the contributions to the “Grundbau-Taschenbuch”. Despite the importance of earth pressure theories in structural engineering, the current view has not been written yet. Many analytical applications have proved useful for decades. In recent years, the Finite-Element Method has been added as a new tool, and in practice, the displacement dependency of earth pressure has to be considered in more detail. Essentially, this book has three major themes. Firstly, to make a set of working instructions available to civil and structural engineers in construction companies, engineering firms and design departments as well as students. This is supplemented with comments on the current earth pressure standard of 2017 and the collection of samples from 2018. Then current methods for determining earth pressure are presented in detail. However, a basic understanding of today’s common theories and rules is hardly conceivable without a thorough study of history. The first empirical design rules were already known to the Romans; hints can be found in the publications by Vitruvius. Today’s theories began in France more than three centuries ago and are closely associated with French military engineers. The third major theme is therefore dedicated to historical development, complemented by the biographies of selected researchers who have made significant contributions to the subject of earth pressure. Without the support of assistants, it is hardly possible to complete a book. Jan Deutschmann has provided untiring, quick and competent support, as well as

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Preface

Marcel Deckert, Ingmar Zehn and Annette Richter. Furthermore, the publisher Ernst & Sohn supported the idea for the present book and its implementation from the very beginning. Achim Hettler, Karl-Eugen Kurrer Dortmund and Berlin, 2019

References Gudehus, G. (1981). Bodenmechanik. Stuttgart: Enke. Kézdi, A. (1962). Erddrucktheorien. Berlin, Göttingen. Heidelberg: Springer. Weißenbach, A. (1985). Baugruben, Teil II, Berechnungsgrundlagen, 1. Nachdruck. Berlin: Ernst & Sohn.

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Contents

1

Introduction 1 References 2

2

The history of earth pressure theory 3

2.1 2.2 2.2.1 2.2.2 2.2.3 2.2.4 2.3 2.3.1 2.3.2 2.4 2.4.1 2.4.2 2.5 2.5.1 2.5.2 2.5.3 2.5.4 2.6 2.6.1 2.6.2

Retaining walls for fortifications 5 Earth pressure theory as an object of military engineering 9 In the beginning there was the inclined plane 10 From inclined plane to wedge theory 20 Charles Augustin Coulomb 23 A magazine for engineering officers 34 Modifications to Coulomb earth pressure theory 36 The trigonometrisation of earth pressure theory 36 The geometric way 44 The contribution of continuum mechanics 54 The hydrostatic earth pressure model 56 The new earth pressure theory 58 Earth pressure theory from 1875 to 1900 67 Coulomb or Rankine? 68 Earth pressure theory in the form of masonry arch theory 69 Earth pressure theory à la française 71 Kötter’s mathematical earth pressure theory 75 Experimental earth pressure research 78 The precursors of experimental earth pressure research 78 Earth pressure tests at the testing institute for the statics of structures at Berlin Technical University 81 The merry-go-round of discussions of errors 85 The emergence of soil mechanics 87 Earth pressure theory in the discipline-formation period of geotechnical engineering 93 Terzaghi 96 Rendulic 99 Ohde 100 Errors and confusion 101 A hasty reaction in print 103

2.6.3 2.6.4 2.7 2.7.1 2.7.2 2.7.3 2.7.4 2.7.5

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Contents

2.7.6 2.8 2.8.1 2.8.2 2.9 2.9.1 2.9.2 2.9.3 2.9.4

Foundations + soil mechanics = geotechnical engineering 103 Earth pressure theory in the consolidation period of geotechnical engineering 109 New subdisciplines in geotechnical engineering 110 Determining earth pressure in practical theory of structures 111 Earth pressure theory in the integration period of geotechnical engineering 116 Computer-assisted earth pressure calculations 118 Geotechnical continuum models 119 The art of estimating 124 The history of geotechnical engineering as an object of construction history 125 References 128

3

Methods for the determination of earth pressure 145

3.1 3.1.1 3.1.2 3.2 3.3 3.4 3.4.1 3.4.2 3.4.3 3.4.4 3.4.5 3.4.6 3.5 3.5.1 3.5.2 3.5.3

Overview and bound methods 145 Overview of the methods 145 Upper and lower bounds 146 Kinematic mechanism methods for active earth pressure 147 Kinematic mechanism methods for passive earth pressure 150 Static stress field methods 154 Fundamentals 154 Rankine’s solution 155 Theory of Boussinesq/Résal/Caquot 156 Solution by Pregl/Sokolowski 157 Analysis of Goldscheider 157 Approach of Patki/Mandal/Dewaikar 158 Tests and measurements 159 Fundamentals and scaling laws 159 Evaluation of test results and application of scaling laws 163 Example: active earth pressure under plane strain conditions from soil self-weight 164 Example: passive earth pressure under plane strain conditions from soil self-weight 165 Example: spatial earth resistance in front of soldier piles 169 Example: spatial earth resistance in front of square anchor slabs 169 Further examples 170 Finite Element Method 172 General 172 Examples 174 References 189

3.5.4 3.5.5 3.5.6 3.5.7 3.6 3.6.1 3.6.2

4

Active earth pressure under plane strain conditions 195

4.1 4.2

Fundamental considerations 195 Soil self-weight, infinite uniformly distributed surcharges and cohesion 197 Vertical wall, level ground, horizontal earth pressure 198

4.2.1

Contents

4.2.2 4.2.3 4.3 4.3.1 4.3.2 4.3.3 4.3.4 4.4 4.4.1 4.4.2 4.4.3 4.5 4.6 4.7 4.8 4.9

Vertical wall, level ground, inclined earth pressure 198 General case 199 Cohesion, calculated tension and minimum earth pressure 199 Determination of the classic earth pressure 200 Minimum earth pressure in comparison with the resultant of earth pressure 201 Minimum earth pressure in comparison with the earth pressure ordinate 201 Minimum earth pressure and surcharges 202 Vertical line loads and strip loads 203 Introduction 203 Standard slip surface from soil self-weight 204 Analysis of any slip surface angle 206 Horizontal line and strip loads 208 Layered soil 209 Discontinuous ground level 210 Discontinuous wall surfaces 212 Distribution of active earth pressure 212 References 213

5

At-rest earth pressure 215

5.1 5.1.1 5.1.2 5.2

Soil self-weight and infinite uniformly distributed surcharges Horizontal ground 215 Inclined ground 217 Concentrated loads, line loads and strip loads 219 References 223

6

Passive earth pressure under plane strain conditions 225

6.1 6.2

Fundamental considerations 225 Soil self-weight, infinite uniformly distributed surcharges and cohesion with parallel movement 227 Straight slip surfaces 227 Pregl/Sokolowski 229 Comparison 230 Rotation about the top or the toe 231 Distribution of passive earth pressure 232 References 233

6.2.1 6.2.2 6.2.3 6.3 6.4

7.1 7.2 7.3

235 Fundamental considerations 235 Cylindrical surfaces 237 Retaining wall across the slope 239 References 242

8

Spatial passive earth pressure 245

8.1

Overview 245

7

Spatial active earth pressure

215

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8.2 8.3

Passive earth pressure in front of soldier piles according to Weißenbach 245 Procedure according to DIN 4085 for limited wall sections 248 References 249

9

Influence of groundwater on earth pressure 251

9.1 9.2 9.3

Groundwater at rest 251 Flowing groundwater 251 Water pressure in tension crack 253 References 254

10

Compaction effects on earth pressure 255

References 257 11

L- and T-cantilever retaining walls 259

References 261 12

Silo pressure 263

References 264 13

Dynamic loading

265

References 266 14

Particular cases 267

14.1 14.2 14.3 14.4 14.5 14.6 14.7

Repeated quasi-static loading 267 Pipelines 269 Lateral pressure on piles 269 Creep pressure 270 Swelling pressure 270 Heavily fissured rock 270 Active earth pressure within dams 272 References 272

15

Mobilisation of earth pressure 275

15.1 15.2

Overview 275 Limit values of displacement on reaching the active earth pressure 275 Limit values of displacement on reaching the passive earth pressure 276 Mobilisation functions 276 Mobilised active earth pressure 276 Mobilised passive earth pressure 279 Spatial mobilised passive earth pressure 282 References 283

15.3 15.4 15.4.1 15.4.2 15.4.3

Contents

16

Application rules 285

16.1 16.2 16.3 16.4

Earth pressure inclination and angle of wall friction 285 Magnitude of earth pressure depending on the wall displacement 287 Earth pressure redistribution 289 Earth pressure as a favourable action 291 References 292

17

Commentary on DIN 4085:2017-08 293

17.1 17.2 17.3 17.4 17.5 17.6

Overview 293 Active earth pressure 293 Passive earth pressure 295 Earth pressure due to compaction 295 Spatial earth pressure 296 Advices on supplement DIN 4085:2018-12 296 References 297

18

Forty selected brief biographies

References 355 Appendix A Terms, symbols, indices 383

A1: A2: A3:

Terms 383 Symbols 384 Indices 385

Appendix B Earth pressure tables

References 387

387

299

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1

1 Introduction The topic of earth pressure is considered one of the oldest and most extensive chapters in soil mechanics and foundation engineering. It is also one of the three pillars of structural engineering together with arch theory and beam theory. The first written sources, dating back to Vitruvius, are more than 2000 years old and therefore much older than the well-known theories of Coulomb (1773/1776) or Rankine (1857). In the first and sixth volume of his ten books, Vitruvius deals with the mode of action of earth pressure on retaining walls and proposes buttresses. Vauban, one of the greatest engineers in history, already published design tables for retaining walls with heights of up to 15 m in 1684, which cannot be bettered even today. The development of the earth pressure theory is described in detail in chapter 2 which is based on the extended edition of “The History of the Theory of Structures. Searching for Equilibrium” by Kurrer (2018). The present book can only include a limited selection of current design methods. The aim of the book is to provide a set of work instructions for foundation engineers and structural engineers in construction companies, engineering consultancies and in design departments, but also for students. In order to further theoretical understanding, the essential principles for determining earth pressure are initially presented in chapter 3. Chapters 4 to 12 contain the most important methods of determining active and passive earth pressure as well as at-rest earth pressure. In chapters 7 and 8, the spatial effects of earth pressure are taken into account. One concern of this book is to give a short overview of non-everyday questions and to refer to further literature (see chapter 14). In recent years, the displacement dependency of earth pressure has increasingly come into view. This applies not only to passive but also to active cases (see chapter 15). The book offers also instructions for practical application in chapter 16 and is supplemented by earth pressure tables for the most important basic cases. Many questions were submitted to the DIN Committee “calculation methods”, and a selection of these is discussed in the commentary to DIN 4085 in chapter 17. In the last section of this chapter, references are provided to the examples in the supplement to DIN 4085, which was published in December 2018. The history of earth pressure theory in chapter 2 includes a few selected short biographies of scientists and engineers working in the field who have taken up and developed the subject over the centuries, see chapter 18. The book is supplemented by two appendices with terms, symbols and indices (Appendix A) and Earth Pressure, First Edition. Achim Hettler and Karl-Eugen Kurrer. © 2020 Ernst & Sohn Verlag GmbH & Co. KG. Published 2020 by Ernst & Sohn Verlag GmbH & Co. KG.

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1 Introduction

earth pressure tables in Appendix B. For historical reasons, the current terms and formulas in chapters 3 to 17 and in the Appendices may differ from the original terms in chapter 2.

References Coulomb, C.A. (1773/1776). Essai sur une application des règles des Maximis et Minimis à quelques Problèmes de statique relatifs à l’Architecture. In: Mémoires de mathématique & de physique, présentés à l’Académie Royale des Sciences par divers savans, Vol. 7, année 1773, 343–382. Paris. Kurrer, K.-E. (2018). The History of the Theory of Structures. Searching for Equilibrium. Construction History Series (Ed. by K.-E. Kurrer and W. Lorenz). Berlin: Ernst & Sohn. Rankine, W.J.M. (1857). On the Stability of Loose Earth. Philosophical Transactions of the London Royal Society 147: 9–27.

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2 The history of earth pressure theory Digging, piling, tipping, stretching, arching, placing and laying are the archetypal forms of building which, in terms of their historical manifestation, appeared in this sequence and formed and still form the foundation for all great architecture. Even today, the archetypal forms are the basic ways of building (v. Halász, 1988, p. 257). Whereas digging reaches back into the depths of the animal-human transition, the teocalli of the Aztecs were magnificent pyramids built by piling and tipping. In fact, teocalli means “covered by stones” (v. Halász, 1988, p. 257), the core of the pyramid consisting of a pile of earth. Building with earth – earthworks – is, even today, based on three elementary forms of activity: digging, piling and tipping. Moving great bodies of soil to form the embankments, cuttings and cuts required during the building of roads, railways and waterways has changed and still changes not only the relief of the natural landscape, but also the urban landscape (Guillerme, 1995). The evolution of geotechnical engineering up to 1700 has been summarised in an extensive congress paper by Jean Kérisel, who from 1951 to 1969 was honorary professor of soil mechanics at the École Nationale des Ponts et Chaussées in Paris (Kérisel, 1985). In contrast to that work, this chapter will try to trace the theory of earth pressure from its beginnings shortly before the turn of the 18th century right up to the present day from the perspective of the history of theory of structures. Besides original sources, the following historical studies have been consulted: (Corradi, 1995 & 2002), (Chrimes, 2008), (Feld, 1928 & 1948), (Golder, 1948 & 1953), (Guillerme, 1995, pp. 85-145), (Habib, 1991), (Herries & Orme, 1989), (Heyman, 1972), (Jáky, 1937/1938), (Kalle & Zentgraf, 1992), (Kérisel, 1953), (Kötter, 1893), (Llorente, 2015), (Marr, 2003), (Martony de Köszegh, 1828), (Mayniel, 1808), (Mehrtens, 1912, pp. 55-73), (Ohde, 1948-1952), (Peck, 1985), (Reissner, 1910), (Skempton, 1981 & 1985), Verdeyen (1959) and (Winkler, 1872). Around the middle of the 19th century, Alexandre Collin (1808-1890) started to shape the theory of earthworks through his theory of embankments made from cohesive soils backed up by experiments (Collin, 1846). Ten years later, Culmann published his article Ueber die Gleichgewichtsbedingungen von Erdmassen (on the equilibrium conditions of bodies of soil) (Culmann, 1856), which was followed in 1872 by his paper on earthworks (Culmann, 1872). In 1888 Earth Pressure, First Edition. Achim Hettler and Karl-Eugen Kurrer. © 2020 Ernst & Sohn Verlag GmbH & Co. KG. Published 2020 by Ernst & Sohn Verlag GmbH & Co. KG.

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2 The history of earth pressure theory

Karl von Ott, professor of soil mechanics at the German Technical University in Prague, divided his lectures into • • • •

the theory of earthworks (or embankments), the theory of retaining walls, the theory of the masonry arch, and elastic theory and its application to timber and iron structures paying particular attention to roofs and bridges.

What he understood by earthworks was the creation of certain soil forms “known by the names of dams, ramparts, cuttings, cuts, etc., the creation of which requires working the material supplied by the natural soil” (v. Ott, 1888, p. 2). Laws governing the equilibrium of such bodies of earth (Fig. 2.1) were postulated in his book Theorie des Erdbaues oder der Böschungen (theory of earthworks or embankments) (v. Ott, 1888, p. 2). It was August von Kaven who provided a classical summary of the theory of earthworks in the middle of the classical phase of theory of structures (1875-1900) (v. Kaven, 1885). But the theory of earthworks did not gain new momentum (e.g. (Hultin, 1916), (Fellenius, 1927)) until the investigations into the collapse of the quayside at Gothenburg on 5 March 1916 (Petterson, 1916). Earth pressure theory backed up by experimentation started to assert itself as soil mechanics evolved in the 1920s, with Terzaghi pointing the way forward with his seminal work Erdbaumechanik auf bodenphysikalischer Grundlage (mechanics of soil in construction) (Terzaghi, 1925). Today, the theory of embankments and earth pressure theory are part of soil mechanics (Fig. 2.2), which in turn is a subdiscipline of geotechnical engineering. Earth pressure theory can look back on 300 years of history. The first half of that was dominated by French engineering officers, a list of names stretching from Vauban to Bélidor to Coulomb to Poncelet, who were involved with the planning, design, construction and upkeep of fortifications. In the following sections, the thesis postulated is that the Corps du Génie Militaire of the early 18th century not only played a decisive role in the development of modern civil engineering, but Fig. 2.1 Investigating the stability of an embankment loaded through excavation; 𝜓 = angle of slip plane, 𝜌 = angle of internal friction (v. Ott, 1888, p. 20).

2.1 Retaining walls for fortifications

Fig. 2.2 The illustration on this book cover shows a schematic view of the investigation of a slip circle in the subsoil behind a retaining wall (Türke, 1990).

also that the engineering officers of that corps created the first genuine engineering science theory in the form of earth pressure theory, providing civil engineers with a scientific conception for their work. Not until the establishment phase of theory of structures (1850-1875) (Kurrer, 2018, pp. 20-21) would the supremacy of the engineering officer in the field of earth pressure be overtaken by that of the railway engineer. So the building of fortifications, with earth pressure theory providing a scientific tool, marks the birth of modern civil engineering.

2.1 Retaining walls for fortifications The building of fortifications in Europe from the early years of the modern era right up to the completion of the Industrial Revolution in the countries of continental Europe was based on earthworks, which together with masonry resulted in large-format structures that were to leave a mark on towns and cities. One example is Luxembourg, where building works between 1543 and 1867 turned it into one of the strongest fortresses in Europe (Fig. 2.3). One of those who worked on extending Luxembourg’s fortifications was Vauban (Fig. 2.4), who in 1678 had been appointed Commissary General of all French fortifications by Louis XIV and who was in charge of the conquest of the city in 1684. However, Luxembourg is only one small part in the output of

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2 The history of earth pressure theory

Fig. 2.3 Historical map of Luxembourg by First Lieutenant Cederstolpe showing the city’s fortifications, c. 1845 (Reinert & Bruns, 2013, p. 48).

Fig. 2.4 Sebastien le Prestre de Vauban (1633-1707); copy by Antoine Coysevox of the marble bust (since lost) produced by Pietro Marchetti by order of Napoleon I (Neumann, Hartwig, 1984, p. 379).

2.1 Retaining walls for fortifications

this “Ingénieur de France”, as he was called during his lifetime. As is written in the Larousse universel of 1923: “Towards the end of his life, Vauban – who Saint-Simon (1760-1825) described as one of the most virtuous men of his century – published his Projet d’une dixme royale [project for a royal tithe] in which, driven by a genuine philanthropic feeling, he called for fair taxes, which resulted in him falling out of favour with Louis XIV” (cited after (Göggel, 2011, p. 136)). In just a few decades, Vauban built 33 new fortifications and rebuilt about 300; so far, 411 construction measures at 160 locations have been proved to be his work (Neumann, Hartwig, 1984, p. 381). Vauban’s fortification and civil works involved about nine million cubic metres of masonry (Petzsch, 2011, p. 191). According to his own figures, Vauban used more than 3.7 million cubic metres of masonry for retaining walls supporting the ramparts with their bastions at the corners of the star-shaped fortifications and the intermediate masonry walls, the curtain walls, (see (Poncelet, 1844, p. 67)), which corresponds to 41% of the total amount of masonry built. As early as 1684, Vauban published design tables for retaining walls with heights of 3 m < H < 25 m (Kérisel, 1985, p. 55). Three years later, Vauban, in his role as newly appointed Commissary General of all French fortifications, sent his engineers in the Corps du Génie Militaire his Profil général pour les murs de soutènement in which he presented his retaining wall profiles that were later adopted by engineering offers such as Bélidor (1729), Poncelet (1840) and Wheeler (1870) (see (Feld, 1928, p. 64ff.)). This “universal Vauban profile” (Poncelet, 1844, p. 4) was investigated by Poncelet, who compared this “main principle of Vauban’s rules” (Poncelet, 1844, p. 68ff.) with the results of his earth pressure theory. Fig. 2.5, which shows the retaining walls for the fortifications at Ypres, conveys an impression of the Vauban profile, which Vauban drew in an entry in his diary for 1698 (see (Kérisel, 1985, p. 86)). The trapezoidal form of the retaining wall on the right of bastion 63 for the Ypres fortifications has the following dimensions: height H = 11.38 m, width at base b = 3.52 m, width at top

Fig. 2.5 Retaining wall with buttresses for the fortifications at Ypres designed by Vauban in 1699, after a drawing by A. de Caligny (Poncelet, 1844, plate IV, Fig. 35).

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8

2 The history of earth pressure theory

k = 1.62 m, batter of wall on air side m = (3.52-1.62)/11.38 = 1:6, average depth of soil covering to top of masonry h′ = 0.5⋅(2.11+1.35) = 1.75 m. The retaining wall is stiffened by buttresses 16.90 m high every 4.87 m, which themselves have a trapezoidal cross-section with depth h = 3.25 m, width at base bu = 2.60 m and width at top bo = 1.30 m. The buttresses increase the stability enormously. In a 1953 essay on the history of soil mechanics in France, Kérisel mentions a paper by M. Chauvelot which was presented to the Paris-based Académie des Sciences by Gaspard Monge (1746-1818) and Alexandre-Théophile Vandermonde (1735-1796) in 1783 and contains examples (with figures) for Vauban’s design principles. For a retaining wall with buttresses at a spacing of 5.75 m and a batter m = 1:5 on the air side, he gives the following formula for the width of the base of the retaining wall: 1 ⋅ H + 1.48 (2.1) 5 Of course, Vauban based his formula of 1684 on the units of length used at that time, the toise (1 T = 1.95 m) and the pied (1 p = 0.325 m), which, when converted to the metric system, results in the Vauban formula of eq. (2.1) (see (Kérisel, 1985, p. 55)). In eq. (2.1), H is the depth of the earth backfill and k Vauban,1:5 the width of the top of the retaining wall. Kérisel also published the table specified by Chauvelot (Kérisel, 1953, p. 153). Eq. (2.1) can be easily fitted to the retaining wall shown in Fig. 2.5: bVauban,1∶5 = m ⋅ H + kVauban,1∶5 =

1 ⋅ H + 1.625 6 which with H = 11.38 m results, according to eq. (2.2), in a base width of bVauban,1∶6 = m ⋅ H + kVauban,1∶6 =

(2.2)

1 ⋅ 11.38 + 1.625 = 1.90 + 1.625 = 3.52 m. 6 In the case of retaining walls with soil surcharge (see Fig. 2.6) and small buttresses, Vauban apparently proposed this formula: bVauban,1∶6 =

1 ⋅ H + 1.625 (2.3) 5 (see (Feld, 1928, p. 64)); again, this equation (like equations 2.1 and 2.2) has been converted to metric. In Fig. 2.6, h′ = C-G stands for the averaged depth of soil surcharge, H = C-B the height of the retaining wall, or soil backfill, bVauban,surcharge = A-B the width at the base and A-C = 1.625 m the width at the top. According to Audoy, Vauban based his retaining wall profiles on a factor of safety against overturning 𝜈 K,Vauban = 3.8 and a factor of safety against sliding 𝜈 G,Vauban = 4.7 (see (Feld, 1928, p. 65)). However, estimating the stability of Vauban’s retaining wall (Fig. 2.5) according to the calculations in (Kurrer, 2018, pp. 55-58) and using the same soil mechanics parameters results in much lower factors of safety than those given by Audoy: at the base of the wall there is an overturning safety factor 𝜈 K = 2.3, which is > 𝜈 permiss = 1.5, and the sliding safety factor 𝜈 G is nearly 1.6, again > 𝜈 permiss = 1.5. If the buttresses are left out of the equation, the stability of the retaining wall against overturning is 𝜈 K = 1.2 and sliding 𝜈 G = 1.07, which are both just on the safe side. bVauban,surcharge =

2.2 Earth pressure theory as an object of military engineering

Fig. 2.6 Designations for retaining walls with soil surcharge after Poncelet (Poncelet, 1844, plate I, Fig. 1).

According to that, Vauban’s retaining walls with their trapezoidal profile and buttresses cannot be further optimised structurally. Even Poncelet therefore assumed that Vauban’s dimensioning rules – e.g. eqs. (2.1) to (2.3) – handed down to us do not represent empirical rules, instead can be attributed to “an exact geometric theory” (Poncelet, 1844, p. 4). Therefore, the Vauban profiles provided the structural/constructional reference for more than 150 years. And it was against this that earth pressure theories had to measure their modelling quality and practicability.

2.2 Earth pressure theory as an object of military engineering More than 2,000 years ago, Vitruvius – for many years responsible for the building of military engines in the armies of Caesar and Augustus – investigated the phenomenon of earth pressure and how to deal with it in structural and constructional terms. In chapter V, “The City Walls”, in Book I of his Ten Books on Architecture, Vitruvius writes about the walls between the towers, which require a “comb-like arrangement” of buttresses between them which are filled with earth (Fig. 2.7): “With this form of construction, the enormous burden of earth will be Fig. 2.7 Horizontal section through fortifications after Vitruvius (Vitruvius, 1981, Fig. 6).

Ditch Outer wall 1 2

2

1 2

2

2

Inner wall 1 = Earth fill

2 = Cross-wall

2

9

10

2 The history of earth pressure theory

distributed into small bodies, and will not lie with all its weight in one crushing mass so as to thrust out the substructures” (Vitruvius, 1981, p. 59). In chapter VIII “On Foundations and Substructures” in Book VI, Vitruvius describes earth pressure not only in qualitative terms, but also tells us how to calculate the earth pressure for the retaining walls of Fig. 2.7: “Particular pains, too, must be taken with substructures, for here an endless amount of harm is usually done by the earth used as filling. This [earth fill] cannot always remain of the same weight that it usually has in summer, but in winter time it increases in weight and bulk by taking up a great deal of rain water, and then it bursts its enclosing walls and thrusts them out ... The following means must be taken to provide against such a defect. First, let the walls be given a thickness proportionate to the amount of filling” (Vitruvius, 1981, p. 297). Vitruvius then proposes rules for dimensioning the system of retaining walls and explains that “to meet the mass of earth, there should be saw-shaped constructions attached to the wall” and “with this arrangement, the teeth and diagonal structures will not allow the filling to thrust with all its force against the wall, but will check and distribute the pressure” (Vitruvius, 1981, p. 299). These quotes are the oldest known references to the nature and effect of earth pressure. Like those involved with building had condensed the nature of masonry arch thrust into structural and constructional knowledge in the form of a structural theory in a lengthy historical process through their observations, own experiences during construction and many years of checking structures in use, so the knowledge of the phenomenon of earth pressure at the end of the 17th century culminated in Vauban’s design theory for retaining walls. The beginnings of the changeover from empiricism to theory took place in masonry arches (see section 4.3.1) as it did in earthworks under the auspices of the Académie Royale d’Architecture de Paris (Kurrer, 2018, p. 212). Whereas La Hire proposed that the règles de l’art for the masonry arch problem be based on classical mechanics, Pierre Bullet (1639-1716) was the first (in 1691) to attempt to model physically and quantify earth pressure on retaining walls (Bullet, 1691, pp. 159-177). Both La Hire and Bullet were committed to the rationalism of René Descartes. It is therefore the classical rationalism of Descartes and Leibniz that formed their scientific theory and epistemological sounding board at the transition from the orientation phase (1575-1700) to the application phase (1700-1775) of theory of structures (Kurrer, 2018, pp. 15-16). The inductive structural theory ideas of Leonardo da Vinci and other engineers of the Renaissance was to be replaced by the deductive method (Polónyi, 1982), which to date shapes the way that this fundamental engineering science discipline sees itself. The difference between masonry arch theory and earth pressure theory in the application phase right up to the end of the constitution phase (1825-1850) of theory of structures (Kurrer, 2018, pp. 19-20) is that earth pressure theory is not the work of civil engineers, but essentially military engineers. 2.2.1

In the beginning there was the inclined plane

The first earth pressure theories were based on the model of the inclined plane (Fig. 2.8), which Stevin had cleverly used as long as go as 1586 for his equilibrium

2.2 Earth pressure theory as an object of military engineering

Fig. 2.8 Determining the earth pressure according to the fundamental model of the inclined plane.

k

X Terrain line

H

Wall line

a

Sρ +

R=μ.N

G

φ=ρ m

n ine

N

el

op

Sl

T

d b

observations (Kurrer, 2018, pp. 29-30). The starting point for these studies was the observation that when loose cohesionless materials are tipped out, they form a conical pile, the slant line of which forms a natural slope and the angle of the slope line with respect to the horizontal φ corresponds to the angle of internal friction 𝜌 of this soil type. If further material is tipped out on top of this, it rolls downwards and in this case a retaining wall must be built upwards from point d to resist the descending material. This resistance was interpreted as earth pressure. In the standard model of the first earth pressure theories, the wedge of soil bounded by slope line d-n, wall line d-a and terrain line a-n was considered as a rigid body with weight G which slides without friction parallel with the slope line. The components of G acting perpendicular N and parallel T to the slope line can be determined from the similarity between triangle d-a-n and the triangle of forces (Fig. 2.8): x x N = ⇒N=G⋅ = G ⋅ cos φ (2.4) G dn dn H H T = ⇒T=G⋅ = G ⋅ sin φ (2.5) G dn dn The force T acting parallel with the slope according to eq. (2.5) functions as earth pressure E on the retaining wall. If, however, the slope line is affected by friction, then the earth pressure is reduced to E = T − R.

(2.6)

In his Mémoire de l’Académie Royale of 19 December 1699, which described the design of waterwheels, Guillaume Amontons (1663-1705) realised that the friction force R is proportional to the normal force N and independent of the contact area. He assumed a value 1/3 for the proportionality factor 𝜇 (Amontons, 1699/1718). The fundamental model of the inclined plane modified to include friction force R E = T − R = G ⋅ sin φ − μ ⋅ N = G ⋅ (sin φ − μ ⋅ cos φ)

(2.7)

for earth pressure was already in use for finding dimensions for retaining walls in the first half of the 18th century by way of diverse simplifications. These earth pressure theory approaches differ in the first place in the figures assumed for the slope angle φ = 𝜌, the magnitude of the friction force and the definition of the point of application of E.

11

12

2 The history of earth pressure theory

Bullet

Bullet modelled the cohesionless soil material, e.g. sand, as a regular pile with small, spherical particles with a theoretical slope angle φ = 60∘ (Fig. 2.9). For reasons of safety, his further studies were based on a slope angle φ = 45∘ (Fig. 2.10). In the next step, Bullet determined the force at the inclined plane that prevents a particle of weight G′ from rolling downwards: √ 2 5 ′ (2.8) ⋅ G′ ≈ ⋅ G′ . E = 2 7 Of course, this relationship also applies to the entire earth pressure wedge with weight G (see Fig. 2.8): √ 2 5 5 (2.9) EBullet = ⋅ G ≈ ⋅ G = ⋅ 0.5 ⋅ γE ⋅ H2 = 0.35 ⋅ γE ⋅ H2 2 7 7 Eq. (2.9) can also be found from eq. (2.5) with φ = 𝜌 = 45∘ . As an example, Bullet now calculated the area of the earth pressure wedge with leg lengths x = 6 toisen as AG = 0.5⋅6⋅6 = 18 square toisen. As G is proportional to EBullet , then according to eq. (2.9), AE = (5/7)⋅18 = 13 square toisen is valid for the “area of earth pressure”. Where the earth and the masonry of the retaining wall have the same unit weight 𝛾 E = 𝛾 MW , Bullet can determine the wall’s dimensions from the area AE assumed by him to be equal to the cross-sectional area of the retaining wall AS . Consequently, the width of the base of the retaining wall can be calculated from 5 (2.10) bBullet = ⋅ H − k 7 where H is the height and k the width of the top of the retaining wall. Here, for H = 6 toisen (= 6⋅1.95 = 11.7 m) and k = 10/6 toisen (= 3.25 m), bBullet takes on a value of about 110/42 ≈ 16/6 = 2.66 toisen (= 5.20 m) (Fig. 2.11).

φ = 60°

Fig. 2.9 Natural slope of small spherical grains of sand after Bullet (redrawn and modified after (Bullet, 1691, p. 171)).

Fig. 2.10 Earth pressure determination after Bullet (Bullet, 1691, p. 172).

2.2 Earth pressure theory as an object of military engineering

Fig. 2.11 Retaining wall design according to Bullet (Bullet, 1691, p. 173).

When determining the width of the base of a retaining wall, according to Emil Winkler (1835-1888), Bullet divided the “area of the earth pressure” AE by the height H (Winkler, 1872, p. 59): bBullet,Winkler =

1 1 5 1 5 H2 5 ⋅ AE = ⋅ ⋅ AS = ⋅ ⋅ = ⋅ H ≈ 0.35 ⋅ H H H 7 H 7 2 14 (2.11)

If H = 6 toisen is entered into eq. (2.11), then, according to Winkler, Bullet would have obtained a value of 2.14 toisen for the width of the base. Feld, too, specifies the same formula as Winkler (Feld, 1928, p. 65). From this it follows that both Winkler and Feld have either misunderstood these parts of Bullet’s work or their misunderstanding is down to having adopted secondary sources without criticism. Obtaining the dimensions of retaining walls using Bullet’s method owes more to geometry than it does to statics, because he is only interested in the magnitude of the vectors of the earth pressure with the weight of the retaining wall and does not consider their point of application or direction at all. Gautier

Hubert Gautier (1660-1737) worked with Vauban and was set to make his mark on French engineering in the early days of the Corps des Ingénieurs des Ponts et Chaussées, which was founded in 1716. Gautier became known for his monographs on roadbuilding (1693) and bridge-building (1716), which progressed to become the number one textbooks for modern civil engineering and remained so for a number of decades. He was an inspector of roads and bridges from 1713 to 1731 and therefore was also involved in solving earthworks problems such as those that occur when laying out routes for roads. We have Gautier to thank

13

14

2 The history of earth pressure theory

for the first figures regarding the most important soil parameters. He measured a unit weight of 18.1 kN/m3 and a slope angle of 31∘ for dry, clean sand; the corresponding values for customary, loosened earth fill were, according to Gautier, 13.4 kN/m3 and 45∘ (Gautier, 1717, pp. 37-51). Although Gautier based his dimensions for retaining walls on geometric rules or rules of proportion, his measurement of these two soil parameters laid the foundation for the evolution of a theory of earth pressure. Couplet

In his first Mémoire de l’Académie Royale on earth pressure, Couplet criticised Bullet’s assumptions (Couplet, 1726/1728): • The assumed slope angle of 60∘ is incorrect (see Fig. 2.9). • The pile of spherical particles is not two-dimensional (see Fig. 2.9), but three-dimensional (see Fig. 2.12). • The slope line d-n cannot be understood as an inclined plane down which the wedge of soil a-d-n slides (see Fig. 2.8). • The factor 5/7 in Bullet’s earth pressure equation (2.9) is incorrect because earth pressure E does not act horizontally. Couplet assumed a configuration of frictionless spherical particles in the shape of a tetrahedron (Fig. 2.12), with every sphere making contact with three others and transferring to those three the compressive forces acting perpendicular to the areas of contact. From this, Couplet initially derived a fictitious slope line L-K (Fig. 2.13). So, the sphere on the outside does not roll down C-B, but down L-K. Couplet showed further that his frictionless theory requires a constant horizontal pressure acting on the smooth wall line which is independent of the slope line angle and√proportional to 0.5⋅H 2 . Taking the elementary tetrahedron with side length 2⋅ 3 and triangle A-I-D (Fig. 2.12), Couplet found that √ the ratio of earth pressure E to the weight of the sliding wedge of soil G was 2: 8, i.e. the triangle

Fig. 2.12 Pile of spherical particles in the form of a tetrahedron after Couplet (Couplet, 1726/1728, plate 4, Figs. 10 & 11).

2.2 Earth pressure theory as an object of military engineering

Fig. 2.13 Fictitious slope line after Couplet (Couplet, 1726/1728, plate 4, Fig. 7).

of forces due to soil prism and earth pressure is similar to triangle A-I-D. From this we get the earth pressure γ (2.12) ECouplet,1726 = E ⋅ H2 = 0.125 ⋅ γE ⋅ H2 , 8 which Couplet applies in the upper third of the wall line. The hydrostatic earth pressure is Ehydrostatic = 0.5 ⋅ γE ⋅ H2

(2.13)

and would act in the lower third of the wall line. After Couplet has established moment equilibrium about point m for a retaining wall with a rectangular cross-section (b = k), he obtains the minimum base width using √ γE . (2.14) min bCouplet,1726 = H ⋅ 6 ⋅ γMW Couplet’s equation (2.14) results in the following figure for Bullet’s sample calculation (i.e. 𝛾 E = 𝛾 MW and H = 11.70 m): √ √ γE 1 min bCouplet,1726 = H ⋅ =H⋅ = 0.41 ⋅ H = 0.41 ⋅ 11.70 = 4.78m. 6 ⋅ γMW 6 (2.15) This value is about 8% less than the value bBullet = 5.20 m obtained with eq. (2.10). Further approaches

In the 18th century numerous authors adhered to the fundamental model of the frictionless inclined plane in order to determine earth pressure (see (Kötter, 1893, pp. 79-80)), and most of those were military engineers. The influential professor of mathematics at Göttingen, Abraham Gotthelf Kästner (1719-1800), was also a fan of this earth pressure theory (see (Woltmann, 1794, pp. 152-158)). Blaveau can be mentioned as another example of this type of theory. In his Mémoire de l’Académie Royale of 1767, he broke down the force acting parallel with the slope into vertical and horizontal components (Fig. 2.14) and interpreted the latter as the earth pressure (see Winkler 1872, pp. 60-61): EBlaveau = G ⋅ sin φ ⋅ cos φ = 0.5 ⋅ H ⋅ x ⋅ γE ⋅ sin φ ⋅ cos φ = 0.5 ⋅ H2 ⋅ γE ⋅ cos2 φ (2.16)

15

16

2 The history of earth pressure theory

Fig. 2.14 Earth pressure calculation after Blaveau.

The following relationship is obtained for the special case of φ = 𝜌 = 45∘ : √ 2 (2.17) EBlaveau = ⋅ H2 ⋅ γE = 0.35 ⋅ γE ⋅ H2 . 4 This earth pressure formula agrees with that of Bullet (eq. 2.9). Nevertheless, there were considerable differences between these two approaches when it came to the point of application of the earth pressure. According to Blaveau, the line of action of the force acting parallel with the slope passes through the centre of gravity S of the earth pressure wedge and intersects the wall line at p such that the earth pressure also passes through p, and for the special case of eq. (2.16), line p-d takes on the value H/3. Based on that, with moment equilibrium about point m, a retaining wall with a rectangular cross-section (b = k) would have a minimum base width of √ 𝛾E min bBlaveau,a = H ⋅ (2.18) 4.29 ⋅ 𝛾MW The stabilising effect of the vertical component V of the force acting parallel with the slope is ignored here. If we again use 𝛾 E = 𝛾 MW and H = 11.70 m, then eq. (2.18) results in √ √ 𝛾E 1 min bBlaveau,a = H ⋅ =H⋅ 4.29 ⋅ 𝛾MW 4.29 = 0.48 ⋅ H = 0.48 ⋅ 11.70 = 5.65 m. (2.19) If instead the stabilising effect of V is considered, then using the above figures, the result is min bBlaveau,b = 0.25 ⋅ H = 0.25 ⋅ 11.70 = 2.88 m.

(2.20)

The latter is only possible where the coefficient of friction of the wall line 𝜇 = tan𝜌 = tanφ = tan45∘ = 1, because then, and only then, is the stabilising effect of V fully valid. Therefore, the wall friction angle 𝛿 should be taken as zero. However, that would result in having to assume a minimum base width of 5.65 m – a value that is much higher than the figures calculated by Bullet and Couplet (5.20 m and 4.78 m respectively). So, in the end, Blaveau’s

2.2 Earth pressure theory as an object of military engineering

earth pressure calculation leads to uneconomic cross-sections for retaining walls. Much more serious, however, is the objection that Blaveau’s forces breakdown is arbitrary, because the force acting parallel with the slope line must mobilise the opposing friction force R (see Fig. 2.8), provided the slope line is identical with the slip plane. The question now was: By what amount R can the earth pressure according to eq. (2.6) or (2.7) been reduced? Couplet and Bélidor tried to answer this question by incorporating friction into the fundamental model of the inclined plane for determining earth pressure. Friction reduces earth pressure

Couplet’s second Mémoire de l’Académie Royale on earth pressure included the friction between the soil material and the wall line (Couplet, 1727/1729): “As the facings [= retaining wall – the author] are made up of stones or bricks, lime and sand, which never produce smooth surfaces, it is thus necessary to investigate the magnitude of the pressure of the earth on these sandy and uneven surfaces, and to arrange the facings in such a way that they are able to withstand this pressure” (Couplet, 1795, p. 89). In terms of method, Couplet kept to his first Mémoire on earth pressure. Using the tetrahedron-shaped pile of particles (see Fig. 2.12) and a particle configuration in the shape of a semi-octahedron, Couplet derived the earth pressure acting obliquely downwards. He then obtained the minimum base width minbCouplet,1727 from moment equilibrium about point m (see Fig. 2.8), which led to a cubic equation. Couplet presented the results of his calculations for the inclinations of the slope line... • side face of tetrahedron: inclination of A-K (see Fig. 2.12), • edge of tetrahedron: inclination of A-D (see Fig. 2.12), and • side face of semi-octahedron (similar to height A-K of side surface ABC in Fig. 2.12) in three tables from which it is possible to obtain the minimum base width b for height H and top of wall width k (Couplet, 1795, pp. 124-126). Couplet’s design tables are based on a unit weight ratio 𝛾 E :𝛾 MW = 2:3. For example, table I with the inclination of the side face A-K as slope line for the minimum base width is based on the relationship min bCouplet,1727 = 0.11 ⋅ H (Couplet, 1795, p. 124). According to eq. (2.14), that would result in √ √ γE H 2 min bCouplet,1726 = H ⋅ =H⋅ = 6 ⋅ γMW 6⋅3 3

(2.21)

(2.22)

for the frictionless case. Responding to the low value of the minimum base width given by eq. (2.21), Jacob Feld wrote: “The reason for the extremely low value is that Couplet has actually assumed it to be just as easy to shear the wall in the direction of the plane of rupture as to overturn it. This is not acceptable, for the weakest shear section of a wall is horizontal. This is especially true of the walls of Couplet’s time, built with masonry or brick laid in horizontal layers” (Feld, 1928,

17

18

2 The history of earth pressure theory

p. 67). Couplet’s earth pressure theory did not enjoy widespread use because of the small minimum base widths it gave, which were well below those of Vauban’s design theory. Couplet’s earth pressure theory based on the fundamental model of granular media would have only little influence on the formation of further theories and would only slowly take shape in programs based on the theory of granular media for earthworks within the scope of the discrete element method (DEM) (Cundall & Strack, 1979) at the start of the integration period of geotechnical engineering (1975 to date). What Couplet’s earth pressure theory did achieve was bring about the knowledge that “the pressure at the slope line is not perpendicular to the latter, that tangential resistance forces [= friction force R in direction of slope line – the author] also apply at the slope line” (Kötter, 1893, p. 82). In the end, the complicated construction of Couplet’s earth pressure theory hampered its use in practical engineering. The first earth pressure theory for everyday engineering can be found in Bélidor’s La Science des Ingénieurs (1729). Like Bullet, Bélidor assumed a slope line inclined at φ = 45∘ . He divided the earth prism into a large number of thin slices parallel with the slope line (Fig. 2.15). In the case of frictionless slices and φ = 45∘ , the horizontal component of the earth pressure acting on wall line dE for the slice under consideration must be equal to the weight of this slice dG (Fig. 2.16). Bélidor now introduced friction into the equation.

Fig. 2.15 Stability investigation for a retaining wall after Bélidor (Bélidor, 1813, plate 2, Fig. 4).

2.2 Earth pressure theory as an object of military engineering

Fig. 2.16 Generalisation of earth pressure determination after Bélidor.

k

dξ a

n

H

dz

Iz

dG

φ

dN

z

dE = e φ m

d b

He stipulated that the friction on the slope line halves the earth pressure resulting from the frictionless inclined plane: dEBelidor = 0.5 ⋅ dE ́

(2.23)

The earth pressure for an arbitrary slope angle φ is determined below. The weight of the infinitely thin slice dG = γE ⋅ lz ⋅ d𝜉

(2.24)

and the geometric relationships lz =

1 ⋅ (H − z) sin φ

(2.25)

and dξ = dz ⋅ cos φ

(2.26)

result in dE = dG ⋅ tan φ = γE .(H − z) ⋅ dz

(2.27)

for the specific earth pressure e(z) acting on the wall line of the retaining wall for the earth pressure component of the slice under consideration. The specific earth pressure according to eq. (2.27) is not dependent on the slope angle φ or a linear function via the ordinate z of the wall line between e(z = 0) = 𝛾 E ⋅H and e(z = H) = 0. Integration supplies the total earth pressure z=H

E=

∫z=0

H

dE =

∫0

γE ⋅ (H − z) ⋅ dz = 0.5 ⋅ γE ⋅ H2

(2.28)

as hydrostatic earth pressure according to eq. (2.13), which Bélidor reduces to EBelidor = 0.5 ⋅ E = 0.5 ⋅ 0.5 ⋅ γE ⋅ H2 = 0.25 ⋅ γE ⋅ H2 ́

(2.29)

If eq. (2.29) establishes the moment equilibrium about point m for a rectangular retaining wall (b = k), then the minimum base width is √ γE min bBelidor = H ⋅ , (2.30) ́ 6 ⋅ γMW

19

2 The history of earth pressure theory

X a

n

Fig. 2.17 Determining earth pressure using the model of the frictionless wedge.

S H

20

G

φ m

φ

N

Ehydrostatic

d

which corresponds to that of Couplet according to eq. (2.14). However, this concordance is coincidental, because compared with Bélidor, Couplet only applies half the earth pressure according to eq. (2.12) at level z = (2/3)⋅H, whereas the earth pressure according to Bélidor acts on the wall line at z = (1/3)⋅H. Bélidor’s contribution to earth pressure theory is a methodology consisting of his slices method with which even practical cases such as polygonal terrain lines (see Fig. 2.15) can be analysed as well. Bélidor produced earth pressure tables for H = 3.25 to 32.50 m for use by practising engineers; in those tables the minimum base widths vary between minb(H = 3.25 m) = 0.37⋅H and minb(H = 32.50 m) = 0.355⋅H (Bélidor, 1813, p. 59). The slices method adopted from differential and integral calculus enables every ordinate of the specific earth pressure and its contribution to the active moment about point m to be determined and, in the end, the minimum base width to be quantified based on the equilibrium with the restoring moment. 2.2.2

From inclined plane to wedge theory

The fundamental model of the inclined plane gives us the earth pressure that acts parallel with the slope line, i.e. totally independent of the properties of the wall line of the retaining wall. By modelling the earth prism, a-d-n as a wedge-shaped rigid body whose weight G acts without friction on the slope line as N and on the wall line as E (Fig. 2.17), we know not only the magnitude, but also the direction of the earth pressure E. The model of the frictionless wedge had already been used by La Hire in his masonry arch theory (see Fig. 4-19 in (Kurrer, 2018, p. 213)). For earth pressure, this model leads to the hydrostatic earth pressure E = Ehydrostatic according to eq. (2.13) or eq. (2.28), which is independent of the slope angle φ and is equal to the pressure exerted by a fluid with the same unit weight as the soil. Von Clasen was just one of the many proponents of this theory in the 18th century (v. Clasen, 1781a). As in masonry arch theory, where the model of the frictionless wedge was soon superseded by the model of the wedge affected by friction, this process also gave way to a more complex model in earth pressure theory as well. It was Reinhard Woltman (1757-1837), a waterways engineer based in northern Germany, who

2.2 Earth pressure theory as an object of military engineering

first assumed the model of the inclined plane with friction and discussed this as an alternative to the model of the wedge with friction. Volumes III and IV of his four-volume work Beyträge zur Hydraulischen Architectur (writings on hydraulic architecture) deal with quantifying earth pressure experimentally and theoretically. Of course, he noted that the friction force R reduces the force acting parallel with the slope T according to eq. (2.7) (Woltman, 1794, p. 165). Fig. 2.18 illustrates Woltman’s method I for determining earth pressure. After he has broken down the weight of the earth prism G in the direction (T) of the slip plane at an angle φ and orthogonal (N) to it, he employs eq. (2.7) to subtract the friction force R from the force acting parallel with the slope T and determines the horizontal component of this differential force as the earth pressure EW,I (φ) = (T − R) ⋅ cos φ = G ⋅ cos φ ⋅ (sin φ − μ ⋅ cos φ).

(2.31)

He determines the unknown inclination of the slip plane φ from the condition that the earth pressure according to eq. (2.31) is a maximum dEW,I (φ) dφ

=0

(2.32)

and substitutes φ = 𝜗 into eq. (2.31): EW,I (φ = ϑW,I ) = G ⋅ cos ϑW,I ⋅ (sin ϑW,I − μ ⋅ cos ϑW,I ).

(2.33)

Unfortunately, the necessary condition for the extreme value of eq. (2.32) leads to a trigonometric equation for calculating the angle of the slip plane 𝜗W,I , which has no closed-form solution. Writing about method I, Woltman notes that “this first calculation only deals with the relationship of the forces taking into account their horizontal direction but not their whole magnitude or complete equilibrium” (Woltman, 1799, p. 299). Woltman’s method II overcomes this shortcoming, assumes that the equilibrium at the wedge with friction is taken into account (Fig. 2.19) and determines the earth pressure, which he calls “the conserving force in the horizontal direction ... for the case of equilibrium” (Woltman, 1799, p. 299). Applying equilibrium of forces at the as yet unknown slip plane T(φ) = TE (φ) + R(φ)

(2.34)

Fig. 2.18 Determining earth pressure according to Woltman’s method I – the inclined plane with friction (see (Woltman, 1799, p. 298)).

k

x a

n S

H

+

G φ

N T–R

T V φ m

d b

R

E

21

2 The history of earth pressure theory

k

x a

n S G φ

E φ

N –NE T

NE T E m

Fig. 2.19 Determining earth pressure according to Woltman’s method II – the wedge with friction (see (Woltman, 1799, p. 299)).

–N

+

H

22

φ NE d

φ

E R TE

b

Woltman uses the relationships T(φ) = G ⋅ sin φ TE (φ) = E ⋅ cos φ R(φ) = [N(φ) + NE (φ)] ⋅ μ = [G ⋅ cos φ + E ⋅ sin φ] ⋅ μ

(2.35)

to obtain the earth pressure according to Coulomb (see (Coulomb, 1773/1776)) EW,II (φ) = EC (φ) = G ⋅

(sin φ − μ ⋅ cos φ) . (cos φ + μ ⋅ sin φ)

(2.36)

Applying Coulomb’s theory, Woltman determines the angle of the unknown slip plane φ from the condition dEW,II (φ) dφ

=

dEC (φ) = 0, dφ

(2.37)

and finally substitutes the angle obtained φ = 𝜗W,II = 𝜗C = 𝜗 into eq. (2.36): EW,II = EC = E = G ⋅

(sin ϑ − μ ⋅ cos ϑ) (cos ϑ + μ ⋅ sin ϑ)

(2.38)

The earth pressure E, the weight G of the wedge of soil with angle a-d-n = 90∘ – 𝜗, the forces T E and R acting at the slip plane and the normal forces –N E and –N acting orthogonal to the slip plane therefore form a system in equilibrium. However, this system of forces acting on wedge a-d-n only represents a special case of the wedge affected by friction because the friction between the wedge of soil and the retaining wall was neglected by Coulomb and Woltman. Using the relationship μ = tan ρ

(2.39)

introduced by Woltman for the relationship between coefficient of friction 𝜇 and angle of internal friction 𝜌 (Woltman, 1794, p. 165), it is possible to prove that eqs. 2-14 (Kurrer, 2018, p. 53) and 2.38 agree for the special case of wall friction angle 𝛿 = 0∘ and vertical wall line 𝛼 = 0∘ . But here, Woltmann took the angle of internal friction 𝜌 to be identical with the slope angle of the soil material – an assumption that only applies to cohesionless soils such as dry sand and gravel, not to cohesive soils.

2.2 Earth pressure theory as an object of military engineering

As can be seen from a comparison of eqs. (2.33) and (2.38), the following applies: EW,II = EC = E =

EW,I (cos2 ϑ + μ ⋅ sin ϑ ⋅ cos ϑ)

,

(2.40)

i.e. Woltman’s method II, or Coulomb’s method, always leads to larger earth pressure values, as the denominator of eq. (2.40) is (cos2 ϑ + μ ⋅ sin ϑ ⋅ cos ϑ) ≤ 1

(2.41)

and the values given by eqs. (2.33) and (2.38) are only identical for the special case of 𝜇 = tan𝜗. Woltman compared the two methods with earth pressure experiments and noted that “the results of the first calculation [method I – the author] agree far better with the experiments than those of the second [method II – the author]” (Woltman, 1799, p. 301). Nevertheless, he recommended method II but wished to remain unbiased and left it to the mathematical sciences to determine which method should be preferred (see (Woltman, 1799, p. 302)). His reason – after much vacillation between methods I and II – for deciding to favour the latter was that “the first type of calculation gives the pressure of the earth approximately 2/5 parts of the whole smaller than that of the second” (Woltman, 1799, p. 305). He appended critical remarks to this concerning the general reliability of tests. In these, Woltman refers to the influence of particle size and cohesion, neither of which were considered in the two methods. To conclude, Woltman shares a teleological principle that the mathematician C. L. Brünings had communicated to him in a letter which in turn relates to Euler’s treaties on the calculus of variations in elastic theory (Euler, 1744): “Experience teaches us that a certain amount of earth would slide down if the retaining wall did not hold it back. Further, all the sliding earth particles ... would reliably slide with the maximum velocity that their friction and cohesion allow, because the laws of nature always reach their goals by the shortest route. Therefore, the horizontal force that holds them back must also be a maximum” (Brünings, cited after (Woltman, 1799, p. 309)). Brünings’ reasoning implies that both Woltman’s method I and method II after Coulomb in the Woltman version are covered by his teleological principle. As the next section will show, Woltman’s formal application of the extreme value method of differential calculus is misleading in physical terms. Critical here is the physics on which the engineering science model of earth pressure determination is founded. The fundamental model of the wedge with friction proved its superiority at the turn of the 19th century and consigned the fundamental model of the inclined plane with friction to the historical stock of 18th-century earth pressure theories. 2.2.3

Charles Augustin Coulomb

Coulomb’s Mémoire on beam, earth pressure and masonry arch theories (Coulomb, 1773/1776) can be regarded as a bombshell in the initial phase of theory of structures (1775-1825) (Kurrer, 2018, pp. 17-18), with a far-reaching historico-logical effect that can still be felt today in earth pressure calculations.

23

24

2 The history of earth pressure theory

He wrote the Mémoire for his own use during his time spent as an engineer officer on the island of Martinique between 1764 and 1772: “This essay written several years ago was originally only intended for my own private use for works I had to carry out during my service” (Coulomb, 1779, p. 164). We can therefore assume that Coulomb had already formulated his beam, earth pressure and masonry arch theories by about 1770. His three-part Mémoire is divided into 18 sections preceded by a summarising introduction. The first nine sections contain observations on the equilibrium of plane figures (Fig. 4 in Fig. 2.20) and friction, describing his own experiments to determine the tensile and bending strengths of stones (Fig. 1-3 in Fig. 2.20), exemplified by the cantilever subjected to a point load at its end (Fig. 6 in Fig. 2.20) and the analysis of the strength of masonry piers with a quadratic cross-section (Fig. 5 in Fig. 2.20). In section IX, Coulomb rounds off the first part of his Mémoire with his observations on the strength of masonry piers. The second part of his Mémoire begins with the continuation of section IX and thus represents, so to speak, a link between his strength analysis of masonry piers (Fig. 5 in Fig. 2.20) and earth pressure theory (Fig. 7 in Fig. 2.20). Section IX is crucial to understanding Coulomb’s approach to theory of structures. Whereas Coulomb develops his earth pressure theory in the second part, the third part of his Mémoire (sections XVI to XVIII) provides the solution to the masonry arch problem in the form of his voussoir rotation theory – already discussed in (Kurrer, 2018, pp. 222-223) in particular. The second part is not just the middle of his three-part Mémoire, it is also the heart of the content of the Mémoire because it is here, in earth pressure theory, that Coulomb manages to express his approach to theory of structures most clearly. Manifestations of adhesion

Up until the beginning of the establishment phase of theory of structures (1850-1875) (Kurrer, 2018, pp. 20-21), strength of materials was based on ultimate load theory. All the failure phenomena of solid bodies were attributed to overcoming the constant force of attraction between two neighbouring particles, which was known as adhesion. Coulomb assumed that adhesion acts between two particles in a state of rest, that it is independent of external forces and exists until the molecular cohesion is disrupted, leading to failure of the body. Cohesion was the name he gave to the internal resistance acting perpendicular to the rupture surface; cohesion is identical with adhesion, proportional to the area of the rupture surface (Coulomb, 1773/1776, p. 348) and corresponds to the tensile strength. Coulomb carried out tests on stones and found that a body loaded parallel with the rupture surface deviates very little from the cohesion, which he equates to adhesion and, expressed in modern terms, corresponds to the shear strength. If, in addition, a compressive force acts on the body perpendicular to the hypothetical rupture surface, then according to Coulomb, the internal resistance is made up of the cohesion plus the frictional resistance. So, for both types of loading, adhesion is the internal resistance that has to be overcome in order to reach the failure state. Adhesion is therefore the sum of cohesion and friction. Coulomb’s assumption that the cohesion and the frictional resistance

2.2 Earth pressure theory as an object of military engineering

have to be overcome simultaneously can be attributed to his observation that for certain types of stone, the compressive strength was about four times the tensile strength (Coulomb, 1773/1776, p. 355). Therefore, Coulomb had to add a further internal resistance to cohesion: friction. Failure behaviour of masonry piers

Coulomb demonstrated the combination of cohesion and friction through his analysis of the failure of a masonry pier with a quadratic cross-section (Fig. 5

Fig. 2.20 Coulomb’s beam, earth pressure and masonry arch theories (Coulomb, 1773/1776, plate I).

25

2 The history of earth pressure theory

Fig. 2.21 Failure analysis of a masonry pier after Coulomb.

P

d

a φP

NP TP

90°–φP

P

XP

26

H 90°–φP n f

e

in Fig. 2.20). This failure analysis is crucial to understanding his earth pressure theory (see also (Freund, 1924, pp. 101-103) and (Gillmor, 1971, pp. 94-100)) – it is, so to speak, the prolegomenon of classical earth pressure theory. Coulomb assumed a concentrically loaded masonry pier with a quadratic cross-section and was looking for the inclination of the rupture surface d-n with the associated failure force P. Fig. 2.21 shows the relationships for the length of the plane of rupture dn =

H , cos(90∘ − φP )

(2.42)

the resistance to cohesion c or shear 𝜏 Tc = c ⋅ H ⋅ dn = τfail ⋅ H ⋅ dn =

τfail ⋅ H2 , cos(90∘ − φP )

the compressive force orthogonal to the plane of rupture N = P ⋅ cos(90∘ − φ ), P

P

(2.43)

(2.44)

the frictional resistance R = μ ⋅ NP = μ ⋅ P ⋅ cos(90∘ − φP )

(2.45)

and the shear force parallel to the plane of rupture T = P ⋅ sin(90∘ − φ ). P

P

(2.46)

A Using the equilibrium condition parallel to the plane of rupture TP = Tc + R = P ⋅ sin(90∘ − φP ) =

τfail ⋅ H2 + μ ⋅ P ⋅ cos(90∘ − φP ) cos(90∘ − φP ) (2.47)

2.2 Earth pressure theory as an object of military engineering

Coulomb was able to determine the failure load P=

τfail ⋅ H2 ∘ cos(90 − φP ) ⋅ [sin(90∘ − φP ) − μ ⋅ cos(90∘ − φP )]

(2.48)

and by exploiting the condition that P must be a minimum dP(90∘ − φP ) =0 d(90∘ − φP ) he could determine the inclination of the plane of rupture 1 . tan(90∘ − φP ) = √ [ (1 + μ2 ) − μ]

(2.49)

For clay bricks, Coulomb assumed a coefficient of friction 𝜇 = 0.75, and so was able to calculate the angle of the plane of rupture 90∘ - φP = 63.45∘ and the failure load P = 4⋅𝜏 fail ⋅H 2 (Coulomb, 1773/1776, p. 355). As Coulomb equated the tensile strength with the cohesion c, or shear strength 𝜏, it follows that the compressive failure load of the masonry pier was four times the tensile failure load. If friction is neglected, i.e. 𝜇 taken to be zero, then eq. (2.49) results in a plane of rupture angle 90∘ - φP = 45∘ and eq. (2.48) gives us a failure load P = 2⋅𝜏 fail ⋅H 2 ; Coulomb regarded this P value as too low. The plane of rupture angle according to eq. (2.49) does not depend on the cohesion c, or shear strength 𝜏 fail , of the masonry, “so from the theoretical viewpoint, precisely the uncertain part of the internal forces is solely crucial. In earth pressure theory, where the same result repeats later on, Coulomb draws attention to this. Here, though, the friction is, from the theoretical viewpoint, precisely the certain part of the internal forces, meaning that the result is less conspicuous within the scope of earth pressure theory” (Freund, 1924, p. 102). Albert Freund criticised Coulomb’s assumption of the cohesion and friction working together prior to failure, regarding it as inappropriate, because Coulomb ignored the difference between the tensile and compressive strengths of masonry units as an elementary physical fact and instead only took a statics/mathematical relationship as his basis. Freund concluded thus: “We now know what we should think of Coulomb’s premises regarding the theory of masonry piers, which at the same time are the premises of earth pressure theory” (Freund, 1924, p. 102). So, what is the connection between masonry pier theory and earth pressure theory? The transition to earth pressure theory

Freund developed his hypothesis in four steps (Freund, 1924, pp. 102-103): 1. The starting point is Coulomb’s theory of structures model of the masonry pier according to Fig. 2.21. 2. Rotation of the masonry pier shown in Fig. 2.21 through 90∘ anticlockwise. 3. Taking account of the self-weight G of the downward-sliding masonry prism a-d-n. The masonry pier is represented by dotted lines in Fig. 2.22. Self-weight G acts perpendicular to P and is a function of angle φP . This theory of structures model of the masonry pier was now interpreted by Freund as a model of earth pressure (Fig. 2.22): Force P acting on earth wedge a-d-n tries to push

27

2 The history of earth pressure theory

XP a

n f 90°–φP

P

H

G

φP

e

d

Fig. 2.22 The third step in the transition from masonry pier theory to Coulomb’s earth pressure theory (redrawn after (Freund, 1924, Fig. 9)). x′′ x′ n′

a

G′ G′′

E = P′

n′′

Fig. 2.23 The fourth step in the transition from masonry pier theory to Coulomb’s earth pressure theory (redrawn after (Freund, 1924, Fig. 10)).

H

28

Ep = P′′ φ′′ = ϑ′′ d φ′ = ϑ′

said wedge upwards. So, force P acting on a-d-n is nothing more than the passive earth pressure EP . 4. The model shown in Fig. 2.23, which shows the active earth pressure E = P′ with the slip plane d′ -n′ at an angle φ′ = 𝜗′ and the passive earth pressure EP = P′′ with the inclined slip plane d-n′′ at an angle φ′′ = 𝜗′′ , completes the transition to Coulomb’s earth pressure theory. Fig. 2.23 shows that Coulomb distinguishes between two states of equilibrium. In the first case earth wedge a-d-n′ slides downwards (active earth pressure E = P′ ); on the other hand, in the second case, force P is increased until it reaches the value P′′ and earth wedge a-d-n′′ moves upwards (passive earth pressure EP = P′′ ). In both cases, force P′ (or P′′ ) is in equilibrium with the self-weight of the earth wedge G′ (or G′′ ) and the resistance of the slip plane inclined at an angle 𝜗′ (or 𝜗′′ ), which Coulomb assembles from the cohesion force T c (eq. 2.43) and the frictional resistance R (eq. 2.45). Only the equilibrium conditions are different. As in the case of active earth pressure the earth pressure wedge moves downwards, the sense of the frictional resistance is upwards from d to n′ . In the case

2.2 Earth pressure theory as an object of military engineering

of passive earth pressure, the earth wedge moves upwards, so the sense of the frictional resistance is downwards from n′′ to d. The lower P′ and upper P′′ limit values enclose a range within which all forces P acting horizontally on wall surface a-d must lie without disturbing the state of rest when cohesion and friction contribute simultaneously to maintaining the state of rest (see (Freund, 1924, p. 108)). The earth pressure at the state of rest, i.e. within P′ < P < P′′ , is called the earth pressure at rest Eo and, normally, cannot be determined: E < Eo < EP . Expressed mathematically, the passive earth pressure is the least upper bound, whereas the active earth pressure represents the greatest lower bound. According to Freund, Coulomb’s analysis of the masonry pier can be interpreted as the determination of the passive earth pressure P′′ = Ep , whereas his calculation of the active earth pressure P′ = E represents the innovation in his earth pressure theory (Freund, 1924, p. 103). Active earth pressure (see section 3.2)

Coulomb begins the second part of section IX of his Mémoire by deriving the formula for active earth pressure for the case of an earth wedge with cohesion and friction acting at the slip plane d-n′ and a frictionless wall line d-a (Fig. 2.24). The designations in Fig. 2.24 are based on those of Fig. 2.23, but they do not match those used in Coulomb’s equations. The weight of the earth wedge G′ = G is broken down by Coulomb into a component in the direction of the slip plane d-n′ G⋅H |T| = √ (2.50) (H2 + x′2 ) and a component perpendicular to that G ⋅ x′ |N| = √ . (H2 + x′2 )

(2.51)

He proceeds similarly with the active earth pressure P′ = E: E ⋅ x′ |TE | = √ (H2 + x′2 )

(2.52)

Fig. 2.24 Determining the active earth pressure P′ = E after Coulomb.

x′ x F a

n′

TE

P′ = E

T Tc + R

φ = ϑ′ = ϑ d

H

N G′ = G NE

29

30

2 The history of earth pressure theory

|NE | = √

E⋅H (H2 + x′2 )

(2.53)

D As the earth wedge moves downwards on slip plane d-n′ , so – according to Coulomb – an opposing friction force R and cohesion force T c are mobilised. He calculates the active earth pressure from the equilibrium of the forces in the direction of the slip plane (Coulomb, 1773/1776, p. 358) as follows: P′ = E = G ⋅

(H − x′ ⋅ μ) (H2 + x′2 ) −c⋅ ′ . ′ (x + H ⋅ μ) (x + H ⋅ μ)

(2.54)

In eq. (2.54), 𝜇 is the coefficient of friction of the earth wedge on slip plane d-n′ (which Coulomb only refers to indirectly as quotient 1/n = 𝜇) and c is the cohesion, or shear strength 𝜏 E , of the soil material. Coulomb specifies the equation E = (G + F) ⋅

(H − x′ ⋅ μ) (H2 + x′2 ) − c ⋅ (x′ + H ⋅ μ) (x′ + H ⋅ μ)

(2.55)

for the modified case with surcharge F (Coulomb, 1773/1776, p. 363). Taking the necessary condition for the extreme value of the earth pressure of an arbitrary earth wedge dE(x) =0 dx Coulomb is able to derive the equation √ x′ = H ⋅ [ (1 + μ2 ) − μ]

(2.56)

(Coulomb, 1773/1776, p. 360). He points out that the cohesion c has no influence on x′ (Coulomb, 1773/1776, p. 360). Prism of maximum pressure

Later, Rebhann would point out that the earth pressure E obtained for the true (most unfavourable) slip plane inclined at an angle φ = 𝜗 is of course also valid for the other sectional planes at angles φ ≠ 𝜗 (Rebhann, 1870/1871, pp. 44-45). Purely theoretically, somewhat lower earth pressures would be obtained for these sectional planes, applying the full sliding resistance in each case, than would be the case for the actual slip plane (angle of inclination 𝜗). However, as the earth pressure cannot exhibit these different values simultaneously, instead merely a single value E(𝜗), it follows that the sliding resistance at the other sectional planes (φ ≠ 𝜗) cannot be fully effective. “The slip plane is, in the first place, characterised by the fact that the resistance to sliding [= frictional resistance plus cohesion – the author] takes on a maximum value here. So, we get the ‘prism of maximum pressure’ as a subsidiary principle to Coulomb’s main principle” [Ohde, 1948-1952, p. 122]. This Coulomb main principle was introduced into the specialist literature in the German language by Martony de Köszegh, who called it the “Prisma des größten Druckes” (= prism of maximum pressure) (Martony de Köszegh, 1828, p. 10) – a mathematical mode of speaking that concealed the physical nature of earth pressure and which later created confusion and had to be clarified by Kötter (Kötter, 1893, pp. 86-87).

2.2 Earth pressure theory as an object of military engineering

A current misinterpretation

Norbert Giesler also geometrises the physical nature of earth pressure in his book, but rather idiosyncratically (Giesler, 2017, pp. 53-55), which in the end misleads him. Coulomb, with reference to Fig. 7 in Fig. 2.20, actually refers to the point of application and position of earth pressure E, denoting it A, as follows: “Si l’on suppose qu’un triangle CBa rectangle, solide & pesant, est soutenu sur la ligne Ba par une force A [= earth pressure E – the author] appliquée en F, perpendiculairement à la verticale CB” (Coulomb, 1773/1776, p. 357). So Coulomb allows the earth pressure E to be applied perpendicular to the wall line at point F. All the same, Coulomb very wisely does not use his definition to design retaining walls, as shown in section Design. Nevertheless, Giesler takes him by his word and establishes that in his comparative sample calculation using the prevailing theory, which, like masonry arch theory, assumes the middle-third rule (see (Kurrer, 2018, pp. 232-232)) – i.e. the point of application of E is taken to be H/3 measured from point d (see Fig. 2.24) –, the overturning moment turns out to be much smaller. This underdesign was Giesler’s reason for “working out a new way of calculating on the basis of classical earth pressure theory according to Coulomb” (Giesler, 2017, p. 40). Unfortunately, his new earth pressure theory lacks the physics foundation it claims to have. This begins with Giesler’s interpretation of the distribution of the specific earth pressure over the wall line as a “horizontal wedge of soil” (Giesler, 2017, p. 39). Giesler’s “horizontal wedge of soil” is a geometrical mirage that includes the basis of the physical earth pressure model of the prevailing theory in formal terms only, but in terms of physics misses its target. So Giesler’s new earth pressure theory joins the phalanx of unsuccessful approaches to finding the equilibrium between the prism of maximum pressure and the retaining wall. He forgets that the crucial point of Coulomb’s earth pressure theory is precisely that one limit state – which, in reality, does not have to occur – can be found mathematically from an infinite number of possible states of equilibrium with the help of the extreme value calculation of differential calculus. This case, too, teaches us that studying the original sources is indispensable if we are to avoid one-sided interpretations. Earth pressure as a function of slip plane angle 𝝑

It is remarkable that Coulomb developed his earth pressure theory without trigonometric relationships. For example, eq. (2.56) with tan𝜗 = H/x′ is easily rewritten as eq. (2.49). Therefore, eq. (2.55) can also be expressed as a function of the slip plane angle 𝜗: E = (G + F) ⋅

(sin ϑ − μ ⋅ cos ϑ) H −c⋅ (cos ϑ + μ ⋅ sin ϑ) sin ϑ ⋅ (cos ϑ + μ ⋅ sin ϑ)

(2.57)

From eq. (2.57) we can see that the cohesion of the body of soil reduces the active earth pressure and therefore Coulomb uses c = 0 for the backfill behind the retaining wall (Coulomb, 1773/1776, p. 364). If there is no surcharge F, then eq. (2.57) is rewritten in Woltmann’s notation for the active earth pressure after Coulomb for cohesionless soils, i.e. eq. (2.38).

31

32

2 The history of earth pressure theory

Influence of wall friction

Finally, Coulomb considers the friction between wall line d-a and the earth wedge and generalises eq. (2.54) as follows (Coulomb, 1773/1776, p. 364): ) (H − x′ ⋅ μ) ( (H2 + x′2 ) 1 ⋅ ′ −c⋅ ′ . (2.58) E= G−E⋅ v (x + H ⋅ μ) (x + H ⋅ μ) In eq. (2.58), 1/v is the coefficient of friction between the retaining wall and the earth wedge 1/v = tan𝛿, where 𝛿 is the wall friction angle (see Fig. 2-41 in (Kurrer, 2018, p. 53)). This means that the earth pressure not only has a horizontal component, but a vertical one E⋅(1/v), too, acting at wall line d-a. In the next step, Coulomb solves eq. (2.58) for E and uses the extreme value condition to determine length x′ , which in turn can be written as a trigonometric expression of the slip plane angle. For the special case of c = 0 and 𝜇 = 1/v (coefficient of friction earth-earth = wall-earth), Coulomb noted in his sample calculation that the minimum thickness of the retaining wall at the base was very much smaller “than those [thicknesses] that seem to have become established in practice” (Coulomb, 1779, p. 192), which is why he pleads for ignoring the wall friction angle, i.e. for assuming that the earth pressure is perpendicular to wall line d-a. A current generalisation

As in the past, the Coulomb earth pressure theory is used as the basis for calculating the active earth pressure. One example is the new edition of DIN 4085 “Subsoil – Calculation of earth pressure” published in 2017 (see chapter 17). DIN 4085 includes a formula for calculating the active earth pressure which generalises Coulomb’s earth pressure model for the case of • inclined wall and terrain lines, • vertical load, and • horizontal force. Of course, in this calculation the friction between the wall line and the body of soil and the cohesion along the slip plane are also taken into account. The horizontal force in DIN 4085 mentioned above represents the case of a horizontal flow towards the wall (Hettler, 2017b, p. 465). Michael Goldscheider has published a seminal work on presenting hydrostatic pressures in the soil with flowing groundwater and its application to earth pressure and ground failure calculations (Goldscheider, 2015), and the earth pressure formula of DIN 4085 is based on his proposals. Passive earth pressure (see section 3.3)

Following on directly from the development of his formula for active earth pressure (eq. 2.54), Coulomb specifies the passive earth pressure (Fig. 2.25) using P′′ = Ep = Gp ⋅

(H + x′′ ⋅ μ) (H2 + x′′2 ) + c ⋅ (x′′ − H ⋅ μ) (x′′ − H ⋅ μ)

(2.59)

albeit without deriving this formula (Coulomb, 1773/1776, p. 358). The retaining wall forces the earth wedge a-d-n′′ upwards in such a way that this mobilises a downward resistance force T c,p + Rp acting on slip plane d-n′′ .

2.2 Earth pressure theory as an object of military engineering

Fig. 2.25 Determining the passive earth pressure P′′ = E p after Coulomb.

x′′ x a

n′′

G′′ = Gp Np

Tc,p + Rp H

P′′ = Ep NE,p

Tp TE,p

φ = ϑ′′ = ϑp d

If the trigonometric relationship x′′ = H⋅cot𝜗p is substituted into eq. (2.59), then, following considerable transformation, we finally arrive at a formula for the passive earth pressure: Ep = Gp ⋅

(sin ϑp + μ ⋅ cos ϑp ) (cos ϑp − μ ⋅ sin ϑp )

+c⋅

H . sin ϑp ⋅ (cos ϑp − μ ⋅ sin ϑp )

(2.60)

Coulomb did not pursue passive earth pressure any further, and this variable merely enabled him to limit the earth pressure at rest E < Eo < EP (Coulomb, 1773/1776, p. 358). Referring to Fig. 7 in Fig. 2.20, he noted: “It is therefore proved that in the case where cohesion and friction contribute to the state of rest, the limits of the force applied at F perpendicular to CB which does not set the triangle [= wedges of active or passive earth pressure – the author] in motion, lie between A [= value of active earth pressure E – the author] and A′ [= value of passive earth pressure Ep – the author]” (Coulomb, 1779, p. 182). Design

By substituting eq. (2.56) into eq. (2.54), Coulomb obtains the active earth pressure (Coulomb, 1773/1776, p. 360) E = γE ⋅ H2 ⋅ m(μ) − c ⋅ H ⋅ l(μ) with the parameters √ √ [ (1 + μ2 ) − μ] ⋅ {1 − μ ⋅ [ (1 + μ2 ) − μ]} m(μ) = √ 2 ⋅ (1 + μ2 ) and

√ {1 + [ (1 + μ2 ) − μ]2 } l(μ) = . √ {μ + [ (1 + μ2 ) − μ]}

(2.61)

(2.62)

(2.63)

In the next step, Coulomb considers the infinitesimal element of the active earth pressure dE(z) dependent on the level z on the wall line (origin of coordinates at C) which acts on B-B′ (Fig. 7 in Fig. 2.20). If the height H is replaced by z in eq. (2.61) and the equation solved for z, then dE(z) = [γE ⋅ 2 ⋅ z ⋅ m(μ) − c ⋅ l(μ)] ⋅ dz.

(2.64)

33

34

2 The history of earth pressure theory

This infinitesimal earth pressure dE(z) generates the moment dM = dE⋅(H-z) with respect to point E (Fig. 7 in Fig. 2.20) which integrated over level z results in the value z=H

M=

∫z=0

H

(H − z) ⋅ dE =

∫0

(H − z) ⋅ [γE ⋅ 2 ⋅ z ⋅ m(μ) − c ⋅ l(μ)] ⋅ dz

H3 (2.65) − c ⋅ l(μ) ⋅ H2 3 (Coulomb, 1773/1776, p. 361). From moment equilibrium about E, it then follows that for a retaining wall of constant thickness (b = k) √ 2 ⋅ m(μ) c ⋅ l(μ) min bCoulomb = γE ⋅ ⋅ H2 − ⋅ H. (2.66) 3 ⋅ γMW γMW M = γE ⋅ m(μ) ⋅

Coulomb specifies eq. (2.66) for the special case of cohesionless soil (c = 0) only, with coefficient of friction 𝜇 = 1 and 𝛾 E = 𝛾 MW : √ √ 2 ⋅ γE ⋅ 0.086 2 ⋅ m(μ) min bCoulomb = H ⋅ γE ⋅ =H⋅ 3 ⋅ γMW 3 ⋅ γMW √ γE min bCoulomb = 0.587 ⋅ H ⋅ = 0.24 ⋅ H (2.67) 6 ⋅ γMW (Coulomb, 1773/1776, p. 361). The minimum width of the base of the retaining wall according to Coulomb is therefore only 58.7% of that obtained by Bélidor using eq. (2.30). In the end, Coulomb recommends a width of 1 ⋅ H + 1.625, (2.68) 7 for trapezoidal retaining wall cross-sections with a batter of 1:6 on the air side, a width at the top k = 5⋅0.325 = 1.625 m, a height H = 11.38 m and factor of safety against overturning 𝜈 = 1.25. With H = 11.38 m, this results in b = 3.25 m – a value that lies a little below that given by the design rule of Vauban according to eq. (2.2). Coulomb explicitly highlights this good agreement with Vauban practice (Coulomb, 1773/1776, p. 362). b=

2.2.4

A magazine for engineering officers

The German translation of Coulomb’s beam, earth pressure and masonry arch theories by J. M. Geuß, professor of mathematics in Copenhagen, appeared in the fifth volume of the Magazin für Ingenieur und Artilleristen (hereinafter referred to as Magazin) (Coulomb, 1779) three years after the theories were published in the Memoires of the Académie Royale des Sciences (Coulomb, 1773/1776). However, Geuß only translated the first 28 pages of the 40-page Memoire (Coulomb, 1773/1776, pp. 343-370), omitting the last part on masonry arch theory (Coulomb, 1773/1776, pp. 370-382). The reason for this incomplete translation was that Andreas Böhm (1720-1790), as editor of the Magazin, was mainly interested in calculations for retaining walls because this was crucial to

2.2 Earth pressure theory as an object of military engineering

Fig. 2.26 Title page of the first volume of Böhm’s Magazin für Ingenieur und Artilleristen.

the everyday work of military engineers engaged on fortifications and at the same time reflected his understanding of his discipline. Andreas Böhm, professor of philosophy and mathematics in Gießen, published the first volume of his Magazin (Fig. 2.26) in 1777; 12 volumes had been published by 1795. The second volume appeared in the founding year of the Magazin, the third and fourth volumes followed in 1778 and Böhm published the subsequent volumes each year between 1779 and 1783. However, it was not until 1787 and 1789 that the 10th and 11th volumes appeared to continue the series. The 12th and final volume of Böhm’s Magazin was published by Johann Carl Friedrich Hauff (1766-1846), professor of mathematics in Marburg, in 1795. Böhm’s Magazin was the first German-language periodical in which the state of knowledge for engineering officers was compiled, edited and archived.

35

36

2 The history of earth pressure theory

Böhm named five reasons for founding the Magazin: • To make available again short, excellent articles from “engineering and artillery science”. • To present extracts from larger relevant works published many years ago and in limited numbers. • To present German translations of articles in foreign languages published by the academies. • To present handwritten manuscripts in printed form. • To publish original works. With these goals in mind, it was the editor’s “intention to promote the continuance of engineering and artillery science ... according to his ability” (Böhm, 1777, preliminary report). Recurring themes in Böhm’s Magazin were the determination of earth pressure and the design of retaining walls. Besides Coulomb’s earth pressure theory, the Magazin also contained a translation of Couplet’s work (Couplet, 1795) and other publications on this subject written by engineering officers and mathematics professors such as Kinsky (1778/Vol. III & 1795/Vol. XII), Ypey (1778/Vol. IV), Lorgna (1778/Vol. IV), Clasen (1779/Vol. V & 1781/Vol. VII), Heurlin (1778/Vol. IV) and Stahlswerd (1778/Vol. IV, 1779/Vol. V & 1781/Vol. VII). Almost all of the articles in that list lagged behind the state of knowledge of French engineering officers and were still in the tradition of the application phase of theory of structures (1700-1775) (Kurrer, 2018, p. 17). An exception was the treatise Vom Drucke der Erde auf Futtermauern (of the pressure of the earth on retaining walls) by Count Kinsky (Kinsky, 1795/Vol. XII). Kinsky based his article on the earth pressure theory of Bélidor (see Fig. 2.15), assumed, like him, a slope angle φ = 45∘ , used the fundamental elements of analysis in doing so and explained the theory of the outstanding French engineering officer by way of 11 tasks taken from everyday engineering. This was how this earth pressure theory entered the specialist German literature.

2.3 Modifications to Coulomb earth pressure theory Theory of structures had been working through Coulomb’s earth pressure theory since the start of the 19th century. It retained its function as a reference theory for determining earth pressure into the innovation phase of theory of structures (1950-1975) (Kurrer, 2018, pp. 25-26). The state of this theory and its generalisation around the middle of this phase has been described by Árpád Kézdi (1919-1983) in his monograph (Kézdi, 1962, pp. 170-215). 2.3.1

The trigonometrisation of earth pressure theory

The first person to modify Coulomb’s earth pressure theory for sloping wall lines was the civil engineer, mathematician and director of the École Nationale des Ponts et Chaussées, Gaspard de Prony (1755-1839) (Prony 1802). David Gilly

2.3 Modifications to Coulomb earth pressure theory

(1748-1808) and Johann Albert Eytelwein (1764-1848) followed him three years later (Gilly & Eytelwein, 1805, pp. 101-130). The problem with both studies is, however, that they assume the direction of the earth pressure to be horizontal, as for a frictionless, vertical wall line. Jean-Henri Mayniel (1760-1809) provided a consistent solution for the case of the frictionless but sloping wall line by taking the direction of the earth pressure to be perpendicular to the wall line (Mayniel, 1808, pp. 112-120). But in the case of wall lines affected by friction, Mayniel made the mistake of transferring the method pursued by Coulomb for vertical wall lines to the more general case without the necessary modifications (Kötter, 1893, p. 92). Formal progress compared with Coulomb earth pressure theory took place upon the introduction of trigonometric functions, which eased the generalisation of the theory considerably. Woltman was at the start of this development (see section 2.2.2), which was continued by Prony and Mayniel and was brought to a relative conclusion by Jacques-Frédéric Français (1775-1833) at the transition from the initial phase (1775-1825) to the constitution phase (1825-1850) of theory of structures (Kurrer, 2018, pp. 17-20). Prony

Together with Adrien Marie Legendre (1752-1833), Lazare Carnot (1753-1823) and other mathematicians, Prony produced logarithmic and trigonometric tables within the scope of introducing the metric system between 1792 and 1801. The detailed calculations were carried out by about 80 assistants, who were organised according to the example of Adam Smith (1723-1790) with an extreme division of labour like a production line, for which Smith used the example of the production of dressmaking pins (Smith, 1776). For Prony it was therefore obvious to express the active earth pressure in the form of trigonometric relationships. In doing so, he discovered a relationship for the retaining wall with frictionless, vertical wall line and horizontal terrain line: The slip plane bisects the angle between the wall and slope lines (Fig. 2.27) to give the slip plane angle ρ (2.69) ϑ = 45∘ + 2 (Prony, 1802a, p. 7). This relationship is called Prony’s theorem in the following, which, incidentally, is also valid for cohesive soil material (c ≠ 0). If eq. (2.69) is substituted into the formula for the active earth pressure of cohesionless soil material (c = 0), eq. 2-14 (see in (Kurrer, 2018, p. 53)), then for the special case under consideration (wall friction angle 𝛿 = 0∘ and inclination of wall Fig. 2.27 Prony’s theorem.

e

ne

e li

p Slo ρ

ϑ = 45° +

ρ 2

H

lan Sli pp

Wall line

Terrain line

37

2 The history of earth pressure theory

line 𝛼 = 0∘ ) we get – following trigonometric transformations – the following simple relationship: ( ρ) . (2.70) E = G ⋅ tan 45∘ − 2 Substituting the weight of the sliding earth prism taking into account eq. (2.69) G=

1 H2 1 H2 ⋅ ⋅ γE = ⋅ ) ⋅ γE ( 2 tan ϑ 2 tan 45∘ + ρ 2

(2.71)

into eq. (2.70) finally leads – following trigonometric transformations – to ( ρ) 1 1 = ⋅ H2 ⋅ γE ⋅ λa = Ehydrostatic ⋅ λa (2.72) E = ⋅ H2 ⋅ γE ⋅ tan2 45∘ − 2 2 2 with the coefficient of active earth pressure 𝜆a introduced by Krey (see eq. 3.5 in section 3.2). The earth pressure E for cohesionless soil material is thus nothing more than the product of hydrostatic earth pressure according to eq. (2.13) and the earth pressure coefficient 𝜆a . Similarly, it is possible to derive the formula ( ρ) 1 1 = ⋅ H2 ⋅ γE ⋅ λp (2.73) Ep = ⋅ H2 ⋅ γE ⋅ tan2 45∘ + 2 2 2 for passive earth pressure, where 𝜆p is the coefficient of passive earth pressure (see eqs. 6.7 & 6.10 in section 6.2.1). Français would later modify Prony’s theorem for the wall line at an angle 𝛼 – again valid for cohesive soil material as well (c ≠ 0) (Français, 1820, p. 163) (Fig. 2.28): 1 ⋅ (ρ − α) 2

ϑ = 45∘ +

(2.74)

Eq. (2.74) is designated the modified Prony’s theorem here. Once again, using eq. 2-14 (see (Kurrer, 2018, p. 53)) with eq. (2.57) for 𝛿 = 0∘ and c = 0, we get the active earth pressure sin[45∘ − 0.5 ⋅ (ρ + α)] , sin[45∘ + 0.5 ⋅ (ρ − α)]

E=G⋅

(2.75)

which is transformed into eq. (2.70) for 𝛼 = 0∘ . Here again, the weight of the sliding wedge G can be expressed as a function of H and trigonometric relationships for angles 𝛼 and 𝜌. Fig. 2.28 The modified Prony’s theorem (Français).

Terrain line α

pe

Slo

line

ρ

ϑ = 45° +

1 (ρ – α) 2

H

Wall lin e Sli pp lan e

38

2.3 Modifications to Coulomb earth pressure theory

Mayniel

On 8 September 1807 the head of the battalion at the military engineering corps, Mayniel, submitted a commemorative article with the following title to the Comité Central des Fortifications: Mémoire sur la Poussée des Terres, d’après la Théorie donnée en 1773 par M. Coulomb, officier du génie et membre de l’Académie des Sciences, et d’après des expériences exécutées à Juliers, en novembre et décembre 1806, et en janvier 1807. Shortly after that, Mayniel sent his manuscript Traité Expérimental et Analytique de la Poussée des Terres contre les murs de revêtement to the same committee. Both works were approved by the Comité Central des Fortifications and were highly recommended to the Minister of War Jean Baptiste Bernadotte (1763-1844) in a report dated 27 February 1808. Mayniel’s work appeared in printed form in that same year (Fig. 2.29). Mayniel divided his work into four books. In the first book (Mayniel, 1808, pp. 1-40) he discusses the earth pressure experiments of Gadroy (1746), d’Antony (1778), Gauthey (1784 & 1785) and Rondelet (1805), and also presents his own experiments that he had carried out in Alessandria (then within the French Empire, now Italy) in 1805 and in Juliers, France (now Jülich, Germany), in 1806/1807. Whereas his predecessors measured the magnitude of the earth pressure using small models, Mayniel determined the earth pressure with a much larger test apparatus that was later adopted by Carl Martony de Köszegh (see section Martony de Köszegh). His second book (Mayniel, 1808, pp. 41-128) is dedicated to a critical examination of the earth pressure theories of the 18th century. At the end of this book the author reviews Coulomb’s earth pressure theory and presents it in the form of trigonometric expressions. Thereupon, Mayniel compares Coulomb earth pressure theory with his own earth pressure experiments in Juliers in the third book (Mayniel, 1808, pp. 129-236). Finally, in the fourth book (Mayniel, 1808, pp. 237-312) he works through numerous examples from practical fortification engineering applications. Sadly, Mayniel’s work on earth pressure was to go no further, as he died on 17 April 1809 during the second siege of Saragossa during the Peninsular War (1808-1813). Nonetheless, Mayniel’s contribution was not only the first comprehensive historico-critical presentation of the earth pressure theories of the 18th century, but also the first detailed examination of Coulomb earth pressure theory systematically backed up by experimentation. It was to be a grand overture to the evolution of earth pressure theories in the 19th century. Français, Audoy and Navier

Français – or to give him his full title, Professeur d’art militaire et de fortification à l’École Royale de l’Artillerie et du Génie de Metz – made significant modifications to Coulomb earth pressure theory already added to by Prony and Mayniel. His ground-breaking contribution to earth pressure theory was published in the fourth volume of Mémorial de l’officier du Génie (Français, 1820), a series of papers founded in 1803 and published under the auspices of the Ministry of War by the Comité Central des Fortifications. That edition of the publication contains not only his modified version of Prony’s theorem (see Fig. 2.28), but also the case of the terrain line inclined at an angle 𝛽 with respect to the horizontal – a case that is important for the ramparts of fortifications, for instance.

39

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2 The history of earth pressure theory

Fig. 2.29 Title page of Mayniel’s Traité on earth pressure on retaining walls.

However, Français’ approach “contains idiosyncrasies; for example, the slip plane is assumed as if for the case of flat terrain [= horizontal terrain line 𝛽 = 0∘ – the author] with regard to the direction of the rear surface of the wall [= inclined wall line at angle 𝛼 ≠ 0∘ – the author]” (Kötter, 1893, p. 93). Later, the future Brigadier-General Jean-Victor Audoy supplied the correct solution for the case illustrated in Fig. 2.30. It is important to note here that the inclination of the terrain line 𝛽 is at least equal to or greater than the angle of the slope line 𝜌;

2.3 Modifications to Coulomb earth pressure theory

Fig. 2.30 Determining the earth pressure for the earth surcharge case after Audoy (redrawn after (Audoy, 1832)).

k

n terrain line

ke

k′

H′

o Br

H

Slip

pla

ne

β

ϑ

ρ

H 3

E

b

therefore, its stability must be ensured by way of fascines, planting or other measures. Audoy developed his modified earth pressure theory in a commentary to a work by Michaux which filled 26 pages (Audoy 1832)! Audoy had already tried to modify earth pressure theory in 1820, but with less success. He derived the earth pressure for the unit weight, friction and cohesion varying with the level y, but confined himself to the case of a horizontal terrain line (Audoy, 1820). However, his generalisation had no practical value “because the law of variability with depth is not known exactly or is itself very variable” (Winkler, 1872, p. 70). Nevertheless, Lahmeyer does explore Audoy’s cumbersome modification in the appendix to his translation of Poncelet’s earth pressure theory (Poncelet, 1844, pp. 233-242), but then continues Audoy’s observations for soil materials with constant unit weight, friction and cohesion using Français’ example taking cohesion into account. Français uses his elegant trigonometric formulation of earth pressure theory to determine cohesion c as well. Eq. (2.72) expanded by the term for cohesion gives us the active earth pressure ( ( ρ) ρ) 1 − 2 ⋅ c ⋅ H ⋅ tan 45∘ − . (2.76) E = ⋅ H2 ⋅ γE ⋅ tan2 45∘ − 2 2 2 Français takes an earth pressure E = 0 and therefore obtains the height H c at which a vertically separated body of earth remains stable without a retaining wall (Français, 1820, p. 165): ( ρ) 4⋅c (2.77) ⋅ cot 45∘ − Hc = γE 2 This equation for the critical height H c “can be used to determine the cohesion c for a type of soil and has indeed been used to do this” (Kötter, 1893, p. 93). For this reason, H c was determined experimentally such that the vertically separated body of soil was just in equilibrium and the cohesion c can be determined directly with eq. (2.77). Français also specifies the correct formula for the general case of a freestanding body of soil at an angle 𝛼 ≠ 0∘ . Ferdinand Löwe would later prove, through his own series of experiments, that the method according to eq. (2.77) is inadequate for the majority of cases (Löwe, 1872, p. 5). Using the critical height H c , Français derived the point of application of the earth pressure, which lies a little below H/3. From that, he developed formulas for the minimum base width of retaining walls based on moment equilibrium. For the simplest case of a retaining wall of constant thickness (b = k) in cohesionless

41

2 The history of earth pressure theory

Fig. 2.31 Earth pressure for a uniformly distributed load p after Navier (redrawn after (Navier, 1826)).

p α pe

Slo

line

H

Wall lin e

42

ρ

soil material (c = 0) with a horizontal terrain line (𝛽 = 0∘ ), Français obtained the following: √ ( γE ρ) ∘ ⋅ min bFrançais = H ⋅ tan 45 − . (2.78) 2 3 ⋅ γMW If – like Coulomb – we use 𝜇 = tan𝜌 = 1, i.e. 𝜌 = 45∘ and 𝛾 = 𝛾 , then eq. (2.78) E

MW

gives us min bFrançais = 0.24 ⋅ H,

(2.79)

hence, the Coulomb minimum base width according to eq. (2.67). The trigonometric formulation of Coulomb earth pressure theory à la Français is thus inconsistent in itself. Summing up, Français notes that “according to the theory, the question asked was fully answered by taking into account all the physical actions that occur, excepting the friction on the inner face of the wall, which is, however, so insignificant that we – according to the examples of Coulomb and Prony – all the more believed we could ignore this, because neglecting this favours stability and increases the thickness of the wall somewhat” (Français, cited after (Martony de Köszegh, 1828, p. 24)). As early as 1826, Navier integrated the most important results of earth pressure theory supplied by Français into the body of theory of structures (Navier, 1826, pp. 103-120). He therefore instigated the emancipation of earth pressure theory from its traditional applications, i.e. the building of fortifications. This process of “civilisation” can be seen in the fact that earth pressure theory examples no longer involved fortifications and the specialist terminology of fortifications disappeared from the relevant writings. This was the case with Navier, too. He followed the example of Français and specified the following earth pressure equation for, in practical terms, the most important case of the retaining wall with an inclined wall line and horizontal terrain line with uniformly distributed load p (Fig. 2.31) (see (Navier, 1826, pp. 109-110)): ( ( ρ) ρ) 1 − 2 ⋅ c ⋅ H ⋅ tan 45∘ − E = ⋅ H ⋅ γE ⋅ (H + 2 ⋅ p) ⋅ tan2 45∘ − 2 2 2 (2.80) The modified Prony’s theorem also applies to the case illustrated in Fig. 2.31, which is why eq. (2.74) was considered when deriving eq. (2.80). Despite the incipient “civilisation” of earth pressure theories during the constitution phase of theory of structures (1825-1850) (Kurrer, 2018, pp. 19-20), it was still engineering officers who achieved the crucial progress in the determination of earth pressure.

2.3 Modifications to Coulomb earth pressure theory

Martony de Köszegh

Carl von Martony de Köszegh (1784-1848), an Austrian engineering officer and later in charge of the building of the Franzensfeste Fortress at Brixen (1833-1838), published the results of many experiments concerning earth pressure, which he carried out in Vienna in 1827 “upon the highest order of General Genie Director Archduke Johann” (Martony de Köszegh, 1828). According to Martony de Köszegh, the driving force behind the development of theories by the phalanx of French engineering officers was achieving a reduction in the thickness of retaining walls for fortifications, so that earth pressure theories could be used to help achieve “significant savings in time and construction costs” (Martony de Köszegh, 1828, p. 3). Nevertheless, he criticised the widely used approach of Bélidor which led to oversized retaining walls: “… although according to the latest theories these walls could be kept significantly thinner for the same circumstances. The reason for this is the lack of practical proofs for the correctness of these theories in their application, and in the case of a large structure, no one wishes to take responsibility for a potential failure. Further, it cannot be denied that many things can act to increase or decrease the pressure of the earth, likewise the greater or lesser resistance of the masonry, and that it is therefore difficult to apply a theory before adequately knowing these things and the magnitude of their possible effects” (Martony de Köszegh, 1828, pp. 4-5). Martony de Köszegh also criticised the engineering officers Prony and Français because they had strengthened the results of the theory for applications in such a way – hence had almost fallen back on Bélidor – that they had destroyed the extremely useful consequences of their theoretical efforts. He writes: “Therefore, where the results of the theory could not be convincingly proved to agree with the manifestations in nature, it could be anticipated that fear and doubts would deprive the state of the benefits that a correct theory would offer.” He concludes that only experiments on a larger scale than those undertaken hitherto are needed and must be extended to different types of soil “in order to find out about all physical causes and their effects ... which have an influence on earth pressure and which, although mentioned in the theory, cannot be measured” (Martony de Köszegh, 1828, p. 5). After Martony de Köszegh has presented the “theory of M. Coulomb” and a translation of the “general solution by M. Français” in the first section of his work (Martony de Köszegh, 1828, pp. 7-41), in the second section he writes a critical review of the earth pressure experiments of Gadroy, d’Antony, Gauthey, Rondelet and Mayniel (Martony de Köszegh, 1828, pp. 42-74). His work – adhering formally to Mayniel’s Traité – focuses on his own experiments (Martony de Köszegh, 1828, pp. 74-166), for which he modified the test apparatus devised by Mayniel somewhat (Fig. 2.32). The box filled with soil has internal dimensions of 2.85 x 0.95 x 1.90 m (l x b x h) and has a movable wall A-B, which is connected to box E via a bar at level h/3. Box E is loaded with lead weights to such an extent that the friction force generated on the wooden bearing plate exceeds the earth pressure transmitted by the bar. Above box E there is another box F filled with fine grit, and this grit escapes from the box through an opening in the side until the friction force is just in equilibrium with the earth pressure. Like Mayniel, Martony de Köszegh

43

44

2 The history of earth pressure theory B F E

I

D

C A

H

G

123

Fig. 2.32 Test apparatus for determining earth pressure as devised by Martony de Köszegh (taken from (Winkler, 1872, p. 123)).

also measured the earth pressure through the gradual loading of the weighing pan H connected to box E via a rope. Wall A-B was moved via two symmetrically arranged handwheels I. Soil escaping from the box at A fell into the masonry pit G (for further details see (Martony de Köszegh, 1828, p. 74ff.)). Using this apparatus, Martony de Köszegh carried out tests on topsoil, sand, pure yellow loam and ballast in characteristic states (e.g. dry, earth-damp and saturated) and measured the earth pressure. He applied his findings to the rebuilding of part of the fortifications around Vienna which had been destroyed in 1809. The series of tests constitute the second part of Martony de Köszegh’s work, and so in the third part he discusses to what extent his tests agree with the theory (Martony de Köszegh, 1828, pp. 167-199). The fourth and last part contains specific criticism of the excessive dimensions of retaining walls (Martony de Köszegh, 1828, pp. 200-221). Taking the example of the fortifications around Vienna partly rebuilt in 1827, he compares his dimensions with those of Bélidor and comes to the conclusion that his design requires only half the amount of masonry compared with that of Bélidor while achieving the same stability. The work of Martony de Köszegh enabled the Coulomb earth pressure theory, in the trigonometric form by Français, to enter German engineering literature and thus lay the foundation for the formation of an independent theory during the establishment phase of theory of structures (1850-1875) (Kurrer, 2018, pp. 20-21). 2.3.2

The geometric way

Prony had specified a graphical method for designing retaining walls back in 1802 [Prony, 1802a, pp. 31-33], which André Guillerme summarised on one printed page (Guillerme, 1995, p. 108). The Prony method was prescribed in the form of instructions [Prony, 1802b & 1809] that became established in

2.3 Modifications to Coulomb earth pressure theory

practical design work for French fortifications in the first three decades of the 19th century. Nevertheless, the method did not cover many important cases encountered in the building of fortifications, e.g. retaining walls with a soil surcharge (see Figs. 2.6 and 2.30). A look at the analytical earth pressure determination of Audoy for raised bodies of soil (Fig. 2.30) clearly reveals this (Audoy, 1832). The awkwardness of Audoy’s formulas for designing retaining walls encouraged Poncelet in 1835 to translate these into the language of geometry. His geometrisation of masonry arch theory served him as a model (Poncelet, 1835). That work, together with Poncelet’s graphical earth pressure theory in the last decade of the constitution phase of theory of structures (1825-1850) (Kurrer, 2018, pp. 19-20), formed the historico-logical introduction to graphical statics, which was given its classical form by Karl Culmann in the 1860s. This was also the period in which fortifications had to give way to railways as the primary application for earth pressure theory. Jean-Victor Poncelet

Earth pressure theory as applied to the building of fortifications experienced a magnificent finale in the shape of Poncelet’s 264-page Mémoire sur la stabilité des revêtements et de leurs fondations (Poncelet, 1840) published in 1840 in the 13th volume of Mémorial de l’officier du Génie. At the same time, it crossed over to other areas of application such as the planning of railway lines, which, so to speak, formed the iron network of the Industrial Revolution in continental Europe in the middle decades of the 19th century and provided earthworks in general and earth pressure theory in particular with new quantitative and qualitative challenges. Poncelet’s earth pressure theory fulfilled this twin function through its systematic use of geometry to determine active and passive earth pressures. This was the first time that structural analysis was moulded into one with working drawings – hence, structural and constructional thinking converged to a structural/constructional attitude towards design. Poncelet divided his Mémoire into three sections. In the first section he looks at how Audoy determines the earth pressure for earth surcharges (Figs. 2.6 and 2.30) and derives tables for dimensions of retaining walls (Poncelet, 1844, pp. 19, 28 & 41). With that in mind, he finally analyses Vauban’s design theory for retaining walls, which has already been explored in section 2.1. Not until the second section does Poncelet move on to the geometrisation of the simple case (𝛽 = 𝜌), followed by the more general case of 𝛽 ≠ 𝜌 (Fig. 2.30), which leads to convoluted trigonometric equations for active pressure and the slip plane angle 𝜗 (Poncelet, 1844, p. 93) but which he transforms elegantly into graphical form. A geometric theory of passive earth pressure then follows (Poncelet, 1844, pp. 103-120), which he called “butée des terres” and which was translated into German by Lahmeyer as “Hebkraft der Erde” (“uplift force of the earth”). So Poncelet achieved the first systematic presentation of passive earth pressure. The standard case for active earth pressure with inclined terrain line (𝛽 < 𝜌) and sloping wall line (𝛼 ≠ 0∘ ) is shown in Fig. 2.33; he also takes into account friction on the wall line (𝛿 ≠ 0∘ ) but neglects cohesion c. A description of “Poncelet’s drawing” will not be provided this point because this has already been explained in section 2.4.2 (see (Kurrer, 2018, pp. 54-55)) and

45

46

2 The history of earth pressure theory

Fig. 2.33 Graphical earth pressure determination after Poncelet (taken from (Poncelet, 1844, plate II, Fig. 23)).

amounts to finding the geometric mean of distances. His graphical determination of the passive earth pressure is carried out similarly. He also proves that the “modified Prony’s theorem” is valid for both active and passive earth pressure – but not for inclined terrain lines (𝛽 ≠ 0∘ ) (Poncelet, 1844, p. 125). Finally, Poncelet also provides graphical solutions for broken terrain lines, which are common in fortifications: “Terraces in the form of cavaliers, or those with a circular walkway, glacis with larger or smaller sloping areas” (Poncelet, 1844, p. 5). So in the form of “Poncelet’s drawing”, the “structural figure” (Kurrer, 2015) had already set foot on the historical stage in 1840 and enhanced the stock of intellectual tools available to engineers. Even today, this “structural figure”, owing to its wonderful simplicity, is ever present in the mind’s eye of the civil engineer. The third section of the Mémoire covers the design of foundations (Poncelet, 1844, pp. 144-228). It also contains remarks on the analytical relationships between active and passive earth pressures (Poncelet, 1844, pp. 216-228). Poncelet’s earth pressure theory in Mémorial de l’officier du Génie, which was not sold via bookshops, was translated into German (Poncelet, 1844, pp. 1-228) in 1844 by Johann Wilhelm Lahmeyer (1818-1859), who would later be a member of the Hannover General Directorate of Waterways. The reason for this translation was financial: Lahmeyer had to wait to join the building authority of the Kingdom of Hannover and in the meantime had to earn his living through providing private customers with drawings and translations. He added an appendix to his translation (Lahmeyer, 1844, pp. 229-270) in which he reviewed the determination of earth pressure by Français (Français, 1820), hydraulics engineer Woltman

2.3 Modifications to Coulomb earth pressure theory

(Woltman, 1794 & 1799) and Gotthilf Hagen (1797-1884) (Hagen, 1833) plus the experiments of Martony de Köszegh (Martony de Köszegh, 1828). By comparing the Coulomb and Hagen earth pressure theories with tests, Lahmeyer discovered good agreement between the results of the tests and Coulomb’s earth pressure theory: “In my opinion, its correctness, likewise the incorrectness of the Hagen theory, can no longer be doubted” (Lahmeyer, 1844, pp. 269-270). In the view of a number of contemporaries who liked to believe in authority, Poncelet’s earth pressure theory – on account of its author – was sacrosanct and therefore earth pressure theory was complete. For instance, in 1857 a report in the Comptes rendus de l’académie des Sciences concerning a paper by Saint-Guilhem entitled Sur la pousée des terres contains the following remark: “Nobody has added anything worthwhile to that said by Poncelet on this matter ... Those who have investigated earth pressures since then have merely repeated Poncelet’s words in a different form, or did not even manage that” (cited after (Scheffler, 1857, p. IX)). But it was precisely Saint-Guilhem who developed Poncelet’s earth pressure theory further analytically and graphically by considering, for example, an arbitrary line load on the broken terrain line (Saint-Guilhem, 1858). Hermann Scheffler’s criticism of Poncelet

It was in 1857 that Hermann Scheffler (1820-1903) adapted earth pressure theory systematically to the practical needs of railway construction for the first time (Scheffler, 1857, pp. 291-374). This is connected with masonry arch theory through the joint book on the “principle of least resistance” (see (Kurrer, 2018, pp. 230-232)) postulated by Moseley and defined more precisely by Scheffler, “which is extremely important for many other applications in engineering” (Scheffler, 1857, p. IX). Besides the applicability of this principle to practical engineering issues, Scheffler describes the foundation of his earth pressure theory as follows: “It appears equally desirable to precede the theory of retaining walls with a treatise on earth pressure in general, because the treatises on this matter already available, i.e. on the distribution of this pressure, on the prism of greatest thrust [= earth pressure wedge with slip plane angle 𝜗 for determining active earth pressure – the author], on the effect of a body of soil on rough wall and smooth ground surfaces, are nowhere near exhausted, in parts even unreliable and erroneous” (Scheffler, 1857, p. IX). Scheffler is particularly hard on Poncelet. For example, Scheffler criticises his way of determining earth pressure for a raised body of soil according to Fig. 2.30. He complains that in this case Poncelet places the line of action of the earth pressure orthogonal to the wall line. More serious, however, is the fact that Poncelet’s earth pressure wedge can only be correct when the intersection between the slip plane and the terrain line is to the right of the change of slope (see Fig. 2.30). Both of Scheffler’s objections are correct. Nevertheless, the other case, where the slip plane intersects the sloping part of the terrain line, only applies to very high embankments or very shallow slopes. However, very high embankments are very common in the building of railways. Therefore, Scheffler’s earth pressure theory also covers this case, although only for a vertical wall line and bodies of soil that cover the top of the wall k completely (k′ = k, see Fig. 2.30). At the end of his

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Fig. 2.34 Scheffler’s approach to designing simple retaining walls with a raised embankment (Scheffler, 1857, p. 361).

explanations, Scheffler provides practical instructions for determining the base widths of retaining walls (Fig. 2.34) which “apply to the construction of retaining walls within the remit of the Duke of Brunswick Railway Authority” (Scheffler, 1857, p. 361). Remarkably, Scheffler does not say anything about Poncelet’s solution for the standard case (Fig. 2.33). Nor does he tackle Poncelet’s geometrisation of earth pressure theory. That would be left for Karl Culmann in 1866. Karl Culmann

In his book Graphische Statik (graphical statics), Culmann, a former Bavarian State Railways engineer, acknowledges Poncelet’s achievements regarding “geometric solutions for the different tasks that present themselves in engineering”, but criticises the fact that they are “only ever translations of analytical expressions developed previously” and therefore constitute indirect solutions. Culmann proposes a strategy to solve this, “which uses the line figures given by the task itself as the foundation from which the solution can be developed with simple geometry” (Culmann, 1864/1866, pp. V-VI). The term “line figures” comes from Karl Georg Christian von Staudt’s (1798-1867) Geometrie der Lage (geometry of position) (Staudt, 1847), also known as “higher geometry”, “synthetic geometry” or “projective geometry”. Starting with this geometry, Culmann created his graphical statics as an attempt “to solve … tasks from the field of engineering which are accessible to a geometric treatment” (Culmann, 1864/1866, p. VI). In eight sections, Culmann touches on the entire world of theory of structures in the middle of its establishment phase (1850-1875) (Kurrer, 2018, pp. 20-21): • • • • • • • •

1st section: Graphical calculations 2nd section: Graphical statics 3rd section: The beam 4th section: The continuous beam 5th section: The trussed framework 6th section: The arch 7th section: The value of the structures 8th section: Theory of retaining walls

The starting point for Culmann’s Graphische Statik was the theory of retaining walls: “We started by working on this; following the theories described in

2.3 Modifications to Coulomb earth pressure theory

Memorial de l’officier du génie, we first learned the advantages of graphical methods and decided to extend these. However, whereas Poncelet only used geometry where he had to, but otherwise preferred analysis, we turned this on its head and applied geometry where we could and thus achieved the results that all expect when undertaking work in an as yet unknown field for the first time” (Culmann, 1864/1866, p. XII). So it was not beam theory, arch theory, trussed framework theory or the theory of the continuous beam that advanced to become the reference theory of graphical statics, but rather earth pressure theory. Culmann’s formulation of earth pressure theory in particular, like theory of structures in general, in the language of projective geometry brought with it – as he emphasised himself – a heuristic potential that was to unfold in detail in the final decades of the 19th century. Fig. 2.35 shows an example of the new quality of the geometrisation of earth pressure theory on Culmann’s higher level of projective geometry. Here we see the Culmann E (= earth pressure) line for a retaining wall with raised body of soil, uniformly distributed load and cohesive soil material (c ≠ 0). This hyperbola, which plays a key role in Culmann earth pressure theory, had already been specified by the French engineering officer P. J. Ardant (Ardant, 1848). The reader is referred to section 4.7 for details of the construction of the Culmann E line for cohesionless soils (c = 0). Whereas the “structural figure” of Poncelet (“Poncelet’s drawing”) was only a translation of analytical expressions from the past, Culmann developed his “structural figures” directly from geometry, as Fig. 2.35 shows. So for Culmann the world of the “structural figure” was graphical statics and he did justice to his claim that drawing, as the language of the engineer, does not reach its limits in two-dimensional presentations of

Fig. 2.35 Graphical study of a retaining wall with raised embankment subjected to a partial uniformly distributed load (Culmann, 1864/1866, plate 32) (from ETH Library, Zurich, Old & Rare Prints Dept.).

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three-dimensional artefacts (descriptive geometry), but experiences significant further progress through the computational figure of statics. Georg Rebhann

Only a few civil engineers will still remember Rebhann’s theorem with which earth pressures on retaining walls could be determined graphically with the elegance inherent to geometry. Georg Rebhann formulated this theorem in his monograph on earth pressure theory, which appeared in six booklets (Rebhann, 1870/1871); publishing was delayed owing to a strike by typesetters, with the third booklet not appearing until the summer of 1870. Even Timoshenko called Rebhann’s theorem a very useful graphical method and explained it with a drawing (Timoshenko, 1953, p. 326). For the special case of a retaining wall with inclined (𝛼 ≠ 0) but smooth wall line (𝛿 = 0∘ ), horizontal terrain line (𝛽 = 0∘ ) and cohesionless soil material (c = 0), Rebhann specified a geometrical proportion at an early stage from which the earth pressure could be determined graphically (Rebhann, 1850 & 1870/1871, pp. 94-96). He used the modified Prony’s theorem in this publication but did not say so as such. A professor of waterways and road construction from Karlsruhe who copied this Rebhann construction in his Allgemeine Baukunde des Ingenieurs (general building knowledge for engineers) almost word for word (Becker, 1853, pp. 189-191) also omitted to mention the source. Rebhann complained about this at the end of the preface to his Theorie der Holz- und Eisen-Constructionen (theory of timber and iron structures) (Rebhann, 1856, p. X), but it made no difference: the source is still missing in the third edition of Becker’s book (Becker, 1865, pp. 210-211). Rebhann introduced the designations of active earth pressure (Fig. 2.36) and passive earth pressure (Fig. 2.37) into German engineering language. With a small reduction in weight Q′ , the slip plane A-a is established in the body of soil and the sliding body of soil A-D-a moves the retaining wall A-D-C to A-D′ -C′ (Fig. 2.36): “As in this case the earth pressure is active, so to speak, while the retaining wall behaves passively, we can call the former the active earth pressure” (Rebhann, 1870/1871, p. 16). If on the other hand Q′′ is increased a little, the slip plane takes on the shape A-b and the sliding body of soil A-D′′ -b is moved Fig. 2.36 Illustration of active earth pressure Q′ = E a (Rebhann, 1870/1871, p. 16).

2.3 Modifications to Coulomb earth pressure theory

Fig. 2.37 Illustration of passive earth pressure Q′′ = E p (Rebhann, 1870/1871, p. 17).

inwards by retaining wall A-D-C (Fig. 2.37): “As in this case the retaining wall is active, as it were, while the body of soil behaves passively, we can call the ensuing counterpressure of the earth the passive earth pressure” (Rebhann, 1870/1871, p. 17). However, the upward-curving slip plane A-b drawn by Rebhann in Fig. 2.37 does in fact also curve downwards for passive earth pressure – as for active earth pressure. For simplicity, Rebhann called the active earth pressure simply “pressure” or “earth pressure”, whereas he introduced the term “resistance of the earth” for the passive earth pressure (Rebhann, 1870/1871, p. 17). Rebhann’s concept not only satisfied the grammatical rules of the active and passive voices of the verb, but also created the conditions for the structure of earth pressure theory at the end of the establishment phase of theory of structures (1850-1875) (Kurrer, 2018, pp. 20-21). In the “first principal element” of his earth pressure theory (Rebhann, 1870/1871, pp. 24-387), which is divided into three sections, Rebhann determines the active earth pressure firstly for a frictionless (𝛿 = 0∘ ) wall line and then for a wall line with friction (𝛿 ≠ 0∘ ); he also takes cohesion into account in the former case (c ≠ 0). He formulates a theorem for these cases, an example of which is shown in Fig. 2.38. Based on the trigonometric relationships for the weight of the sliding wedge of soil A-F-E and the active earth pressure Ea , Rebhann discovers that slip plane A-E bisects area A-F-E-r: AΔAFE = AΔAEr

(2.81)

The area equivalence expressed by eq. (2.81) is known as Rebhann’s theorem. From the similarity of triangle A-E-r with the triangle of forces for Q, Ea = E and G (see Fig. 2-41 in (Kurrer, 2018, p. 53)) Ea ∶ G = Eq ∶ Aq

(2.82)

it follows that, using eq. (2.81), the active earth pressure Ea = 0.5 ⋅ Pq ⋅ PE ⋅ γE

(2.83)

is equal to the area of the isosceles triangle E-r-q (shaded in Fig. 2.38). The area of E-r-q is therefore the “earth pressure triangle” and distance r-q the “earth pressure magnitude”.

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Fig. 2.38 Determining the active earth pressure for wall lines with friction using Rebhann’s theorem (Rebhann, 1870/1871, p. 309).

In a similar way, in the “second principal element” of his earth pressure theory, Rebhann determines the passive earth pressure, or earth resistance, initially for a frictionless (𝛿 = 0∘ ) wall line and then for a wall line with friction (𝛿 ≠ 0∘ ) (Rebhann, 1870/1871, pp. 387-442). Incidentally, it is easy to see that Prony’s theorem can be derived from Rebhann’s theorem (eq. 2.81) for the former’s limited range of validity (see Fig. 2.27); the modified Prony’s theorem (see Fig. 2.28) is a special case of Rebhann’s theorem. Whereas Rebhann develops the determination of active and passive earth pressures in the two principal elements subsumed in the first section, in the second section he turns to the design of retaining walls (Rebhann, 1870/1871, pp. 443-543). This is where the reader finds the stress and stability analyses, examples of which were given in (Kurrer, 2018, pp. 55-58). Rebhann succeeded in giving earth pressure theory a clear, well-structured geometric form with which engineers could easily solve practical problems in earthworks. He significantly extended the world of the “structural figure” for earth pressure theory. Starting with the Rebhann earth pressure theory, P. Grumblat compiled tables for quantifying the slip plane and the earth pressure (Grumblat, 1920). Finally, taking Rebhann’s theorem as a starting point, Otto Mund developed his method for determining earth pressure (Mund, 1936), which was the object of a scientific controversy (see (Ohde, 1938b) and (Mund, 1938a,b,c & 1939)). Compelling contradictions

When calculating the minimum base width minb of retaining walls, it is not just the dimension that is important, but also the point of application of the active earth pressure Ea on the wall line and its direction (position of Ea ). But these three decisive aspects are input into the moment acting about the toe m of the retaining wall, which with a factor of safety against overturning 𝜈 K,m = 1 must

2.3 Modifications to Coulomb earth pressure theory

be equal to the restoring moment (see eq. 2-18 in (Kurrer, 2018, p. 57). Bélidor (see Fig. 2.16) and Coulomb (see Fig. 7 in Fig. 2.20 and section Design) bypassed the direct determination of the position of Ea by using their method of slices to calculate the acting moment. That method implies that the entire sliding wedge of soil is penetrated by a group of slip planes at the slip plane angle 𝜗 at the limit state of equilibrium. But that means they infringe the rigid body model of the sliding wedge on which wedge theory is based (first principal assumption) and transcend the discipline of the mechanics of rigid bodies. So without knowing it, Bélidor and Coulomb had entered the realm of continuum mechanics, which was already gaining a foothold in earth pressure theory during the application phase of mechanics (1700-1775) in the form of hydrostatics – with Bélidor (section Friction reduces earth pressure) and von Clasen (1781a), for instance. In the special case of a horizontal terrain and vertical wall line, a linear course of the specific earth pressure e(y) really is established over the wall line in the case of cohesionless soil such that the active earth pressure Ea intersects with the wall line orthogonally and at the upper end of the lower third, so the position of Ea is fully known: “It was also precisely this agreement with the pressures of fluids that certainly made a major contribution to protecting the determination of the point of application given here against criticism” (Kötter, 1893, p. 103). But the Coulomb earth pressure theory and its modifications contained a further contradiction closely related to the hydrostatic continuum model and which Johann Ohde (1905-1953) called a “blemish” [Ohde, 1948-1952, p. 122]: forces Ea , G and Q do not intersect at one point (Fig. 2.39). If the points of application of the active earth pressure Ea and the slip plane compressive force Q are assumed to be at the respective third points d-f = 1/3⋅(d-a) of the wall line or d-q = 1/3⋅(d-n) of the slip plane, i.e. a linear stress distribution is implied, then the result is generally the non-central system of forces Ea , G und Q; this was pointed out first of all by Karl Culmann (1864/1866, p. 629) and then by Otto Mohr (1871, p. 494). Although any non-central system of forces will satisfy translational equilibrium, it will not satisfy moment equilibrium (see eq. 2-11 in (Kurrer, 2018, p. 30)). This contradiction, which later turned out to be merely a “blemish”, formed the main line of Mohr’s criticism of classical earth pressure theory. Fig. 2.39 Section through retaining wall and force diagram for determining the active earth pressure according to the model of the wedge with friction.

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Fig. 2.40 Curved slip plane after Scheffler (Scheffler, 1857, p. 303).

Coulomb clearly highlighted the fact that modelling the slip plane as a straight line is a simplification in earth pressure theory: “I first of all assume that the curved line of the greatest pressure [= active earth pressure – the author] is the straight line. This agrees with experience, which shows that the earth that slides down behind failed retaining walls is virtually triangular in form” (Coulomb, 1779, p. 184). The second principal assumption of the earth pressure theories based on the wedge model is therefore only an engineering-type approximation of the reality. Nonetheless, even Coulomb investigated curved slip planes (Fig. 7 in Fig. 2.20,). Later, Scheffler would establish that the slip plane is generally not a straight line, but a curve (Fig. 2.40). For the special case of a retaining wall with frictionless wall line and horizontal terrain line, he proved – with the help of calculus of variations – that the slip line must be straight (Scheffler, 1851). Like Coulomb, Scheffler also realised that the earth pressure in common cases of the second principal assumption does not differ very significantly from the reality (Scheffler, 1857, p. 303). Therefore, Scheffler, too, assumed straight slip planes when developing his earth pressure theory. A style of theory that owed more to continuum mechanics than rigid body mechanics gradually took hold in the establishment phase of theory of structures (1850-1875) (Kurrer, 2018, pp. 20-21) in both masonry arch theory and earth pressure theory. Ernst Hellinger supplied a valid definition of its basic model: “The general continuous medium expanded in three dimensions ... means – with abstraction of all the individual properties of the material – the totality of material particles that, firstly, can be distinguished from each other and, secondly, should always fill the space or a permanently limited part of a space” (Hellinger, 1914, p. 606). Two-dimensional media separated by the terrain line will be dealt with below.

2.4 The contribution of continuum mechanics Cauchy presented his work Recherches sur la l’équilibre et le mouvement intérieur des corps solides ou fluides, élastiques ou non élastiques, extracts from which appeared in the Bulletin des Sciences par la Société Philomatique, to the Académie Royal des Sciences in Paris on 30 September 1822 (Cauchy, 1823). This study contains not one single formula but does constitute the “foundation of continuum

2.4 The contribution of continuum mechanics

mechanics” (Szabó, 1977, p. 395). As Cauchy writes: “If one focuses on a fixed element in a solid, elastic or non-elastic body bounded by any surfaces and loaded in any arbitrary fashion, so this element experiences a (tensile or compressive) stress at every point on its surface. This stress is similar to that occurring in fluids, the only difference being that the hydrostatic pressure at a point is always perpendicular to the arbitrarily oriented surface at that point, whereas the stress at a given point in a solid body will generally be at an angle to the surface element passing through this point and dependent on the position of the surface element. This stress can be very easily deduced from the stresses occurring in the three planes of coordinates” ((Cauchy, 1822, p. 10) after (Szabó, 1977, p. 395)). Later, the primus inter pares of rational mechanics, C. A. Truesdell, would acknowledge the apparently small step from Euler’s method of sections in hydromechanics, where the stresses always act perpendicular to the released surfaces of the element under consideration, to the method of sections of the mechanics of continua as “an achievement of truly Eulerian proportions and clarity” (Truesdell, 1956, p. 325). Cauchy’s simple yet ingenious fundamental idea was to omit the restriction that the stress vector must be perpendicular to the respective sectional surface. The fact that the findings of continuum mechanics (created in the 1820s) were not adopted by earth pressure theory in the constitution phase of theory of structures (1825-1850) (Kurrer, 2018, pp. 19-20) was due to the following reasons: • Mathematics was split into geometry and analysis in France during the years of the Bourbon Restoration (1815-1830). • Cauchy, a royalist, rejected the projective geometry of Poncelet, a republican, and thwarted the publication of this in the 1820s under the auspices of the Académie Royal des Sciences. • As the first authority on earth pressure theory, Poncelet employed geometrisation to exclude modelling by continuum mechanics. • Continuum mechanics focused its epistemological interest on the behaviour of elastic bodies, but not on the much more complex semi-fluid substances such as soils. • Finally, the formation of a genuine engineering science theory style must be mentioned as well, which was clearly demarcated from the theory style of the exact sciences. It was none other than Poncelet who called for “a special investigation into the physical properties of the soils considered as semi-fluids”. In the same breath, he praised the first continuum mechanics approach of Garidel (1839), an engineering officer, but then complained in a footnote that “his results would be difficult for engineers to accept” (Poncelet, 1844, p. 3). In Germany, Ortmann (1847) attempted the modelling of sand based on continuum mechanics but applied a misguided hydrostatic method, which Emil Winkler rejected harshly: “Ortmann deals with a great many questions on the basis of his seemingly very academic study ... with the same ease as is possible in hydrostatics. If we are to believe the selection of brilliant conclusions, then we will not need to ask for any further information” (Winkler, 1872, p. 99).

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Both of these unsuccessful, mathematically overloaded continuum mechanics attempts were quickly followed by Scheffler’s coherent modelling (Scheffler, 1851), which he published in abridged form in his popular book (Scheffler, 1857). 2.4.1

The hydrostatic earth pressure model

Scheffler assumed a body of soil infinite in the horizontal direction which is only bounded by a horizontal terrain line (Fig. 2.41). The soil material is incompressible, permanently distributed and displaceable. In a first step, he considers the earth column a-b-f-e and notes that the vertical force at section e-f is equal to the weight of the earth column F 0 such that the compressive stress 𝜎 zz (z) increases linearly from the terrain line z = 0 to the depth of soil z = H: σzz (z) = γE ⋅ z

(2.84)

For reasons of symmetry, the horizontal compressive stress acting on the body of soil must be the same at every horizontal section. In a second step, Scheffler determines the slip plane angle 𝜗 and the specific earth pressure 𝜎 xx (z) = ex (z) from the equilibrium conditions for the x and z directions (Fig. 2.42). Force F z = 𝜎 zz ⋅Δx = 𝛾 E ⋅z⋅Δx, force F x = 𝜎 xx ⋅Δz and force Q inclined at the friction angle (= slope angle 𝜌) act on the triangular element A-B-C. Scheffler still considers the cohesive force acting at section A-B as a shear Fig. 2.41 Scheffler’s hydrostatic earth pressure model (Scheffler, 1857, p. 292).

Fig. 2.42 Scheffler’s derivation of the specific earth pressure 𝜎 xx (z) = ex (z).

2.4 The contribution of continuum mechanics

force c⋅[Δz/sinφ], but for reasons of clarity, the influence of cohesion c is not considered in Fig. 2.42. Once Scheffler has presented the specific earth pressure ex (z) as a function of φ, he derives the latter and sets the resulting expression to zero. He determines the slip plane angle φ = 𝜗 (see eq. 2.69) from this necessary condition of the maximum for ex (z) and thus obtains the specific (active) earth pressure for cohesive soils: ( cos𝜌 𝜌) −c⋅ (2.85) 𝜎xx (z) = ex (z) = 𝛾E ⋅ z ⋅ tan2 45∘ − ( 𝜌). 2 cos2 45∘ − 2 Scheffler calls the specific earth pressure the “true horizontal pressure” (Scheffler, 1857, p. 297). In the case of non-cohesive soils, eq. (2.85) is simplified to ( ( 𝜌) 𝜌) 𝜎xx (z) = ex (z) = 𝛾E ⋅ z ⋅ tan2 45∘ − = 𝜎zz (z) ⋅ tan2 45∘ − . (2.86) 2 2 Integrating eq. (2.86) z=H H ( 𝜌) ⋅ dz (2.87) E= ex (z) ⋅ dz = 𝛾E ⋅ z ⋅ tan2 45∘ − ∫z=0 ∫0 2 results in eq. (2.72). This proves that Scheffler’s hydrostatic earth pressure model can be transformed into the case investigated by Prony of the retaining wall with frictionless, vertical wall line and horizontal terrain line (see section Prony). Fig. 2.43 shows the stress ellipse for the convergence point K of the infinitesimal element dx⋅dz with the semi-major axis according to eq. (2.84) and the semi-minor axis to eq. (2.86). Scheffler’s hydrostatic earth pressure model is distinguished by the fact that the stress ellipses for z = const. are identical and their major axes always parallel or orthogonal to the terrain line. The parallelism of the terrain line and the direction of the active earth pressure E follow directly from this. In the purely hydrostatic state, the stress ellipses degenerate to stress circles with a radius corresponding to the respective hydrostatic pressure. Although a very capable mathematician, Scheffler did not pursue the stress ellipses concept any further. Deeper insights into the stress states of bodies of soil therefore remained hidden from him. Fig. 2.43 The stress ellipse of the hydrostatic earth pressure model.

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2.4.2

The new earth pressure theory

Cauchy derived the nine stress components contained in the equations of motion in the third volume of his Exercises de Mathématique (Cauchy, 1828, p. 166). Fig. 2.44 shows Cauchy’s specialised equations for the equilibrium of the planar continuum mechanics model of earth pressure with soil material of constant unit weight (𝛾 E = const.) for the static case. The three equilibrium conditions (see eq. 2-11 in (Kurrer, 2018, p. 30)) take on the following form for the infinitesimal element A-B-C-D (Fig. 2.44): ( ) ( ) 𝜕σ 𝜕τ ΣFx = 0 = σxx ⋅ dz − σxx + xx ⋅ dx ⋅ dz − τzx + zx ⋅ dz ⋅ dx + τzx ⋅ dx 𝜕x 𝜕z ( ) ( ) 𝜕σzz 𝜕τxz ⋅ dz ⋅ dx − τxz + ⋅ dx ⋅ dz + τxz ⋅ dz + γE ⋅ dx ⋅ dz ΣFz = 0 = σzz ⋅ dx − σzz + 𝜕z 𝜕x ( ) ( ) 𝜕τzx 𝜕τ dz dx dz dx ΣMS = 0 = τzx ⋅ dx ⋅ + τzx + ⋅ dz ⋅ dx ⋅ − τxz ⋅ dz ⋅ − τxz + xz ⋅ dx ⋅ dz ⋅ 2 𝜕z 2 2 𝜕x 2

If the terms with the quadratic differentials (dx)2 and (dz)2 are ignored, then the equilibrium conditions are transformed into 𝜕σxx 𝜕τzx + =0 𝜕x 𝜕z 𝜕σzz 𝜕τxz + = γE 𝜕z 𝜕x τzx = τxz

(2.88)

The third equation is the known law of associated shear stresses which, substituted into the first equation, results in Cauchy’s equilibrium equations for the continuum mechanics model of earth pressure with soil material of constant unit weight (𝛾 E = const.) (see eq. 3.13 in section 3.4.1): 𝜕σxx 𝜕τxz + =0 𝜕x 𝜕z 𝜕σzz 𝜕τxz + = γE 𝜕z 𝜕x

(2.89)

x

0

dx

z

x

σzz D σxx

dz

58

z

τzx

C

∂τxz dx ∂x ∂σ σxx + xx dx ∂x

τxz+

S τxz dG = γE·dx · dz A B ∂τzx ∂σzz τ + σzz + dz zx ∂z dz ∂z

Fig. 2.44 Derivation of Cauchy’s equations for the continuum mechanics model of earth pressure.

2.4 The contribution of continuum mechanics

The two partial differential equations linked via the shear stress 𝜏 xz = 𝜏 contain three unknown stress components: • the compressive stress in the x direction 𝜎 xx (x,z), • the compressive stress in the z direction 𝜎 zz (x,z), and • the shear stresses in the x and z directions 𝜏 xz (x,z). Therefore, a third condition is required for the solution, which generates a functional relationship between the compressive and shear stresses. This is the Coulomb-Mohr yield (or failure) criterion (see eq. 3.12 in section 3.4.1) τBruch = f(σ) = σ ⋅ tan ρ + c,

(2.90)

where 𝜌 is the angle of internal friction (see eq. 2.39) and c the cohesion of the soil material. The system of equations, eq. (2.88) or (2.89), was published by Carl Holtzmann indirectly via the concept of the stress ellipsoids that can be traced back to Lamé (Holtzmann, 1856), but without mentioning this authorship; on the other hand, William John Macquorn Rankine (1857) and Emil Winkler (1861, 1871a & 1872) assumed eq. (2.88) or (2.89) directly. All three developed their continuum mechanics analyses of the earth pressure problem independently of each other. To conclude, it is necessary to draw attention to the publication of the British mathematician James Joseph Sylvester (1814-1897), who determined the stress ellipse from the variation in the stresses (principle of virtual forces) (Sylvester, 1860); unfortunately, this original contribution to earth pressure theory went unnoticed (Feld, 1928, p. 28). The continuum mechanics model of earth pressure formed the basis of the “new theory of earth pressure” (Winkler, 1872, p. 3), which was promoted by Maurice Lévy (1869/1870), Armand Considère (1870), Otto Mohr (1871 & 1872) and Johann Jakob Weyrauch (1880) in particular. Incidentally, the latter also generalised Rebhann’s earth pressure theory by formulating conditional equations from which he determined analytically the angle of inclination of the earth pressure Ea with respect to the horizontal ((Weyrauch, 1878), see also (Ritter & Mörsch, 1906)). Carl Holtzmann

In a little-known work (Fig. 2.45), Carl Holtzmann (1811-1865), professor of physics and mathematics at the Polytechnic School in Stuttgart (now the University of Stuttgart), investigated Cauchy’s equations for the three-dimensional continuum for the static case. He derived the stress ellipsoid in a vector presentation and calculated its major axes, which he called “principal pressures” and are nothing other than the principal stresses. Holtzmann initially specialised his “pressure ellipsoid” for hydrostatics and the planar hydrostatic earth pressure model (Holtzmann, 1856, pp. 9-12), which he extended to the calculation of the “uplift force of the earth”, i.e. the passive earth pressure. Therefore, he not only derived the specific active earth pressure (eq. 2.85), but also specified the specific passive earth pressure: ( cos ρ ρ) +c⋅ (2.91) σxx,p (z) = ex,p (z) = γE ⋅ z ⋅ tan2 45∘ + ( ρ). 2 cos2 45∘ + 2

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Fig. 2.45 Title page of Holtzmann’s treatise on Cauchy’s equations and their applications.

Holtzmann was the first to apply the concept of the stress ellipse to earth pressure theory (see also Fig. 2.43). Like Cauchy, Holtzmann, too, obviously despised illustrations; his publication contains not one single explanatory drawing, which makes it heavy going for civil and mechanical engineers. It therefore remained a solitary, special attempt to exploit the ingenious consequences resulting from the general Cauchy equations in the form of the stress ellipsoid for earth pressure theory. Only Jacob Feld refers to Holtzmann’s contribution to earth pressure

2.4 The contribution of continuum mechanics

Fig. 2.46 Rankine’s earth pressure theory.

ds

β

90° – ρ Terrain line x

h

dg

τ τ σxx

σzz

t′ t

ϑ

ρ q

ϑ ϑ z

theory, but without reviewing his work in detail, because it is a special case of Rankine’s earth pressure theory (Feld, 1928, p. 83). Rankine’s stroke of genius (see section 3.4.2)

The Scottish civil engineer William John Macquorn Rankine submitted his earth pressure theory for cohesionless soils to the Royal Society in London on 10 June 1856. Nine days later it was presented at a meeting and thereafter published in the Transactions on 1 January 1857 (Rankine, 1857). In contrast to Holtzmann, Rankine directly assumed the partial differential equations (eq. 2.89) (Rankine, 1857, p. 14), and by applying the conditions 𝜕𝜎 xx (x,z)/ 𝜕x = 0 and 𝜕𝜏 xz (x,z)/ 𝜕x = 0 derived the angle of inclination 𝜗 of the slip planes for the terrain line inclined at an angle 𝛽 (Fig. 2.46): cos(2 ⋅ ϑ − β − ρ) =

sin β sin ρ

(2.92)

It is possible to calculate the directions of the main axes of the stress ellipse using eq. (2.92). Its semi-major axis – the principal stress 𝜎 1 – is inclined at an angle 𝜗+(45∘ -𝜌/2) to the horizontal; the principal stress 𝜎 3 is orthogonal to this at an angle 𝜗-(45∘ +𝜌/2). With a horizontal terrain line, eq. (2.92) is transformed into Prony’s theorem (eq. 2.69). Besides the law of friction for cohesionless soils with an angle of internal friction 𝜌 (eq. 2.90 for c = 0), the only assumption of Rankine’s earth pressure theory is an infinite extent in the horizontal direction. The slip planes resulting from this Rankine continuum as families of straight lines are not only hypothetical, but exact. The groups of slip planes are two parallel families that intersect at an angle 90∘ - 𝜌 (see Fig. 2.46). Rankine’s most important finding, however, was the fact that his earth pressure theory gives the direction of the earth pressure, which no longer has to be assumed (see section Compelling contradictions). He proved that the lines of equal stress t′ are parallel to the terrain line and the vertical stress t due to the weight dg = 𝛾 E ⋅h⋅ds⋅cos𝛽 (see Fig. 2.46) is t = γE ⋅ h ⋅ cos β

(2.93)

The angle of inclination of stress t and the associated conjugate stress t′ with reference to the respective section corresponds to the angle 𝛽 of the terrain line.

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This special limit equilibrium is characterised by the fact that it supplies the same stresses, i.e. t = const. and t′ = const., for h = const. Working independently of Rankine, it was the French civil engineers Lévy (1869/1870) and Considère (1870) who recognised the specifics of this state of equilibrium. The stress t′ conjugate to t can be determined in the 𝜎-𝜏 system of coordinates with the help of the Coulomb-Mohr yield criterion (eq. 2.90) and Mohr’s stress circle (see, for example, (Caquot & Kérisel, 1967, pp. 262-263)). By integrating t′ (z) between z = 0 and z = H of the vertical section under consideration (x = const.), we obtain the total magnitude of the earth pressure. For the simple case of a vertical wall line (𝛼 = 0∘ ) and horizontal terrain line (𝛽 = 0∘ ), Rankine’s earth pressure theory leads to the active earth pressure given by Prony according to eq. (2.72). Rankine appeared to have found a definitive and consistent solution to the earth pressure problem for cohesionless soils, as, on the one hand, it corresponds to the fundamental laws of mechanics, e.g. all equilibrium conditions, and, on the other hand, allows earth pressure to be determined unequivocally according to magnitude, direction and point of application (at the lower third point of the wall line). But its Achilles heel was the “introduction of a half-continuum divided by a wall and hence the assumption of planar slip planes” (Jáky, 1937/1938, p. 193). The Rankine continuum disturbed by the wall line triggered scientific debates again and again. Not until 15 years after its publication was Rankine’s earth pressure theory noticed and adopted in German engineering literature, first of all by Winkler (1872, pp. 102-103) and, shortly afterwards, by Mohr (1872, pp. 265-266).

Emil Winkler

Emil Winkler submitted his 32-page, handwritten dissertation Über den Druck im Innern von Erdmassen (on the pressure within bodies of soil) to the University of Leipzig in 1860, which was accepted in 1861 (Winkler, 1861). He divided his dissertation into 13 paragraphs: 1: Earth pressure (pp. 2-3) 2: The limits of earth pressure (pp. 3-5) 3: The equilibrium of an infinitely small parallelepiped (pp. 5-8) 4: Earth pressure on a surface element in any position (pp. 8-10) 5: The ellipsoid of earth pressure (pp. 10-12) 6: Determining the principal pressures (pp. 12-15) 7: The principal earth thrusts (pp. 15-16) 8: Relationships between the principal earth pressures (pp. 16-22) 9: Earth pressure on a surface element in any position (pp. 22-24) 10: Differential equations for earth pressure (pp. 24-27) 11: The body of earth is only bounded at the top by a horizontal plane (pp. 27-29) 12: The body of earth is only bounded at the top by a plane in any position (pp. 29-31) 13: The surface of the body of earth forms a natural slope (pp. 31-32)

2.4 The contribution of continuum mechanics

By determining the principal compressive stresses, Winkler succeeded in adding the third conditional equation (see eq. 3.12 in section 3.4.1) √ (σzz − σxx )2 + 4 ⋅ τ2xz − (σzz + σxx ) ⋅ sin ρ = 2 ⋅ c ⋅ cos ρ (2.94) to the system of differential equations (eq. 2.89) (Fig. 2.47) such that the partial differential equations (eq. 2.89) form a system of equations with the algebraic equation (eq. 2.94), the solution of which results in the stresses 𝜎 xx (x,z), 𝜎 zz (x,z) and 𝜏 xz (x,z) for the failure state of the body of soil. Unfortunately, this system of equations can only be solved for special cases, e.g. the (undisturbed) Rankine continuum (c = 0) according to Fig. 2.46 (see (Winkler, 1861, pp. 29-31)). Using this system of equations (eqs. 2.89 and 2.94), the 25-year-old PhD candidate was able to formulate a complete continuum mechanics model of earth pressure for cohesive soils. Therefore, from now on, eqs. (2.89) and (2.94) were known as Winkler’s system of equations for cohesive soils. Rankine (1857), Lévy (1869/1870) and Boussinesq (1876 & 1882), all working independently of Winkler, specified this system of equations for the special case of cohesionless soils. Winkler’s dissertation formed the theoretical backbone to his later publications on this subject (Winkler, 1871b to 1872), with which he wanted to assure his priority. He knew that the solution to his system of equations was difficult. Winkler therefore tried out several academic boundary conditions that led to families of Fig. 2.47 Winkler’s system of equations for determining the stresses in a body of earth (Winkler, 1861, p. 24) (Note: In the second partial differential equation, the zero on the right-hand side must be replaced by the unit weight 𝛾 E ).

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Fig. 2.48 Curved slip plane for 𝛿 = 𝜌 with the stress ellipse for a point on the wall (Winkler, 1872, p. 37).

straight slip planes (Winkler, 1872, pp. 31-36). Even in the case of a vertical wall line (𝛼 = 0∘ ) with a wall friction angle 𝛿 = 𝜌 (= angle of internal friction of the soil) and a horizontal terrain line (𝛽 = 0∘ ), he noted, even in his dissertation, that the solution to the system of equations must supply the necessary curved slip planes (Winkler, 1861, pp. 26-27); in doing so, the wall line forms a slip plane and the conjugate slip plane is inclined at an angle 90∘ -𝜌 to the wall line. Fig. 2.48 reproduces this case with the stress ellipse for the principal stresses at an arbitrary point on wall line C-E. The curved slip plane has the inclination 𝜌 at point E on the wall line and the inclination 45∘ +𝜌/2 at point D on the terrain line (Fig. 2.48). Winkler wrote the following regarding the application of his continuum mechanics model for determining earth pressure: “One assumes the theory of the unbounded body of earth to be also correct if a wall is present, i.e. disregards the condition that the direction of the pressure on the wall must fulfil” (Winkler, 1872, p. 37). According to Winkler, the wall friction angle 𝛿 must lie in the range 0 ≤ 𝛿 ≤ 𝜌 in this situation. As an alternative, Winkler considered the customary wedge theory of earth pressure, which presumes a straight slip plane. Compared with the continuum mechanics model of earth pressure, the wedge theory agrees better with his test results for the static case of horizontal terrain line and vertical wall line, and so Winkler pleaded for applying the customary earth pressure theory (Winkler, 1872, p. 38). So Winkler regarded the new theory of earth pressure he had helped to create as less suitable for the determination of earth pressure. Otto Mohr would object to this. Otto Mohr

Mohr introduced his criticism by way of a clear analysis of the contradiction in the customary earth pressure theory (Fig. 2.49). He concluded from the infringement of the moment equilibrium of the system of forces Ea , G and Q (see section Compelling contradictions) that the “conditions for the theory of earth pressure hitherto are incorrect” (Mohr, 1871, p. 346). In particular, his criticism was aimed at the earth pressure theory of Rebhann (1870/1871), which “perhaps possesses the advantage of greater clarity”, but is not so very different from the older theories and would lead to the same results. Mohr’s verdict was that Rebhann’s work did not achieve any significant progress (Mohr, 1871, p. 346). But he didn’t stop

2.4 The contribution of continuum mechanics

Fig. 2.49 Graphical determination of earth pressure for cohesionless soils using the stress circle (redrawn after (Mohr, 1872, p. 248)).

rain

Ter β

B H

α

ρ N

R S

V

line

C

e Wall lin

vertical

A

J

L

there – he also criticised the continuum mechanics models of earth pressure of Lévy (1869/1870), Considère (1870) and Winkler (1871a). Although these three authors achieve “exactly the same results ... they differ in their application of this theory for determining the earth pressure acting on retaining walls and solve this neither completely nor correctly ... for the most important task in practice” (Mohr, 1871, p. 346). Furthermore, Mohr criticised the “difficulty of the calculations” and regarded it as appropriate “to treat the same artefact by means of graphical statics” and, in conjunction with this, “to communicate (his] other views regarding the application of the theory” (Mohr, 1871, p. 346). Mohr translated the continuum mechanics model of earth pressure into the language of graphical statics. To do this he presented the planar stress state of an arbitrary point in a two-dimensional continuum by way of a circle, as Culmann had already done, in particular, within the scope of beam theory (Culmann, 1864/1866, pp. 226-231). The fact that the stress state at a point in a two-dimensional continuum is described completely by the stress ellipse was something that Holtzmann (1856) had already worked out clearly. However, the fact that, using the theory of conic sections, a circle can be assigned to each of these stress ellipses, this is the mathematical foundation of the concept of the stress circle. The originality of Mohr’s idea is that he applied the concept of the stress circle to the earth continuum with and without cohesion and linked it graphically to the Coulomb friction law, extended by the term c. In the graphical earth pressure theory, Mohr anticipated his visualisation of the stress and deformation states of a body element in the continuum (Mohr, 1882), which in the planar case leads to circles in the 𝜎-𝜏 system of coordinates and would later be called the Mohr stress circle.

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Fig. 2.49 shows Mohr’s simplified construction for determining the compressive stress at point A on wall line A-B, which he had already developed one year earlier (Mohr, 1871, p. 366). The wall line inclined at an angle 𝛽 is drawn through any point H, the perpendicular H-L set up on this and the vertical line H-J drawn. The distance H-L is then made equal to 100 mm (corresponding to principal stress 𝜎 1 ) and ( ρ) (2.95) HN = σ3 = σ1 ⋅ tan2 45∘ − 2 marked on the perpendicular H-L. This defines the stress circle with radius 0.5⋅(𝜎 1 -𝜎 3 ) and tangent H-R. The tangent is nothing other than the straight failure plane (eq. 2.90) for cohesionless soils (c = 0) inclined at the angle of internal friction 𝜌 with respect to H-L. The perpendicular on H-L passing through R then supplies point S. The line parallel to wall line A-B drawn through J intersects the stress circle at V , and extending V-S finally results in point C on the stress circle. Straight line J-C therefore defines the direction of the compressive stress on wall line A-B at point A; its magnitude can be found using the following equation (Mohr, 1871, p. 366, & 1872, p. 248): eA = 0.5 ⋅ HV ⋅ AB.

(2.96)

Winkler’s most important objection to Mohr’s earth pressure theory was that for the case of “friction on the wall equal to friction between the soil particles”, the new theory “is only applicable when the wall surface coincides with a slip plane or has an even shallower position” (Winkler, 1871b, p. 494). He therefore identified the Achilles heel of the continuum mechanics model of earth pressure, which became a theme in his book (Winkler, 1872, pp. 102-103). In the end, the earth pressure determined according to Mohr “cannot be proved by experiments” (Winkler, 1871b, p. 495). Mohr rejected this criticism in his statement and was of the opinion that earth pressure experiments only verify the preconceived views of those carrying out the experiments. Despite this difference of opinion, Mohr took the view that when designing retaining walls, the difference between the conventional and the new earth pressure theories was of “only secondary importance” (Mohr, 1872, p. 73). Concluding, Mohr disclosed his credo. He is looking for the scientific value of the new earth pressure theory in the following properties: “1. that the new theory avoids the arbitrary and unfounded presumptions of the older theory with respect to the direction of the earth pressure and the form of the slip plane, and “2. that the new theory allows a simpler, more elegant presentation consistent with the principles of statics.” (Mohr, 1872, p. 74) George Filmore Swain (1857-1931) presented Mohr’s graphical earth pressure theory to his American colleagues in the Journal of the Franklin Institute (Swain, 1882). Starting with his graphical analysis of the earth pressure problem, Mohr succeeded in generalising his concept of the stress circle for the strength of materials in the form of the Mohr strength hypothesis (Mohr, 1882 & 1900). In the process, the synthesis of the yield condition (eq. 2.90) with the theory of the stress circle

2.5 Earth pressure theory from 1875 to 1900

τ

σ1 α

σ3

c

τ

τ

c Slip plane an ρ + t σ σ1 – σ3 τ= = sinρ P σ1 + σ3 + 2 · c · cotρ σ

0 c · cot ρ

ρ σ3 σ

α = 45°+

2α = 90° + ρ

ρ 0 1 2

σ

σ1

Fig. 2.50 Coulomb-Mohr yield condition for cohesive soils (redrawn after (Kézdi, 1962, p. 42)).

(Mohr, 1906, pp. 220-240) was only a special application for earth pressure theory (Fig. 2.50). So Mohr’s presentation of the continuum mechanics model of earth pressure (Fig. 2.50) points well beyond the classical phase of theory of structures (1875-1900) (Kurrer, 2018, pp. 21-22). Even today, it is a reference point for soil mechanics (Kézdi, 1964, p. 295) and the mechanics of bulk goods (Schwedes & Schulze, 2003, pp. 1144-1145).

2.5 Earth pressure theory from 1875 to 1900 At the start of the classical phase of theory of structures (1875-1900), earth pressure theory finally broke free of the construction of fortifications and arrived at “a certain, albeit imperfect, conclusion” (Schäffer, 1878, p. 527). However, the momentum that the building of railways generated for new theories also started to run out of steam because the agglomeration and overlapping of urban infrastructure systems, e.g. water supplies, sewers, transport routes and energy supplies, called for the civil engineer to become more involved with the ground and the soil, which led to a multitude of diverse foundation measures, the like of which had never been seen before. So earth pressure theory with its commitment to graphical statics gradually adapted to the more complex boundary conditions. Typical examples of this are: • using graphical statics to investigate masonry arches with an earth backfill where the abutment walls were located entirely within the soil (Wittmann, 1878), • geometric earth pressure theory (Engesser, 1880b), and • determining earth pressure on broken and curved wall lines (Winkler, 1885). On the other hand, empirical methods were trying to ascertain the earth pressure realistically according to magnitude, point of application and direction. In particular, the question of the magnitude of the wall friction angle 𝛿 was increasingly the focus of attention for civil engineers. However, the continuum mechanics model of earth pressure was, so to speak, also put to the test, and could therefore be further developed; this is proved impressively by relevant

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publications by the scientific school around Saint-Venant (see (Corradi, 2002)). Those were the studies by Joseph Valentin Boussinesq (1842-1929), who set the course for new theories in France for many decades and whose work was the basis for the creative contributions of outstanding civil engineers such as Jean Résal (1854-1919) and Albert Caquot (1881-1976). In Germany it was the mathematician Fritz Kötter (1857-1912) who achieved a comprehensive historico-logical presentation of earth pressure theory (Kötter, 1893), reappraised the theory in terms of calculus of variations and postulated the differential equation for the slip plane (Kötter, 1888, 1903 & 1908). Nevertheless, both the fundamentals and the usability of the continuum mechanics model of earth pressure remained contentious. Practising engineers therefore continued to make use of the simplified determination of earth pressure in the tradition of Coulomb or rely completely on empirical methods, e.g. the influential British civil engineer Sir Benjamin Baker (Baker, 1881, p. 184). 2.5.1

Coulomb or Rankine?

Rankine’s earth pressure theory leads to results for descending terrain lines which contradict experience (Fig. 2.51). According to Rankine, the earth pressure Ea acts at the lower third point of the wall line for a retaining wall of height H and width b. The lever arm of the moment of Ea acting about m is ( ) H rm = − b ⋅ tan β ⋅ cos β (2.97) 3 for the ascending terrain line (Fig. 2.51a), and ( ) H Rm = + b ⋅ tan β ⋅ cos β. (2.98) 3 for the descending terrain line (Fig. 2.51b). Eqs. (2.97) and (2.98) state that the overturning moment for a descending terrain line is greater than that for an ascending terrain line and the ensuing minimum width is minb2.98 ≥ minb2.97 . “This now contradicts all experience and thus shows that pressure distribution determined [according to Rankine – the author] for the infinite body of soil agrees at best within certain limits that prevail in the soil behind the wall at the limit state of equilibrium [according to Coulomb – the Terrain Ea a)

line

rm

m Terrain

–β line

Ea b)

β

m

Rm

Fig. 2.51 Stability of retaining walls with a rectangular cross-section according to Rankine for a) ascending, and b) descending terrain lines.

2.5 Earth pressure theory from 1875 to 1900

author].” For this reason, Rankine’s earth pressure theory was “only to be used to determine masonry thicknesses under certain conditions” (Kötter, 1893, pp. 116-117). Therefore, Mohr only used Rankine’s earth pressure theory for those cases where the line of action of the greatest principal compressive stress passing through the base of the wall line remains within the body of soil and intersects the terrain line (see Fig. 2.49, for instance). He thus excluded undercut retaining walls, i.e. those with a wall line inclined towards the soil (Fig. 2.31), and, of course, retaining walls with a descending terrain line (Fig. 2.51b). Mohr turned to Coulomb earth pressure theory for such cases, and in so doing, he gladly assumed that the earth pressure is orthogonal to the wall line and hence a wall friction angle 𝛿 = 0. The discussion surrounding the magnitude of the wall friction angle and its influence on the determination of active and passive earth pressures did not subside in the first years of the consolidation period of theory of structures (1900-1950) (Kurrer, 2018, pp. 22-24). As was shown in section Emil Winkler, Winkler, too, restricted the applicability of the continuum mechanics model of earth pressure. A completely new path was taken with the amalgamation of the earth pressure determination by Mohr and Winkler, based on Rankine and Coulomb, which Engesser prompted with his geometrical earth pressure theory. 2.5.2

Earth pressure theory in the form of masonry arch theory

Taking the theory of the voussoir arch with joints affected by friction (see (Kurrer, 2018, pp. 211-215)), Friedrich Engesser modelled the homogeneous body of soil as a system of rigid wedges with friction (Engesser, 1880b). At the limit state of equilibrium, force Qi acts at friction angle 𝜌 on wedge i with weight Gi (Fig. 2.52). The envelope for the lines of action of Qi then results from the force diagram.

P2

P3

P4 p

a

G1

G2

G3

Q2

Q4 Q5

m

d

Ea

G1 G2

ρ

Q line

P2 G3

Q3 Ea

G5

ρ

Q1

δ

G4

ρ ρ

ρ

P3 G4 P4 G5

Fig. 2.52 Determining the active earth pressure E a after Engesser.

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2 The history of earth pressure theory

If the direction of the earth pressure Ea is prescribed, e.g. by the wall friction angle 𝛿, then its magnitude results from the force diagram. As, for reasons of equilibrium, the endpoint of the earth pressure vector must lie on a Q line, the magnitude of Ea follows as the distance of the point of intersection of the known earth pressure direction with the envelope and the starting point of G1 . Finally, the angle of the slip plane 𝜗 can be determined from the direction of the tangent to the point of intersection of the earth pressure vector with the envelope. Similarly, it is possible to find the arch thrust with joints affected by friction when Ea is used for the arch thrust at the impost joint and Gi for the weight of the ith voussoir; in this situation the wall line corresponds to the impost joint and the boundary of the wedges of soil are the joints. From this it follows that Engesser’s Q line corresponds to the line of action of the compression, i.e. Moseley’s “line of pressure” or the inverted catenary (see Fig. 4-38 in (Kurrer, 2018, p. 230)). The slip plane angle 𝜗 of the wedge of soil corresponds to the complementary angle of failure 90∘ -φ1 of case 1 of the limit equilibrium of masonry arches (Fig. 4-30 in (Kurrer, 2018, p. 223)). As Engesser specifies the direction and point of application of the earth pressure Ea , he transforms the earth pressure problem into a statically determinate problem. Here again, he adheres to that type of masonry arch theory where the position parameters of the arch thrust are deliberately fixed at the impost joint in order to be able to analyse the masonry arch solely with the equilibrium conditions (see (Kurrer, 2018, p. 230-232)). Point and line loads on the terrain line can also be taken into account with Engesser’s geometrical earth pressure theory (Fig. 2.52). Finally, he broadens his theory to include cohesive soils as well. He clearly realises that an infinite number of states of equilibrium are possible in the soil, so the earth pressure problem is statically indeterminate. The limit state of equilibrium shown in Fig. 2.52 was denoted by Engesser as unstable. According to him, a stable state of equilibrium is established in the reality because the forces Q must always lie inside the friction cone defined by the angle of friction 𝜌. Important for him was not just determining the limit state of equilibrium, but also ascertaining the earth pressure at rest. Engesser concluded that “for all practical cases where a compressible material is involved ... in the end the state of equilibrium established in the supported body of soil is the one corresponding to the smallest horizontal thrust” (Engesser, 1880b, p. 208). So the minimal principle of statics formulated by Moseley (see (Kurrer, 2018, p. 230-232)) was the initial hypothesis behind Engesser’s geometrical earth pressure theory for determining the true state of equilibrium of the body of soil as well. As in masonry arch theory, where in the search for the true line of thrust the Moseley minimal principle was replaced by the law of elasticity and transformed into elastic arch theory, in earth pressure theory as well, there was a major attempt to integrate it into the fundamental theory. So in earth pressure theory, too, this was the hour of French mathematicians and civil engineers who were fully familiar with elastic theory.

2.5 Earth pressure theory from 1875 to 1900

2.5.3

Earth pressure theory à la française

Boussinesq published a new earth pressure theory in 1876, “which provides a direct connection between the statics of sand-type substances and elastic theory and hydrostatics” (Kötter, 1893, p. 135) and places soil material at the phase transition between solids and liquids. He was therefore looking for the general determination of those states of equilibrium of the earth continuum which lie between the two limiting cases of equilibrium. Boussinesq assumed that the compressive stresses in the three-dimensional earth continuum cause the deformations u, v and w and the stresses can be expressed as a function of u, v and w. The equations he developed are valid irrespective of whether or not the limit states of equilibrium (active and passive earth pressure) are reached. These limit states represent a plastic theory problem. In contrast to elastic theory, in the case of plastic material behaviour, the deformations u, v and w are not generally included in the mechanics model, which means that it is possible to determine the stress state for the rupture surfaces present in the soil and therefore the problem is determinate. Boussinesq therefore formulated the continuum mechanics model of earth pressure for the standard case of the wall line with friction (𝛿 ≠ 0∘ ) inclined at an angle 𝛼 and the terrain line at an angle 𝛽 and specified a general solution on the basis of plastic theory (Boussinesq, 1876). Fig. 2.53 illustrates this simple case of the cohesionless soil (c = 0) with horizontal terrain line (𝛽 = 0∘ ) and vertical wall line with friction (𝛼 = 0∘ and 𝛿 ≠ 0∘ ). Boussinesq’s general solution is outlined below by way of this particular case. His principal assumption was that he regards point a (Fig. 2.53) as a centre of similitude for the stress ellipses that characterise the state of stress on rays radiating from a. This means that all stress components at every point on a ray are proportional to the distance r from point a and all resultant stresses have the same direction. Taking this principal assumption, Boussinesq derived a system of equations from the equilibrium conditions of a wedge-shaped element with included angle dφ inclined at an angle φ with respect to the z axis (see (Kézdi, 1962, pp. 216-218)). He formulated his treatment of equilibrium in the system of r-φ polar coordinates in such a way that a system of partial differential equations Fig. 2.53 Earth pressure determination according to Boussinesq (1882 & 1885).

a

n x

x– z

ϑ = 45° +

H

0 ε= . tan

δ

u

Ea z m

d

ε = 45° –

ρ 2

ρ 2

71

72

2 The history of earth pressure theory

must ensue. The fact that Boussinesq can specify the system of simultaneous differential equations dσ(φ) = 3 ⋅ τ(φ) − sin φ dφ dτ(φ) = m ⋅ σ(φ) − cos φ dφ

(2.99)

can be attributed to the proportionality of the stresses with respect to the polar coordinate r on the ray; he thus reduced the earth pressure problem to the variable φ. The system of differential equations (eq. 2.99) is non-homogeneous and m is dependent on 𝜏(φ)/𝜎(φ). Closed-form integration is not generally possible for such systems with non-constant coefficients m, i.e. only approximate solutions are possible. Boussinesq determined such an approximate solution for positive and small angles of the terrain line 𝛽 and wall line 𝛼 and tabulated them. Later, Résal used the finite different method to calculate the earth pressure coefficients (see eq. 2.72) for negative 𝛽 and 𝛼 values as well – but only for specific wall friction angles 𝛿 – and summarised them in earth pressure tables (Résal, 1903 & 1910). Neither Boussinesq nor Résal investigated passive earth pressure Ep , which is crucial when designing foundations, for example. This was left to Albert Caquot (1934), who developed the Boussinesq continuum mechanics model of earth pressure further and, together with his student Jean Kérisel, prepared this for foundation engineers, albeit in French only (Caquot & Kérisel, 1948 & 1956). But let us return to Boussinesq. The year 1882 saw him solve Winkler’s system of equations (eqs. 2.89 and 2.94) for the simple case of cohesionless soil (c = 0) with a horizontal terrain line (𝛽 = 0∘ ) and vertical wall line with friction (𝛼 = 0∘ , 𝛿 ≠ 0∘ ) (Fig. 2.53). Taking into account the boundary conditions τ(z = 0) = 0 und σzz (z = 0) = 0 plus tan δ =

τ(x = 0) σxx (x = 0)

(2.100)

Boussinesq derived the stresses in the Rankine continuum a-u-n (x ≥ z⋅tan(45∘ -𝜌/2)): ( ρ) σxx (x, z) = γE ⋅ z ⋅ tan2 45∘ − 2 σzz (x, z) = γE ⋅ z

(2.101)

τ(x, z) = 0 The stress equations (eq. 2.101) satisfy Winkler’s system of equations for c = 0 and the first two boundary conditions of eq. (2.100). On the other hand, the stress

2.5 Earth pressure theory from 1875 to 1900

equations set up by Boussinesq for the area a-u-d, i.e. x ≤ z⋅tan(45∘ -𝜌/2) ( 𝜌) ⋅ (z + x ⋅ tan 𝛿) tan2 45∘ − 2 𝜎xx (x, z) = 𝛾E ⋅ ( 𝜌) ⋅ tan 𝛿 1 + tan 45∘ − 2 z + x ⋅ tan 𝛿 𝜎zz (x, z) = 𝛾E ⋅ ( 𝜌) ⋅ tan 𝛿 1 + tan 45∘ − 2 ) [( ( ] ( 𝜌 )) 𝜌 ⋅ tan 𝛿 ⋅ z ⋅ tan 45∘ − −x tan 45∘ − 2 2 𝜏(x, z) = −𝛾E ⋅ ( 𝜌) ∘ ⋅ tan 𝛿 1 + tan 45 − 2

(2.102)

only satisfy the Cauchy differential equations (eq. 2.89) for the homogeneous earth continuum (𝛾 E = const.) and the secondary boundary condition (eq. 2.100). The stress field is bisected by the discontinuity line a-u in the areas x ≥ z⋅tan(45∘ -𝜌/2) and x ≤ z⋅tan(45∘ -𝜌/2); this straight line x-z⋅tan(45∘ -𝜌/2) = 0 divides the homogeneous earth continuum into the Rankine and Boussinesq continua. How did Boussinesq take the yield criterion (eq. 2.94) into account? He did this by substituting the stress equations into eq. 2.94 and hence defining a fictitious angle of internal friction 𝜌fict for the Boussinesq continuum (see (Heyman, 1972, p. 153)): sin2 𝜌fict (x, z) = sin2 𝜌 + (1 − sin 𝜌)2 ⋅ tan2 𝛿 [ ( ]2 𝜌) z ⋅ tan 45∘ − −x 2 ⋅[ ) ( ]2 ( 𝜌) 𝜌 + x ⋅ tan 45∘ − ⋅ tan 𝛿 z ⋅ tan 45∘ − 2 2

(2.103)

For small wall friction angles 𝛿, 𝜌fict is only slightly greater than the angle of internal friction 𝜌. Boussinesq discussed his solution of the specific case according to Fig. 2.53 by way of extensive numerical calculations. Alec Westley Skempton (1914-2001) employed comparative calculations to check the quality of Boussinesq’s earth pressure theory (Skempton, 1984). In doing so, he assumed the horizontal projection of the earth pressure Ea (see Fig. 2.53) Ea,x = Ea ⋅ cos δ =

1 1 ⋅ γ ⋅ H2 ⋅ λa ⋅ cos δ = ⋅ γE ⋅ H2 ⋅ λa,x 2 E 2

(2.104)

with the earth pressure coefficient 𝜆a (see eq. 2.72). Skempton used the projection of Ea , or 𝜆a,x , so that he could include the Rankine earth pressure theory, because in that theory the direction of Ea is always parallel to the terrain line and is therefore horizontal in the case of a horizontal terrain line Ea , hence, the wall friction angle 𝛿 must vanish. Taking the case shown in Fig. 2.53, the result according to

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Coulomb is Ea,Coulomb ⋅ cos δ =

=

cos2 ρ 1 ⋅ γE ⋅ H2 ⋅ [ (√ 2 1 + sin ρ ⋅ 1+

tan δ tan ρ

)]2

1 ⋅ γ ⋅ H2 ⋅ λa,x,Coulomb , 2 E

and according to Rankine (see eq. 2.72) ( ρ) 1 1 = ⋅ γE ⋅ H2 ⋅ λa,x,Rankine Ea,Rankine = ⋅ γE ⋅ H2 ⋅ tan2 45∘ − 2 2 2

(2.105)

(2.106)

and according to Boussinesq (1882) ( ρ) tan2 45∘ − 1 2 Ea,Boussinesq,1882 ⋅ cos δ = ⋅ γE ⋅ H2 ⋅ [ ) ] ( ρ 2 ⋅ tan δ 1 + tan 45∘ − 2 1 2 = ⋅ γE ⋅ H ⋅ λa,x,Boussinesq,1882 . 2

(2.107)

The earth pressure according to eq. (2.107) follows directly from the integration of the first equation of eq. (2.102) between z = 0 and z = H. In a second approximation, Boussinesq (1885) managed to improve his earth pressure formula (eq. 2.107) yet further on the basis of the system of differential equations (eq. 2.99): 1 ⋅ γ ⋅ H2 ⋅ λa,x,Boussinesq,1882 ⋅ C 2 E 1 = ⋅ γE ⋅ H2 ⋅ λa,x,Boussinesq,1885 . 2

Ea,Boussinesq,1882 ⋅ cos δ =

(2.108)

with 2 ) ( ⎧ tan 45∘ − ρ ⋅ tan δ ⎫ ( ) ⎪ ⎪ 1 4 2 C=1+ ⋅ ln ⋅ ⎨[ ) ] ( ⎬ ρ sin δ ⎪ 1 + tan 45∘ − e ⋅ tan δ ⎪ ⎭ ⎩ 2

(2.109)

The mathematical constant e is included in eq. (2.109). Boussinesq assumed that his second approximation would represent the upper limit of the active earth pressure. Skempton’s comparative calculation with eqs. (2.105) to (2.109) for the case illustrated in Fig. 2.53 reveals, however, that the horizontal projection of the earth pressure coefficient 𝜆a,x approaches the values according to Caquot and Kérisel (1948) from below and deviates from these by max. 0.5% (Fig. 2.54). The modification of the earth pressure theory of Boussinesq and Résal by Caquot and Kérisel brought the continuum mechanics model of earth pressure à la française to a provisional conclusion at the transition from the consolidation to the integration period of theory of structures around the middle of the 20th century (Kurrer, 2018, pp. 22-25).

2.5 Earth pressure theory from 1875 to 1900

Fig. 2.54 Comparison of earth pressure coefficients 𝜆a,x = 𝜆a ⋅cos𝛿 for retaining walls with vertical wall line, horizontal terrain line and cohesionless soil: (Rankine, 1857), (Coulomb, 1773/1776), (Boussinesq, 1882 & 1885) and (Caquot & Kérisel, 1948) (redrawn after (Skempton, 1984, p. 270)).

2.5.4

Kötter’s mathematical earth pressure theory

Fritz Kötter incorporated Coulomb’s earth pressure theory – based mathematically on the extreme value calculation of differential calculus – by setting up the earth pressure problem within the more general framework of calculus of variations (Kötter, 1893, pp. 125-134). He recognised that, with certain auxiliary conditions, the integral of the unknown earth pressure distribution over wall line a-d (see Fig. 2.53) could be formulated as a minimum or a maximum. Kötter’s calculus of variations is founded on the principle of virtual displacements and, as such, is an equilibrium statement; in this situation the virtual displacement condition must be geometrically possible and small in the sense of differential geometry. The simplest virtual displacement condition for determining the active earth pressure results from the anticlockwise rotation of the rigid retaining wall about point d on the wall line. On the other hand, the passive earth pressure follows from the clockwise rotation about d (see Fig. 2.53). Almost all practical earth pressure theories are based on these kinematics (see Fig. 2-40 in (Kurrer, 2018, p. 52)). Of course, it is possible to find further geometrically possible virtual displacement conditions for rigid retaining walls (Kézdi, 1962, p. 91). Nevertheless, Kötter formulated his earth pressure theory not only for systems with one degree of kinematic indeterminacy, but also very generally for systems with n degrees of kinematic indeterminacy. Thus, the planar model of the rigid retaining wall includes a virtual displacement state with three degrees of kinematic indeterminacy. He showed that the earth pressure distribution over wall line a-d (see Fig. 2.53) corresponding to the maximum or minimum virtual work is the differential quotients of that work done by the displacements. Nevertheless, the fundamental question of earth pressure theory according to Kötter’s calculus of variations for systems with one degree of kinematic indeterminacy, or rather just one force component acting on the wall line, leads to a determinate task. Coulomb reduced the variation problem to the extreme value calculation of differential calculus in his earth pressure theory “by not selecting the most favourable case of all the cases statically possible, but rather the cases with a simple multiplicity of favourable assumptions” … and so Rankine suspected “the infringement of the equilibrium conditions in the infinitesimal” (see Fig. 2.44),

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but also overcame the variation problem “through the arbitrary assumption of a certain limit state in the volume element” (Reissner, 1910, p. 388), as can be seen in Fig. 2.46. Kötter (1888 and 1903) improved the Coulomb and Rankine earth pressure theories by adding the mathematical determination of curved slip planes for cohesionless soil, something that Müller-Breslau would pick up on later (Müller-Breslau, 1906, pp. 107-121). Working independently of Kötter, Müller-Breslau specified an approximation method for determining the active earth pressure for curved slip planes (Müller-Breslau, 1906, pp. 91-106). By considering the equilibrium of the infinitely narrow wedge O-C-C′ lying on the planar slip plane, both proved that the slip plane pressure q is a linear function and the slip plane compressive force Qz acts at the lower third point of slip plane O-C (Fig. 2.55). From this it follows that for all the earth pressure theories in the tradition of Coulomb, it is not only the earth pressure that is specified in terms of direction and point of application, but also the position parameter of the slip plane compressive force Q. This in turn leads to the fact that the forces Ea , G and Q do not intersect at one point (see Fig. 2.39). The contradiction following on from this – the infringement of the equilibrium conditions in the infinitesimal (see Fig. 2.44) – led Kötter and Müller-Breslau to conclude that the slip plane cannot generally be planar (Müller-Breslau, 1906, p. 25). Kötter elevated this contradiction of classical earth pressure theory by describing the curved slip plane mathematically. Taking the basic equations of the continuum mechanics model of earth pressure (eq. 2.89) and the friction law for the slip plane pressure q, Kötter derived the

Fig. 2.55 Distribution of the slip plane pressure q over a planar slip plane O-C (Müller-Breslau, 1906, p. 23).

2.5 Earth pressure theory from 1875 to 1900

Fig. 2.56 Curved slip plane after Kötter (redrawn after (Reissner, 1910, p. 412)).

e s

Terrain lin

ϑ

e

lin Wall

G

δ

Ea

q Q

ρ

differential equation for determining q, or 𝜗, for a curved slip plane (Fig. 2.56): dq d𝜗 (2.110) = 2 ⋅ q ⋅ tan ρ ⋅ + γE ⋅ sin(ϑ − ρ) ds ds Kötter’s differential equation (eq. 2.110) (Kötter, 1888 & 1903) expresses the relationship between the slip plane pressure q(s) and the angle of inclination 𝜗(s) of the slip plane. Its integral for the active earth pressure is s

q(s) = γE ⋅ (e2⋅ϑ⋅tan ρ ) ⋅

∫s=0

[(e−2⋅ϑ⋅tan ρ ) ⋅ sin(ϑ − ρ) ⋅ ds] + qa .

(2.111)

In 1936 Jáky expanded Kötter’s differential equation to cover cohesive soils (c ≠ 0) (Kézdi, 1962, p. 78). As Kötter’s differential equation contains two unknowns in the shape of dq/ds and 𝜗(s), one of these must be assumed. If we assume, for example, the form of the slip plane 𝜗(s), then the equilibrium conditions are generally also infringed. Furthermore, one boundary condition at the end of the slip plane must be known. As the theory was developed further, so the circular arc was used as well as the straight line (e.g. (Fellenius, 1927), (Krey, 1926ab, 1932 & 1936)). Schwedler had first introduced the logarithmic spiral for a slip plane forming in cohesionless soil beneath the cross-section of a longitudinal railway sleeper with a central point load F (Fig. 2.57) as long ago as 1882 – a model that corresponds to the ground failure of a centrally loaded strip footing with cross-sectional width B. Generally, however, assuming the shape of the slip planes leads to approximate solutions only. Therefore, Kötter’s differential equation belongs to the “solution of

Fig. 2.57 Ground failure load F for a longitudinal railway sleeper with cross-section width B after Schwedler (redrawn after (Schwedler, 1891, p. 94)).

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the question without being able to supply a complete answer” (Jáky, 1937/1938, p. 195). Until well into the consolidation period of theory of structures (1900-1950) (Kurrer, 2018, pp. 22-24), Kötter’s differential equation was adopted and further developed by civil engineers for exceptional cases only (Ritter, 1910 & 1936), even though Müller-Breslau, the number one international authority on theory of structures, gave it due praise in his influential book on earth pressure (Müller-Breslau, 1906, pp. 107-121). Neither Kötter’s essay in an engineering journal (Kötter, 1908) nor Max Ritter’s elegant derivation of the Kötter differential equation in the Schweizerische Bauzeitung (Ritter, 1910) helped to popularise Kötter’s mathematical earth pressure theory, which was “accorded primarily theoretical significance” (Kézdi, 1962, p. 78). Only as the area of study of earth pressure theory was systematically extended beyond the determination of the active earth pressure to include more complex earth pressure problems, e.g. the interaction of active and passive earth pressures in earthworks and foundations, did a practical need for earth pressure models arise in which curved slip planes called the tune.

2.6 Experimental earth pressure research A new relationship between empiricism and theory took shape at the transition from the classical phase (1875-1900) to the accumulation phase (1900-1925) of theory of structures (Kurrer, 2018, pp. 21-23). This was described in (Kurrer, 2018, pp. 157-161) as a cooperation between the causal relationship realised in the technical model and the constructional or technological modelling of causal relationships in technical artefacts and techniques (see Figs. 3-6e and 3-6f in (Kurrer, 2018, p. 157)). Therefore, within the scope of forming the system of classical engineering sciences, earthworks laboratories appeared in addition to the materials-testing institutes and the machine and hydraulics laboratories at the technical universities. As soil mechanics became established between 1925 and 1950, so that experimental earth pressure research found a home in this new engineering science discipline. 2.6.1

The precursors of experimental earth pressure research

Even at the start of the classical phase of theory of structures (1875-1900) (Kurrer, 2018, pp. 21-22), earth pressure research committed to empiricism was taking shape, which gradually emancipated itself from the legitimising nature of earth pressure testing, against which Mohr had polemicised so splendidly in 1872 (see section Otto Mohr). This is why Schäffer pleaded for the fact “that in some respects it is better to make use of purely empirical methods” in a paper on earth pressure on retaining walls (Schäffer, 1878, p. 529). Although it was still a case of checking the customary earth pressure formulas by way of earth pressure tests, the next stage, i.e. finding the limits to the validity or the practical applicability of existing earth pressure theories, was definitely on the horizon.

2.6 Experimental earth pressure research

Cramer

One year later, in 1879, the engineer E. Cramer from Breslau (now Wrocław, Poland) reported on his experiments (carried out back in 1863) in which he had tried to answer the question of the form of the slip plane for earth pressure prisms and the magnitude of the active earth pressure Ea acting on wall and terrain lines at any angle in cohesionless soil (Cramer, 1879). He discovered that the angle between the direction of Ea and the wall line was 90∘ -𝜌, hence, the wall friction angle was 𝛿 = 𝜌. Considering equilibrium, Cramer deduced a specific slip curve that changes to a straight line at u and intersects the terrain line at n (see Fig. 2.53). Cramer’s slip curve d-u results from the condition that the angle between the direction of the ray with origin at a and the tangent to the slip curve has the magnitude 90∘ -𝜌 not only at d and u, but that this angle is constant for all points (see Fig. 2.53). From this Cramer found the mathematical form of the slip curve d-u to be a logarithmic spiral (Cramer, 1879, p. 524). He deduced a formula for Ea from the earth pressure prism a-d-u-n (see Fig. 2.53) and calculated for the specific case shown in Fig. 2.53 that Ea lies only 2.7% below the value that would ensue if the slip plane d-n were a straight line. However, Weyrauch had shortly before criticised the usual definition of the wall friction angle 𝛿, which is also found Cramer’s work: “First of all, for about 60 years we had 𝛿 = 0∘ with Coulomb, but since Poncelet (1840) we have jumped to the other extreme and now use 𝛿 = 𝜌 more or less universally” (Weyrauch, 1878, p. 206). He therefore recommended using the value 𝛿 = 𝜌/2 for undercut retaining walls (wall line angle 𝛼 > 0∘ ) and specified an equation to determine 𝛿 for 𝛼 < 0∘ (Weyrauch, 1878, p. 205). Baker

One advocate of radical empiricism in earth pressure research was Benjamin Baker (for a biography see (Hamilton, 1958)), who was able to call on the support of Peter Barlow, the most influential advocate of empirical strength of materials in Great Britain. The latter justified his earth pressure theory – just four pages long – as follows: “The above can only be considered as a very imperfect sketch of the theory of revetments, at least as relates to its practical application, for want of the proper experimental data ... To render the theory complete, with respect to its practical application, it is necessary to institute a course of experiments upon a large scale” (Barlow, 1867, p. 114). Baker took this last comment to heart and surveyed, analysed and evaluated 65 retaining walls, the majority of which served as large structures for infrastructure systems such as trams, and mainly ports, and some of which collapsed. The retaining walls investigated included a number of test walls (Fig. 2.58). Baker’s summary was devastating: The laws of earth pressure were currently not satisfactorily formulated (Baker, 1881, p. 184). Theorists such as Flamant, Boussinesq and Gaudard took part in the discussion following the publication of Baker’s report. Boussinesq used the occasion to publish a 12-page outline of his earth pressure theory (Baker, 1881, discussion, pp. 212-223), which also includes the original of Fig. 2.53. British civil engineers also contributed to the discussion, but they wanted to talk about practical problems concerning retaining walls and did not mention earth pressure theory at all. However, even the theorists ignored

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Fig. 2.58 Large-scale test on retaining walls with loose soil backfill and hydrostatic pressure by General Burgoyne (after (Baker, 1881, p. 150)).

the complex practical engineering tasks involved in the design and construction of retaining walls; theorists and practising engineers were talking at cross purposes. Baker was the only one who tried to see both sides. He had high regard for the theoretical works of Flamant, Boussinesq and Gaudard and rejected the accusation that he despised theory, writing about himself as follows: “His habit of thought and mode of working were entirely opposed to such a feeling, and, indeed, in his opinion, an engineer who did not attempt to classify his practical data with the ultimate aim of elucidating a satisfactory theory was wilfully playing the part of a blind man” (Baker, 1881, discussion, p. 235). Two sentences further on, Baker remarked that – apart from a few exceptions, which included Boussinesq’s contribution to the discussion – writings on earth pressure were misleading and disappointing (Baker, 1881, discussion, p. 236). Donath and Engels

It was not until the earth pressure tests of Adolf Donath were carried out at Berlin Technical University on his testing apparatus completed in 1889 did experimental earth pressure research get underway in Germany, too. Donath’s idea was to use experimentation to answer the questions of the magnitude and direction of the earth pressure and the applicability of the continuum mechanics model of earth pressure. In his opinion the previous earth pressure experiments were “thoroughly useless” (Donath, 1891, p. 492). Answering the question of the applicability of the continuum mechanics model of earth pressure involved “finding a method that renders it possible to measure the pressure that a body of earth exerts on a retaining wall in a state of rest, i.e. which allows one to determine what it is that we can call the pressure of the earth at rest” (Donath, 1891, p. 493). So he was not interested in determining the limiting equilibria of the earth pressure, instead determining the earth pressure at rest, a term that Donath introduced into the specialist literature. Like the statics of masonry arches in the establishment phase of theory of structures (1850-1875) (Kurrer, 2018, pp. 20-21) was increasingly concerned with the search for the true state of equilibrium and gradually became aware of the static indeterminacy, so earth pressure theory was now becoming more interested in the true state of equilibrium – what is and not what could

2.6 Experimental earth pressure research

be. After describing his test apparatus, Donath presented his test results for the simplest case of a vertical wall line, horizontal terrain line and cohesionless soil material. He thus confirmed experimentally the direction of the earth pressure according to Rankine, but not its magnitude. On the other hand, the test results verify the acting moment of the earth pressure (see (Kurrer, 2018, pp. 55-58)) according to Coulomb: the discrepancy was 6–9% (Donath, 1891, p. 518). Unfortunately, Donath could not carry out the planned tests for the inclined wall and terrain lines so important for practice, as he changed his job. Nevertheless, his report provoked a debate about earth pressure research experiments in which Engesser (1893), T. Hoech (1896), H. Zimmermann (1896), L. Brennecke (1896) and E. Beyerhaus (1900) took part. The comprehensive tests of the Dresden-based hydraulics professor Hubert Engels (1854-1945), performed at the Gustav Zeuner Hydraulics Laboratory (Engels, 1896), also prompted criticism. For example, Cramer disputed Engels’ conclusion that the direction of the earth pressure against the vertical wall line is generally horizontal (Cramer 1896, 1898). In his reply, Engels said that “it is worrying to deduce a downward diversion of the earth pressure due to the movement of the masonry itself for all retaining walls without distinction ... [Instead it is] ... more correct and more reliable to refrain from such a contribution of the masonry in the most general sense, i.e. to use the theoretically most unfavourable and not the theoretically most favourable conditions for the stability investigations” (Engels, 1897, p. 145). One year after Engels had written these lines, he was able to open the world’s first permanent river engineering laboratory at Dresden Technical University. For him as a hydraulics engineer, it seemed obvious to allow the earth pressure to act on the wall line horizontally just like hydrostatic pressure. Such a proposal was presented by Otto Franzius (1877-1936), professor of hydraulics in Hannover (Franzius, 1918), – civil engineers would then always be on the safe side when designing retaining walls. On the other hand, the demands of economy require that the wall friction angle 𝛿 be taken into account in order to avoid oversized retaining walls. Experimental work on earth pressure would foster this conflict of aims after 1900.

2.6.2 Earth pressure tests at the testing institute for the statics of structures at Berlin Technical University Just one year after founding the Testing Institute for the Statics of Structures at Berlin Technical University, Müller-Breslau presented his report Über den Druck sandförmiger Massen auf standfeste Mauern (on the pressure of sand-type bodies on stable walls) (anon., 1902, p. 1007) at a session of the physics/mathematics class at the Royal Prussian Academy of Sciences on 30 October 1902. In the report he is critical of the state of the statics of sand-type bodies at that time and shows that restricting the limit states of equilibrium “must be abandoned for a series of important cases” [anon., 1902, p. 1007]. Müller-Breslau stressed the need for earth pressure experiments, presented the testing apparatus for determining earth pressure he had developed together with the Rudolf Fuess company

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from Steglitz, revealed the results of the first tests and submitted an agenda for further testing. The testing apparatus was the outcome of a competition to develop an apparatus for measuring wind pressures on tall structures (Müller-Breslau, 1904, p. 366). From the start it was conceived by Müller-Breslau for measuring other unknown forces such as the flow pressure acting on solid bodies and the earth pressure on retaining walls. Fig. 2.59 shows how Müller-Breslau developed the basic concept of his measuring method. For a planar wall surface loaded non-uniformly by earth pressure, Müller-Breslau fixed the test specimen with four horizontal bars 1, 2, 3 and 6 plus two vertical bars 4 and 5 (Fig. 2.59). This three-dimensional system is statically determinate. Müller-Breslau replaced the earth pressure acting obliquely on the wall surface by the forces E1 , E2 and E3 , the magnitude of which he determined from the three translation equilibrium conditions. On the other hand, the point of application of E1 , i.e. 𝜉 1 , and 𝜂 1 , was derived by Müller-Breslau from two moment equilibrium conditions. He obtained the position 𝜉 2 of the resultant R of E2 and E3 from the moment equilibrium condition about point A4 . All six equilibrium conditions were set up by Müller-Breslau as a system of linear equations with the unknown bar forces S1 to S6 : “This therefore determines the crossing of the two skew forces R and E1 , and which is equivalent to the earth pressure on the wall surface” (Müller-Breslau, 1906, p. 125). Müller-Breslau modelled the earth pressure experimentally via the measurement of the elastic elongation of the bars, which are converted to the readings S1 to S6 by means of the law of elasticity. To do this, measuring rods 385 mm long were inserted into bar axes 1 to 6 (Fig. 2.60). The measuring rods were designed to accommodate compressive forces of up to 200 kg (≈ 2000 N) and resulted in a shortening of the bar of 0.064 mm for 100 kg (≈ 1000 N). Their

Fig. 2.59 Structural system of the testing apparatus for determining earth pressure (Müller-Breslau, 1906, p. 124).

2.6 Experimental earth pressure research

Fig. 2.60 Measuring rod supplied by the Rudolf Fuess company (Müller-Breslau, 1906, p. 126).

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elastic properties were determined and calibrated by the Royal Materials-Testing Institute at Groß-Lichterfelde (Müller-Breslau, 1906, pp. 128-129). Sand was slowly backfilled behind the test wall bearing on six elastic supports (equipped with measuring bars). The earth pressure could be calculated at any time via the measured bar forces S1 to S6 with the help of the system of equations specified by Müller-Breslau. The way that loads on the terrain line influence the earth pressure could also be determined experimentally. In contrast to the earth pressure tests conducted previously, in Müller-Breslau’s tests the wall line was not moved arbitrarily. Instead, the displacements of the test retaining wall were caused by the earth pressure itself. “A similar process,” writes Helmut Neumeuer, “can be seen in nature when a retaining wall founded on subsiding subsoil is subjected to earth pressure” (Neumeuer, 1960, p. 11). Simulating the earth pressure on real retaining walls by a model retaining wall supported on six elastic bars reproduces its true behaviour in a much better way than direct measurement of the earth pressure, which had been carried out with the testing apparatus employed in the past (see Fig. 2.32). Müller-Breslau’s tests on retaining walls with vertical wall line, inclined terrain line and cohesionless soil revealed that the measured values of the active earth pressure Ea were, without exception, slightly larger than those calculated using the modified Coulomb theory, which assumes planar slip planes anyway. According to Müller-Breslau, the main cause of this difference was the curved slip plane, which would lead to higher Ea values (Müller-Breslau, 1906, pp. 151-152). When it came to the direction of Ea , he was unable to confirm the equivalence with the inclination of the terrain line given by the Rankine earth pressure theory. Instead, Müller-Breslau recommended setting the wall friction angle 𝛿 to a maximum of (2/3)⋅𝜌 to (3/4)⋅𝜌 for rough wall lines. In the case of high point loads on the terrain line, he suggested reducing 𝛿 to (1/2)⋅𝜌 for safety reasons. Müller-Breslau therefore called the highly controversial wall friction angle 𝛿 a “value based on experience … about which it is not possible to make any complete statement” (Müller-Breslau, 1906, p. III) 1 . When it came to the point of application of Ea , the tests revealed that it lies between 0.33⋅H and 0.36⋅H (H = height of wall line) for a terrain line without loads and a little below 0.33⋅H for a steeply descending terrain line (𝛽 ≪ 0∘ ). Loads on the terrain line shift the point of application of Ea upwards somewhat. It was therefore proved experimentally that the active earth pressure Ea may be assumed to be applied at the lower third point of the wall line for practical structural analyses of retaining walls (see Fig. 2-42c in (Kurrer, 2018, p. 56)). As Müller-Breslau’s tests say nothing about the distribution of the specific earth pressure e(z), instead merely measured its integral over the wall line, e(z) continued to remain an object of research into earth pressure theory, especially for 1 Max Möller recommended 𝛿 = (1/3)⋅𝜌 for smooth wall lines, 𝛿 = (2/3)⋅𝜌 for moderately rough wall lines and 𝛿 = 𝜌 for very rough wall lines (Möller, 1902, p. 47). The wall friction angle 𝛿 has only a minor influence on the magnitude of the earth pressure. On the other hand, it determines its direction and hence the factor of safety against overturning of the retaining wall. Therefore, it is advisable to select a value of 𝛿 that is not too large and to limit it to (1/2)⋅𝜌 ≤ 𝛿 ≤ (2/3)⋅𝜌. In the case of a smooth wall line and saturated soil (lubricating surfaces), 𝛿 = 0∘ should be used to be on the safe side (Schreyer et al., 1967, pp. 107).

2.6 Experimental earth pressure research

non-rigid retaining structures such as the steel sheet pile wall of Tryggve Larssen (1870-1928), the Norwegian engineer in charge of the building authority in Bremen, who was granted a patent for his wall on 8 January 1904 (Roth, 1992, p. 179). Some 34 years later, Johann Ohde (1905-1953) would publish an earth pressure theory backed up by experiments taking particular account of the earth pressure distribution in a ground-breaking series of essays in the journal Die Bautechnik (Ohde, 1938a). For practical analysis it is sufficient to express e(z) as a linear function of the level z of the wall line (see Fig. 2-42c in (Kurrer, 2018, p. 56)), although Mohr quite rightly criticised that, for rigid bodies, this statement is not covered by physics (Mohr, 1907). On the final pages of his monograph on earth pressure, Müller-Breslau presents a detailed and comprehensive plan for continuing the tests (Müller-Breslau, 1906, pp. 153-157). Among other suggestions, he proposes broadening the tests to cover • • • •

inclined wall lines for various angles of the terrain line, broken wall lines, various types of soil taking cohesion into account, and determining the passive earth pressure Ep .

Müller-Breslau understood his far-reaching testing programme to be a motivation “for founding similar institutes at other technical universities as well in order to create a secure foundation for one of the most important and still least researched areas of the engineering sciences” (Müller-Breslau, 1906, p. 157). After 1910 Hans-Detlef Krey, formerly a scientific assistant to Müller-Breslau, would implement a large part of the programme of tests at the Royal Testing Institute for Hydraulics and Shipbuilding, the results of which were regularly included in the several editions of his monograph Erddruck, Erdwiderstand (earth pressure, earth resistance) (Krey, 1912, 1918, 1926b, 1932 & 1936). As late as 1960, Helmut Neumeuer assessed the earth pressure tests of Müller-Breslau “as an example of carefully prepared, carefully conducted and carefully observed experiments” (Neumeuer, 1960, p. 9). So experimental earth pressure research at last became established through the collaboration of the innovative precision instrument manufacturer Rudolf Fuess (1838-1917) with the person who rounded off classical theory of structures – Müller-Breslau. 2.6.3

The merry-go-round of discussions of errors

Müller-Breslau still continued to regard the modified Coulomb earth pressure theory as an “as yet unsurpassed tool for the scientific determination of earth pressure” (Müller-Breslau, 1906, p. III), although he also gave credit to the results of the Rankine earth pressure theory for their great practical value. However, for more complex tasks in building practice, e.g. non-constant loads on the terrain line or broken wall lines, Müller-Breslau still regarded the modified Coulomb earth pressure theory to be superior. Nevertheless, he regarded its limitation to the underlying assumption of planar slip planes in scientific research into earth pressure as “inadequate, and extending the research to curved slip planes essential” (Müller-Breslau, 1906, p. IV). Müller-Breslau’s pragmatic stance in

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turn challenged Mohr’s criticism; Mohr, too, had developed theoretical ideas regarding the earth pressure determination for curved slip planes (Mohr, 1907). In his reply, Müller-Breslau begins by referring to the relevance of empiricism for the solution of earth pressure problems, examines the weaknesses of Rankine earth pressure theory (see Fig. 2.51) and provides a numerical solution to Kötter’s differential equation (eq. 2.110) (Müller-Breslau, 1908). The discussions of errors surrounding earth pressure theory never subsided completely but reached their climax around 1920. For example, E. Jacoby (1918) tried to disprove Mohr’s criticism – repeated again and again – of the modified Coulomb earth pressure theory regarding the fact that the forces Ea , G and Q acting on the sliding wedge never intersect at one point (see Fig. 2.39). Two years later, A. Freund (1920) investigated Müller-Breslau’s earth pressure tests and claimed that his assumption of curved slip planes does not eliminate all contradictions. He saw another error in the fact that “the elastic properties of the loose earth have not been considered so far” (Freund, 1920, p. 625). Nevertheless, he recommended applying Coulomb to determine the magnitude of the earth pressure with 𝛿 = 0∘ ; but in the design the calculated earth pressure should then be applied at the wall friction angle 𝛿 selected to suit each case. Thereupon, Krey (1921) and Senft (1921) formulated their objections, which Freund (1921a,bc) answered. That led Schwarz (1922) and Krey (1923b) to voice their concerns regarding Freund’s variation on earth pressure theory and occasioned Freund to publish another reply (Freund, 1922 & 1923). Freund brought this discussion to a certain conclusion with his essay entitled Untersuchungen der Erddrucktheorie von Coulomb (investigations of the Coulomb earth pressure theory) (Freund, 1924). So in the second half of the accumulation phase of theory of structures (1900-1925) (Kurrer, 2018, pp. 22-23) the discussions of errors went around in circles and only in isolated circumstances did they add to the knowledge of earth pressure theory. One exception was the contribution by Emil Mörsch. He used Rankine’s earth pressure theory to determine the earth pressure on cantilever retaining walls and developed a sliding wedge theory for these which he derived with the help of projective geometry and, alternatively, on the basis of the continuum mechanics model of earth pressure (Mörsch, 1925ab). What is remarkable here is that his earth pressure theory adopted for cantilever retaining walls was verified by photographic model experiments carried out at the Materials-Testing Institute of Stuttgart Technical University (Fig. 2.61), which were performed on behalf of Wayss & Freytag AG. Fig. 2.61 shows the formation of a sliding wedge for a cantilever retaining wall displaced by 20 mm to the left with a terrain line at an angle 𝛽 = 17∘ at the top of the body of sand. The stress circle added above the photograph was used by Mörsch to determine the direction of the slip planes (indicated by arrows), which agree well with those of the experiment. The angle of the sliding wedge at the tip of the base slab is 90∘ -𝜌. Mörsch now applies the active earth pressure Ea resulting from the equilibrium of the sliding wedge to the lower third point of the left-hand slip plane and together with the soil load on the base slab forms the resultant; it is assumed here that Ea acts parallel to the terrain line. Mörsch’s sliding wedge theory is subjected to the same limitations as Rankine earth pressure theory.

2.6 Experimental earth pressure research

Fig. 2.61 Model experiment on a cantilever retaining wall with internal leg lengths H x L = 40 x 28 cm (Mörsch, 1925a, p. 98).

2.6.4

The emergence of soil mechanics

As a result of his comprehensive literature studies, the Austrian civil engineer Karl von Terzaghi (1883-1963) realised in 1917 that identifying universally applicable relationships for the interaction between structure and subsoil was a hopeless task. After being discharged from the teaching staff at the Ottoman Technical University in Constantinople due to the war, he was able to continue his research in the USA at Robert College, where he set up an earthworks laboratory. He published the knowledge he had gained in Constantinople in a book (Fig. 2.62), which promptly earned him a professorship at MIT. Terzaghi’s book is a synthesising achievement of the very highest quality and, as such, an epic work in civil engineering science in general and geotechnical engineering in particular. Like Navier combined statics with strength of materials to form theory of structures in his Mechanik der Baukunst (mechanics of architecture) in 1826, Terzaghi united earth pressure and soil physics to produce soil mechanics in 1925, which he called the “mechanics of soil in construction” (Terzaghi, 1925, p. 2).

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Fig. 2.62 Title page of what had obviously been Krey’s own copy of Terzaghi’s Erdbaumechanik (mechanics of soil in construction).

According to Terzaghi, “the tasks of the mechanics of soil in construction are to predict the effect of given force systems on the soil and estimate the pressure exerted by the soil on retaining structures; knowledge of the soil types that earthworks engineers will have to deal with is provided by applied geology. The mechanics of soil in construction and applied geology must be regarded as auxiliary sciences to earthworks knowledge” (Terzaghi, 1925, p. 1). Both auxiliary sciences do the foundations engineer “insufficient service and fail ... precisely ... where one needs them most” (Terzaghi, 1925, p. 1). The shortcomings of the auxiliary sciences of earthworks “exist primarily in the absence of a link between geology and the mechanics of soil in construction and in the dogmatic character of the premises on which classical mechanics of earthworks is based” (Terzaghi, 1925, p. 1). Three lines of development

Terzaghi created this link with his Erdbaumechanik, in which three lines of development are subsumed dialectically on a higher level. Firstly: Criticism of the premises of the physics of the soil in classical earth pressure theory during the preparatory period of geotechnical engineering (1700-1925) – particularly during its application phase (1775-1875) –, i.e. that • the disrupted equilibrium in the soil is identical with the shearing of the soil according to a slip plane inclined at an angle 𝜗,

2.6 Experimental earth pressure research

• the resistance of the soil to shearing for any type of soil is uniquely determined by the angle of internal friction 𝜌 and the pressure prevailing on the slip plane Q or q, and • the angle of internal friction 𝜌 is identical with the natural slope angle. “With these premises as a basis, one treated earth pressure problems as statically determinate exercises and investigated neither the deformations of the soil nor those of parts of the structure subjected to the pressure of the soil” (Terzaghi, 1925, p. 2). The aforementioned premises of classical earth pressure theory therefore had to be abandoned and greater consideration given to the physical properties of the soil. This led to “the need to create a mechanics of soil in construction based on the physics of soils” (Terzaghi, 1925, p. 2). Secondly: The critical adoption of investigations carried out by the committee for “codifying the normal permissible stresses in the subsoil and for studying the physical properties of soils which are important for engineering” (Terzaghi, 1925, p. 3) set up by the American Society of Civil Engineers (ASCE) in 1913 and the tests of the US Bureau of Standards in Washington. Thirdly: Terzaghi’s own research since 1917 • on the earth pressure on yielding retaining walls (Terzaghi, 1920), • on the distribution of stress in locally loaded plastic and granular bodies, • on the static effect of flowing water on the sand strata through which it flows, which led to a distinction between erosion and earth pressure ground failures, and, finally, • on the transfer of the mechanical behaviour of sand to clay – especially the analysis of true and apparent cohesion. Unfortunately, Terzaghi only learned of the final report of Sweden’s “Geotekniska Kommission” (Statens Järnvägar, 1922) “at the last minute” through Fellenius (Terzaghi, 1925, p. 390), so he could only mention the ground-breaking work of this commission briefly in the appendix. Nevertheless, he acknowledged the report “as an important stage in the brief history of earthworks engineering research”, the significance of which went well beyond the framework of his programme and represented a “pioneering achievement” (Terzaghi, 1925, p. 391). The final report of the “Geotekniska Kommission” could have formed the fourth line of development of Terzaghi’s Erdbaumechanik. The disciplinary configuration of soil mechanics

Based on the aforementioned three lines of development, Terzaghi created “the physical principles for a scientific treatment of earthworks engineering problems which corresponded better with the purposes of practice” (Terzaghi, 1925, p. 4). In doing this it was clear to him that the strength properties of soils deviate so greatly from Hooke’s law that a strict mathematical treatment of the problems is inconceivable, and, of necessity, soil mechanics must take on the character of a descriptive science. Therefore, the first task of soil mechanics research was “to develop the laboratory methods for investigating soil samples and find the most rational formula for the numerical description of the physical properties”. Terzaghi saw its second task as “the systematic processing of the experiences gained on

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building sites through observations in the field and soil studies in the laboratory” (Terzaghi, 1925, p. 5). Consequently, Terzaghi divided his monograph into six chapters: I: The properties of the soil (pp. 7-29) II: The friction forces in the soil (pp. 30-66) III: The strength properties of soils (pp. 67-110) VI: Hydrodynamic stress phenomena (pp. 111-183) V: The statics of the soil (pp. 184-333) VI: The soil as foundation (pp. 334-386) Including the appendix and the alphabetical index, the book ran to 399 printed pages. Of that, chapter V on earth pressure, with 150 printed pages, accounted for 37.5% of the total. The contours of phenomenological earth pressure theory

Terzaghi understood the determination of earth pressure as the solution of a statically indeterminate problem. His main interest was not the limit equilibrium of a retaining wall, but the state of equilibrium established in use. He therefore asked the question regarding “the magnitude of the earth pressure acting against the wall prior to the onset of sliding”. Consequently, it is not about what could be (kinematic school of statics), but what it is (geometric school of statics). This question, Terzaghi continues, “can only be answered if one knows the degree of yielding. The nature of this question is identical with the question of the lateral pressure that a solid, elastic, loaded body exerts on the yielding obstacle where lateral expansion is partly prevented” (Terzaghi, 1925, p. 308). Therefore, earth pressure theory in 1925 was on the same level as masonry arch theory prior to elastic theory asserting itself in the middle of the establishment phase of theory of structures (1850-1875) (Kurrer, 2018, pp. 20-21). However, whereas the mutual dependence of the force and deformation states – a key parameter of statically indeterminate systems in masonry arches – was later covered by Hooke’s law, more complex material laws prevail in soils, the formulation of which first appeared in the middle of the consolidation period of geotechnical engineering (1950-1975) (Kurrer, 2018, p. 346). Whereas theory of structures had a fundamental theory in the form of elastic theory upon which other theories could build, geotechnical engineering in its preparatory period (1700-1925) was a theoretical non-event in this respect, which – e.g. in earth pressure theory – always had to fall back on simple equilibrium relationships. Terzaghi questioned this tradition radically and, as an alternative, literally conjured up a plan for an engineering science discipline of soil mechanics primarily committed to experimentation, thus dismissing the preparatory period of geotechnical engineering and heralding its discipline-formation period (1925-1950). This dismissal is particularly evident in prototypical form in Terzaghi’s phenomenological earth pressure theory; his pioneering essay had the programmatic title Old Earth-Pressure Theories and New Test Results (Terzaghi, 1920). Fig. 2.63 shows the simple – but not too simple – concept behind Terzaghi’s phenomenological earth pressure theory.

2.6 Experimental earth pressure research

Fig. 2.63 Determining the coefficient of earth pressure at rest 𝜉 0 after Terzaghi (redrawn and modified after (Neumeuer, 1960, p. 5)).

Narrow thin strips are embedded horizontally (layer A) and vertically (layer B) in the body of soil made up of very fine gravel. Terzaghi chose the material of the strips in such a way that the friction to be overcome when pulling out the strips is always proportional to the normal pressure. The relationship between the forces necessary to pull out the strips from positions B and A supplies the coefficient of earth pressure at rest 𝜉 0 , so the horizontal stress is σxx (z) = ξ0 ⋅ σzz (z)

(2.112)

In the experiments, Terzaghi applied uniformly distributed loads p = 8–40 N/cm2 to the terrain line. He discovered that 𝜉 0 always has a value of 0.42 and σxx (z) = ex,0 (z) = ξ0 ⋅ σzz (z) = ξ0 ⋅ γE ⋅ z = 0.42 ⋅ γE ⋅ z

(2.113)

irrespective of the absolute magnitude of p. Like Scheffler, Terzaghi derived the specific earth pressure at rest from the consideration of the infinitesimal element (see Fig. 2.42); eq. 2.113 with 𝛾 E as the unit weight of the soil material can then be expressed as follows: ( ρ ) (2.114) ex,0 (z) = ξ0 ⋅ γE ⋅ z = 0.42 ⋅ γE ⋅ z = γE ⋅ z ⋅ tan2 45∘ − 0 2 With a triangular distribution of the specific earth pressure at rest over z, the earth pressure at rest for the simplest case of a retaining wall of height H with horizontal terrain line and vertical wall line is ( ρ ) 1 1 1 E0,Terzaghi = ⋅ H2 ⋅ γE ⋅ ξ0 = ⋅ H2 ⋅ γE ⋅ tan2 45∘ − 0 = ⋅ H2 ⋅ γE ⋅ 0.42. 2 2 2 2 (2.115)

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In eqs. (2.114) and (2.115), 𝜌0 is the angle of internal friction at rest introduced by Terzaghi, which is a variable derived from the measured value 𝜉 0 – totally in contrast to the angle of internal friction 𝜌, which is used as a material parameter for the soil in conventional earth pressure theory. In 1944 Joseph Jáky, professor of foundations and soil mechanics at Budapest Technical University, published the formula for earth pressure at rest for the standard case which is still accepted today (Türke, 1990, p. 96): E0,Jáky = E0 =

1 ⋅ H2 ⋅ γE ⋅ (1 − sin ρ) 2

(2.116)

If the angle of internal friction for very fine gravel 𝜌 = 35∘ (see Fig. 2.63) is entered in eq. (2.116), then 1 1 (2.117) ⋅ H2 ⋅ γE ⋅ (1 − sin 35∘ ) = ⋅ H2 ⋅ γE ⋅ 0.43 2 2 confirms Terzaghi’s formula (eq. 2.115) for earth pressure at rest. In 1936 Terzaghi published further test results for the coefficient of earth pressure at rest 𝜉 0 (see (Neumeuer, 1960, p. 5)): E0,Jáky = E0 =

• densely bedded sand: 𝜉 0 = 0.40 to 0.45 • loosely bedded sand: 𝜉 0 = 0.45 to 0.50 From measurements of the earth pressure coefficient 𝜉 depending on the movement of the retaining wall, Terzaghi was able to produce diagrams that put the earth pressure from the undisplaced wall (earth pressure at rest with coefficient 𝜉 0 ) up to failure in their historical picture, so to speak. If the retaining wall yields, then 𝜉 first increases considerably and thereafter always less so (Fig. 2.64). “These two phases in the behaviour of the retaining wall-backfill system are designated the first and second phases of active earth pressure” (Terzaghi, 1925, p. 309). In the initial stadium of the second phase, the slip plane starts to form at 𝜉 I ≈ 0.26 and manifests itself through an abrupt forward thrust of the retaining wall which had even been observed by G. H. Darwin (1883); in Fig. 2.64 this thrust appears as an almost vertical branch. According to Terzaghi, the cross-hatched area 0.24 ≤ 𝜉 ≤ 0.26 in Fig. 2.64 represents the area already researched. When 𝜌 = 35∘ , using Prony’s formula (eq. 2.72) the coefficient of active earth pressure is calculated as 𝜉 = 𝜆a = 0.27, which lies roughly in the transition from the first to the second phase. According to Fig. 2.64, initial sliding begins at 𝜉 1 ≈ 0.21, a figure that corresponds to 50% of the earth pressure coefficient at rest (𝜉 0 = 0.42). According to Terzaghi, the modified Coulomb earth pressure theory is suitable for calculating the effective earth pressure in the initial stadium of the second phase. He refers here to the earth pressure experiments of Heinrich Müller-Breslau and Jacob Feld, who proved that the modified Coulomb earth pressure theory was superior to other theories. In the end, Terzaghi poses the question of “whether Coulomb’s principle can also be used for calculating the lower bound of the earth pressure (end of second phase, complete breakdown of equilibrium in backfill)” (Terzaghi, 1925, p. 320). Terzaghi also answers this question with yes and points to the experiment. The prerequisite is, however,

2.7 Earth pressure theory in the discipline-formation period of geotechnical engineering

Fig. 2.64 How the earth pressure coefficient 𝜉 (abscisse) depends on the wall movement (ordinate) (Terzaghi, 1925, p. 309).

that the slope angle is no longer equated with the angle of internal friction 𝜌, and the angle of internal friction has to be measured indirectly for • the boundary between the first and second phases (𝜌I ), • the first slip (𝜌1 ), and • the complete failure of the backfill (𝜌II ). For Terzaghi, the model experiments form “an indispensable tool for researching the physics of earth pressure which has not been adequately acknowledged so far. Such research cannot be avoided, because it is the only way of uncovering the nature of earth pressure phenomena and supplying the principles required to provide a scientific footing for applied research” (Terzaghi, 1925, pp. 326-327). Therefore, Terzaghi’s plan for a phenomenological earth pressure theory progressed to become the model of the style of theory in soil mechanics which several researchers would take as their starting point during the 1930s.

2.7 Earth pressure theory in the discipline-formation period of geotechnical engineering Whereas experimental earth pressure research in the initial phase of geotechnical engineering (1900-1925) (Kurrer, 2018, p. 348) was in the first place committed

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to the principle of checking the theory of earth pressure and demonstrating the limits of its validity, during the discipline-formation period (1925-1950) (Kurrer, 2018, p. 348) the task was to present new findings about the physical behaviour of the soil so that new analytical models could be derived from those findings. “Determining in advance the magnitude of the forces acting on a planned structure” (Neumeuer, 1960, p. 4) became one of the particular tasks of foundation construction. The two latter goals of experimental earth pressure research were pursued not only by Terzaghi and his school in the USA and Austria, but also by civil engineers from the Scandinavian countries, France, Germany, The Netherlands, Japan and the USSR. So earthworks laboratories were set up at the large technical universities with chairs of foundations and soil mechanics: Hannover, Freiberg (Förster & Walde, 2015), Vienna, Stockholm, Zurich, Delft, Cairo, Cambridge (MIT and Harvard) and Berkeley (see Kurrer 2018, p. 349). But governmental building authorities and private industry, too, turned to experimental earth pressure research; examples are the earthworks laboratory founded by Krey in 1927 at the Prussian Testing Institute for Hydraulics and Shipbuilding and also that of the US Bureau of Public Roads. And the earthworks laboratories of the Veritas and Securitas insurance companies in Paris and building contractors such as Société des Sondages (Paris) and Rodio & Co. (Milan) should not go unmentioned. At the end of 1928, representatives from the German Transport Ministry, the building industry and the technical universities in Berlin founded the Deutsche Gesellschaft für Bodenmechanik (Degebo, German Society for Soil Mechanics). So not only did the triadic organisation of engineering science collaboration so typical of Germany (see Fig. 10-9 in (Kurrer, 2018, p. 680)) now apply to soil mechanics, but the technical term Bodenmechanik (= soil mechanics) was now part of the German language. The number of publications on soil mechanics and related areas also started to increase rapidly. A relevant bibliography lists about 2,500 journal papers and books that appeared worldwide between 1925 and mid-1936 (Petermann & Bödeker, 1937). Fig. 2.65 therefore shows only a small extract from the many publications in German during the 1930s. The leading journals such as Die Bautechnik and Der Bauingenieur contained numerous papers on soil mechanics in general and the determination of earth pressure in particular. For example, Terzaghi published a review of subsoil research from 1920 to 1935 in the latter journal, providing brief insights into 12 fields of soil mechanics and acknowledging the work of individual researchers (Terzaghi, 1935): 1. Cohesion and internal friction – especially for clay soils: Arthur Casagrande and Leo Jürgenson 2. Elastic properties and compressibility of soils: Walter Bernatzik, Arthur Casagrande, Joachim Ehrenberg, Leo Jürgenson and Leo Rendulic 3. Effect of vibrations on the soil: August Hertwig/Degebo 4. Soil surveys and soil classification: Arthur Casagrande, Franz Kögler et al. 5. Theory of earth pressure: Leo Rendulic and Karl von Terzaghi 6. Stability of embankments and earth dams: Knut E. Pettersson

2.7 Earth pressure theory in the discipline-formation period of geotechnical engineering

Fig. 2.65 A 1940 advertisement by publishers Ernst & Sohn for books on soil mechanics and related fields (Kurrer, private collection).

7. Theory of stress distribution in locally loaded subsoil: Joseph Valentin Boussinesq, Otto Karl Fröhlich, Emil Gerber, Hans Hugi, Ernst Melan, Franz Kögler, Alfred Scheidig, Ferdinand Schleicher and O. Strohschneider 8. Theory of the settlement of structures on primarily sandy subsoil 9. Settlement of structures on soil strata with thick clay inclusions: Otto Karl Fröhlich and Karl von Terzaghi 10. Settlement of pile foundations 11. Foundations for dam structures on alluvial land: William George Bligh and Karl von Terzaghi 12. Roadbuilding: Karl von Terzaghi

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Terzaghi derived three tasks from his presentation of the state of development in subsoil research. He saw the first task as “clarifying the factors upon which the behaviour of soils are dependent in the event of interventions by man … and the numerical treatment of the influence of these factors on the basis of reasonable, simplified assumptions”. The second task according to Terzaghi was “working out methods to determine soil constants that occur in the formulas”, and the third task “checking the theoretical findings by way of observations of building sites and finished structures” (Terzaghi, 1935, p. 30). It was this last task that provoked Terzaghi’s criticism. He accused the building authorities of believing that they could “procure by way of pure theory all the principles for overcoming the obvious shortcomings of civil engineering regulations” (Terzaghi, 1935, p. 30). He was thinking of earth pressure theory in particular here, where he was calling for adaptations to suit the conditions found in practice, which he saw as the relationships for deformations of the body of soil and the position of the point of application of the earth pressure and its magnitude. In his review, Terzaghi does not mention Krey’s book at all, the third edition of which appeared in 1926 (Fig. 2.66) and the fourth edition posthumously in 1932 (Krey, 1932). In this work, Krey deals not only with earth pressure theory (also by way of experiments), but also numerous other earth pressure issues. The fact that Terzaghi omitted to mention this book can be attributed to the fact that Krey always assumed a linear distribution of the specific earth pressure over the wall line in his investigations and earth pressure tables, which was due to the implied assumption of a wall movement with the centre of rotation at the bottom of the wall line (Fig. 2.67a). Nevertheless, Krey’s book can be regarded as the most influential monograph on earth pressure theory (and determining it in a manner suitable for practice) that had been published by the middle of the discipline-formation period of geotechnical engineering (1925-1950). It was at that time that the first International Conference on Soil Mechanics and Foundation Engineering took place in Cambridge, Massachusetts, from 22 to 26 June 1936, which led to the founding of the International Society for Soil Mechanics and Foundation Engineering with Terzaghi as its president (1936-1957). Terzaghi presented his earth pressure theory and other matters at the conference (Terzaghi, 1936), about which A. Casagrande said that this theory “is the most important contribution to soil mechanics ... which provides an explanation for many hitherto contradictory observations of the magnitude and, in particular, the distribution of the earth pressure on retaining walls and timber shoring in shafts and tunnelling” (cited after (Hertwig, 1939, p. 1)).

2.7.1

Terzaghi

In order to determine the earth pressure distribution according to Fig. 2.67c, Terzaghi divides the sliding prism of soil into rigid slices of thickness dz and finds the pressure dE exerted by these slices on the wall line (Fig. 2.68). Ignoring the shear stresses, Terzaghi obtains dE ⋅ cos δ = dQ ⋅ sin(90∘ − ρ)

(2.118)

2.7 Earth pressure theory in the discipline-formation period of geotechnical engineering

Fig. 2.66 Title page of the standard work by Krey (1926b).

for the equilibrium of the slice element (inclination 𝜀 = 0∘ ) in the horizontal direction and tan(90∘ − ϑ) ⋅ (γE ⋅ z ⋅ dz + z ⋅ dq + q ⋅ dz) − dQ ⋅ cos(90∘ − ρ) − dE ⋅ sin δ = 0. (2.119) in the vertical direction. Instead of the moment equilibrium condition, Terzaghi introduces the following assumption: ) ( z dE ⋅ cos δ = k0 ⋅ 1 + ci ⋅ ⋅ q ⋅ dz (2.120) H

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Fig. 2.67 Wall movements and the associated distribution of the specific earth pressure according to Terzaghi (Ohde, 1938a, p. 177). Fig. 2.68 Earth pressure theory according to Terzaghi (redrawn after (Terzaghi, 1936)).

In eq. (2.120) k0 =

tan(90∘ − ρ) tan δ + cot(90∘ − ρ)

(2.121)

u and ci is a constant that Terzaghi will determine from tests. Taking eqs. (2.118) to (2.120), Terzaghi derives the differential equation ) c dq ( + γE − i ⋅ q = 0 (2.122) dz H the solution to which, taking into account the boundary condition q(z = H) = 0, is [ ( )] H ⋅ γE −c ⋅ 1− Hz ⋅ 1−e i (2.123) q= ci Substituting eq. (2.123) into eq. (2.120) results in the following earth pressure distribution: ( )] ) H⋅γ [ k0 ( dE z −ci ⋅ 1− Hz E ⋅ . (2.124) ⋅ 1−e = ⋅ 1 + ci ⋅ dz cos δ H ci Terzaghi essentially adheres to Coulomb earth pressure theory but deviates from this in one important respect. He “claims, incorrectly, that the Coulomb

2.7 Earth pressure theory in the discipline-formation period of geotechnical engineering

theory requires a further assumption in addition to assuming a planar slip plane in order to determine the distribution of the earth pressure over the wall and the slip plane” (Hertwig, 1939, p. 2). This is, however, unnecessary, because the linear distribution of the earth pressure over the slip plane follows from the assumption of the planar slip plane (Reissner, 1910, p. 409). The Achilles heel of Terzaghi’s “slice element theory” (Walz & Prager, 1979, p. 375) is the arbitrary assumption regarding the relationship between the earth pressure E and the pressure q, i.e. eq. (2.120). Whereas in the Coulomb earth pressure theory the contradiction due to the moment equilibrium not being satisfied is quite clear (see Fig. 2.39), this is “not so clear” (Hertwig, 1939, p. 2) in Terzaghi’s approach. In his critical examination of Terzaghi’s earth pressure theory, Hertwig investigates the case that the slice elements are inclined... • firstly, parallel to the slip plane (𝜀 = 𝜗), • secondly, parallel to the terrain line (𝜀 = 𝛽), and • thirdly, 𝜀 at any angle (Hertwig, 1939). Whereas in the first case Coulomb earth pressure theory is confirmed by a triangular earth pressure distribution, Hertwig considers the two other cases as additional when investigating non-homogeneous soils: “If, for example, a wedge-shaped backfill is installed in layers behind a wall and the sloping in situ soil, then an earth pressure distribution similar to group 2 [second case – the author] and group 3 [third case – the author] will probably ensue” (Hertwig, 1939, p. 8). Hertwig concludes his theoretical deliberations on Terzaghi’s earth pressure theory with the remark that the task of further experiments will be “to check the viability of the additional calculations” (Hertwig, 1939, p. 9). However, Terzaghi’s earth pressure theory would be little used, and only four decades later would it be expanded to solve three-dimensional earth pressure problems (Walz & Prager, 1979). 2.7.2

Rendulic

Leo Rendulic, one of Terzaghi’s many students, undertook an attempt to provide a unified presentation of the earth pressure problem in 1938 (Rendulic, 1938). In doing so, he was able “to incorporate the influence of the nature and magnitude of the retaining wall movement on the earth pressure in the theory of earth pressure and hence show that the old, so-called classical earth pressure theory” does not contradict the new findings, as was believed by many (Rendulic, 1938, p. 7). Rendulic assumed Terzaghi’s earth pressure theory (1936), which quantifies the influence of the retaining wall displacement on the earth pressure distribution over the wall line (Fig. 2.67). To this end, Rendulic investigated three kinematically possible displacement configurations of the wall line: centre of rotation at the base (Fig. 2.67a), centre of rotation at the top (Fig. 2.67c) and parallel displacement (Fig. 2.67b). He established that there was a need to “expand the classical earth pressure theory … for non-yielding walls and a high centre of rotation of the retaining wall movement, but in no way does it [have to be] replaced by a completely new theory” (Rendulic, 1938, p. 7). His modification of the geometric

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earth pressure theory of Engesser (see section 2.5.2) resulted in a consistent, clear and structured solution for earth pressure tasks which would provide engineers working on road and bridge projects in particular with a guideline for their daily workloads. 2.7.3

Ohde

Like Rendulic, Johann Ohde also took Terzaghi’s earth pressure theory as his starting point. In his seven-part series of essays in Die Bautechnik, Ohde provided an exhaustive presentation of the theory of earth pressure, with particular emphasis on the earth pressure distribution over the wall line (Ohde, 1938a). For the funding and publication of this profound scientific work, he was grateful to Joachim Ehrenberg, head of the earthworks laboratory at the Prussian Testing Institute for Hydraulics and Shipbuilding, who also provided funding for the last two editions of Krey’s monograph (Krey, 1932 & 1936). In his series of essays, Ohde – who since 1927 had been an employee of the earthworks laboratory founded by Krey – carried out a systematic reappraisal of the gaps in research that Krey’s book left regarding how kinematically possible displacement configurations of the wall line influence the earth pressure distribution and the slip plane pressure (Fig. 2.69). Even Kötter had pointed out that the distribution of the earth pressure is a function of the wall movement and, accordingly, there is also a relationship between the shape of the slip plane and the earth pressure distribution (Kötter, 1893, p. 128). Ohde assumed Kötter’s mathematical earth pressure theory (see section 2.5.4) in the experimental and theoretical development of the wall movements shown in Fig. 2.69. As it is difficult to obtain an exact solution to Kötter’s differential equation (eq. 2.110), Ohde replaced the exact shape of the slip plane by an approximation function and used Kötter’s law for the slip plane pressure q(s) (eq. 2.111). In doing so, Ohde added the passive earth pressure to Kötter’s mathematical earth pressure, “as here, more than with the [active] earth

Fig. 2.69 Schematic results for three different wall movements (Ohde, 1938a, p. 178).

2.7 Earth pressure theory in the discipline-formation period of geotechnical engineering

pressure, one is obliged to work with curved slip planes” (Ohde, 1938a, p. 242). In the case of active earth pressure, on the other hand, Ohde established that the Coulomb slip plane deviates only marginally from the exact slip plane position (Fig. 2.70) and the exact earth pressure value lies only 3.5% above that supplied by the Coulomb theory (Ohde, 1938a, p. 335). Ohde declined his earth pressure theory for wall movement cases B and C in Fig. 2.69 and considered yet other horizontal wall movement possibilities. Finally, he investigated earth pressure distribution on excavation shoring (soldier pile walls) stiffened with capping beams and also sheet pile walls with and without anchors. Owing to the elaborate formulas, Ohde’s series of essays was not easy reading for the practising engineer. His second seven-part series of essays Zur Erddruck-Lehre (on earth pressure theory), which also appeared in Die Bautechnik (Ohde, 1948-1952), therefore examined more themes, but at the same time focused on the key physical findings of earth pressure theory, which were presented in a more straightforward mathematical form. In terms of content and form, it can be seen as a masterpiece of earth pressure theory at the end of the discipline-formation period of geotechnical engineering (1925-1950) which is still illuminating today. 2.7.4

Errors and confusion

The Berlin-based consulting engineer Alfons Schroeter tried to expand the Coulomb earth pressure theory to cover loads on the terrain line, especially short line loads, in a comprehensive article (Schroeter, 1940). In a letter, Hans J. Stahl criticised Schroeter’s determination of the active earth pressure (Stahl,

Fig. 2.70 Exact position of the slip plane and the pressure distribution for wall movement according to Fig. 2.69a (Ohde, 1938a, p. 335).

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1941). Following a second round of discussions immediately afterwards, further debate was closed by the editor of Die Bautechnik. However, Schroeter held lectures, e.g. in Danzig (now Gda´nsk, Poland) on 29 October 1942, and kept to his principles. Thereupon, Friedrich-Wilhelm Waltking, professor at Danzig Technical University, published an article on the expansion options of the Coulomb earth pressure theory and in which he criticised Schroeter’s basic equation for earth pressure theory (Waltking, 1943, p. 53), which will be explained below by means of a later publication by Schroeter (Fig. 2.71). Schroeter derived the equilibrium for the infinitesimal wedge A-C′ -C′′ by breaking down the weight dGx according to fixed directions of dEx and dQx dEx = dGx ⋅ tan(ϑx − ρ)

(2.125)

and by using 1 (2.126) ⋅ γ ⋅ h ⋅ dx 2 E and integrating over all the infinitesimal wedges to obtain the active earth pressure dGx =

x=BN

1 tan(ϑx − ρ) ⋅ dx ⋅γ ⋅h⋅ ∫x=0 2 E 1 = ⋅ γE ⋅ h ⋅ [−1 + (1 + tan2 ρ) ⋅ ln(1 + cot2 ρ)] (2.127) 2 (Schroeter, 1947, pp. 14-15). Schroeter’s earth pressure formula (eq. 2.127) is wrong because it infringes the transition conditions between two neighbouring wedges n and n+1. The forces acting at the common boundary surface between the two wedges must be equal and opposite (Newton’s third law: action = reaction). But that is not the case with Schroeter’s derivation. So Schroeter had repeated the mistake that Bélidor had already made when adding the frictionless voussoirs to make up the entire masonry arch (see Fig. 4-20 in (Kurrer, 2018, p. 214)). However, Bélidor was very much aware of the fact that he infringed the transition conditions in his masonry arch model, whereas Schroeter’s cardinal error had to be proved to him by a high-profile examination commission of the subsoil committee of the building department of the Nationalsozialistischer Bund Deutscher Technik (NSBDT, National Socialism Association of German Technology) (Hoffmann, 1944, p. 118). Sadly, Rudolf Hoffmann succumbed to concluding the external circumstances of the discussion surrounding Schroeter’s earth pressure theory in the language of Nazi propaganda, writing that “in a time of intense harnessing of all energies for the victory, countless hours of highly Ea =

Fig. 2.71 Schroeter’s derivation of the active earth pressure (Schroeter, 1947, p. 15).

2.7 Earth pressure theory in the discipline-formation period of geotechnical engineering

valuable engineering effort have been expended in futile theoretical debates” (Hoffmann, 1944, p. 118). But Schroeter insisted on his view. In 1947 he summarised his incorrect earth pressure theory in a polemic paper and unduly claimed that his “modified earth thrust theory” completed Coulomb’s earth pressure theory. In his publication, Schroeter called his critics “misguided” and styled them followers of the Nazis, because “the editor during Hitler’s time (Lohmeyer) had forcibly prevented the customary response of the person being criticised and prevented the author from holding further university lectures in Germany and Austria, and the warnings and threats from the main political division of the NSDAP made explanations impossible” (Schroeter, 1947, p. 42). Two years later, Kurt Gaede (1886-1975), professor at Hannover Technical University, criticised Schroeter’s publication, summing up thus: “... Schroeter’s work is an attempt made with completely inadequate means to solve a problem, the crux of which has not been understood by the author. The calculation infringes the elementary laws of mechanics, is unsuitable for answering the question that has been posed and should be utterly rejected” (Gaede, 1949, p. 9). Whether Schroeter realised his mistake is, unfortunately, unknown. 2.7.5

A hasty reaction in print

In 1948 Heinrich Press (1901-1968) published his paper Über die Druckverteilung im Boden hinter Wänden verschiedener Art (on the pressure distribution in soil behind walls of different kinds) (Press, 1948), which the editor of the Bautechnik-Archiv hoped would contribute to “clarifying the earth pressure issue still controversial in many respects” and can be interpreted as a reaction to Schroeter’s incorrect earth pressure theory. In his letter to Press dated 30 June 1948, Ohde formulated 17 critical remarks concerning his paper (Ohde, 1948). For example, he criticised that the introduction on earth pressure theories had been too short and that Krey’s contribution was not just the compilation of earth pressure tables (Fig. 2.72). Ohde also criticised the fact that Press had not reproduced Terzaghi’s earth pressure formula correctly (see eq. (2.124)). Finally, he contested the remark by Press that “as a result of considerable movements in the soil the active earth pressure [can] attain the value of the earth resistance” (Press, 1948, p. 52). Concluding, Ohde writes: “I would be delighted if you were to regard my remarks as not inopportune and you, like me, would have the desire to clarify the above thoughts further in an intense personal discussion. But even if otherwise is the case, I would be very pleased to hear from you again …” (Ohde, 1948, p. 4). Despite Ohde’s scientific criticism of Press’ hasty reaction in print, it is respectful. Scientific discourse in the form of the personal conversation presumes mutual respect: “science results from discussion” – as the physicist Werner Heisenberg (1901-1976) liked to say. 2.7.6

Foundations + soil mechanics = geotechnical engineering

Prussia’s Minister of Building Works Ludwig Brennecke published the first monograph on foundation construction in 1887 (Brennecke, 1887). After Brennecke’s

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Fig. 2.72 Letter from Johann Ohde to Heinrich Press (Ohde, 1948).

death, Erich Lohmeyer revised this book, which survived as far as its 6th edition in 1948. During the discipline-formation period of geotechnical engineering (1925-1950), numerous monographs on foundation construction appeared, such as Agatz’ Der Kampf des Ingenieurs gegen Erde und Wasser im Grundbau (the engineer’s fight against earth and water in foundation construction) (Agatz, 1936) and Terzaghi and Peck’s Soil Mechanics in Engineering practice (Terzaghi & Peck, 1948). These specialist publications for practising engineers were complemented by two books on the theory of soil mechanics: Theoretical soil mechanics

2.7 Earth pressure theory in the discipline-formation period of geotechnical engineering

(Terzaghi, 1943) and Fundamentals of soil mechanics (Taylor, 1948). And in June 1948 the first edition of the journal Géotechnique was published in London under the auspices of the British Geotechnical Society (now the British Geotechnical Association), the Institution of Civil Engineers and publishers Thomas Telford Ltd. Foundation and soil mechanics specialists now had their own journal. The three books mentioned above together with Géotechnique not only rounded off the discipline-formation period of geotechnical engineering in general, but also finally disentangled earth pressure theory from theory of structures in particular. The civil engineer as soldier

The book by Arnold Agatz (1891-1980) (Agatz, 1936) sees the civil engineer marching against the forces of earth and water in foundation construction. Agatz divides his monograph (Fig. 2.73) into four chapters: • • • •

The opponent ‘earth’ (pp. 1-43) The opponent ‘water’ (pp. 43-67) Establishing a defensive position (pp. 67-271) Assessing failures and the consequences of the battle for the future (pp. 271-276)

The first two chapters conclude with a section entitled “Compiling the probable plan of attack”. Fig. 2.73 Title page of the book by Agatz (1936).

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Fig. 2.74 The dynamics of nature acting on a retaining wall (Agatz, 1936, p. 25).

The reader could be tempted to understand the language of Agatz in the context of the “Lingua Tertii Imperii” (Victor Klemperer), the “language of the Third Reich”. However, such an evaluation does not do justice to Agatz’ book. Instead, the language is reminiscent of the military engineering that is a genetic constituent of modern civil engineering, which first appeared in the century of the Enlightenment. Agatz’ aim was to reinstate the close bond between engineer and nature in the structural and constructional treatment of foundation structures. Agatz criticised the fact that, in particular, incorrect appraisals due to the purely structural/constructional treatment of foundation structures lead to failures and the knowledge-of-nature factor had been pushed too far into the background: “It is now time to sum up this development and admit, openly and honestly, what was and is right and wrong, and what options are open to us so that we can take the better route of the many ahead of us” (Agatz, 1936, p. III). The “better route” for foundation construction advocated by Agatz was based on a critical empiricism that analysed failures clearly and drew conclusions from this for practice. Using the example of a retaining wall, he demonstrated what civil engineers must know if they wished to assess the strength and stability of structure and subsoil (Fig. 2.74).

2.7 Earth pressure theory in the discipline-formation period of geotechnical engineering

Agatz summed up as follows: “The current state of our knowledge is that although we have admitted that our theoretical approaches up to now have been inadequate, we are still far from a practical application of the elastic behaviour of the soil to the calculation of the forces acting. The number of our observations is still too few for this, and it will be a task of the future to prepare new calculation options.” (Agatz, 1936, p. 25) In compiling his “probable plan of attack” with relation to the theoretical determination of the earth pressure, Agatz therefore saw the task of the civil engineer as “clarifying the most likely form of the slip plane [in each situation] and comparing it with the simplified slip plane form that is used when calculating slip failure, earth pressure and earth resistance, and establishing – at least approximately – the deviations that increase or decrease safety … In cases of doubt ... the various slip plane forms must be worked through and weighed against each other” (Agatz, 1936, p. 43). Where the theoretical determination of the slip planes did not fully reflect the reality, Agatz regarded such comparative calculations as the only option. Agatz took a one-sided view of the relationship between theory and empiricism, soil mechanics and foundation construction, in favour of a praxeological foundation engineering. Addendum

Eighteen years after his Erdbaumechanik appeared, Terzaghi achieved a second scientific synthesis of soil mechanics in the shape of his Theoretical Soil Mechanics (Terzaghi, 1943). In this book, theory is totally separated from its practical application. He understands theoretical soil mechanics as one of the many branches of applied mechanics which, like reinforced concrete theory, for example, is based on an ideal material whose mechanical properties are obtained in a process of radical simplification through field tests and experiments. “The magnitude of the difference between the performance of real soils under field conditions and the performance predicted on the basis of theory can only be ascertained by field experience. The content of this volume has been limited to theories which have stood the test of experience and which are applicable, under certain conditions and restrictions, to the approximate solution of practical problems” (Terzaghi, 1943, p. VII). This consequential separation of theory and practice enabled Terzaghi to pursue another goal of a pedagogic kind. He writes: “The radical separation between theory and application makes it easy to impress upon the reader the conditions for the validity of the different mental operations known as theories” (Terzaghi, 1943, p. VII). In doing so, he emphasises that the theory must be combined with fundamental knowledge of the physical properties of the real soil material and points out the difference between the behaviour of the soil under laboratory and field conditions – a difference that is characteristic of soil mechanics. Terzaghi divides his monograph into four sections: • Section A, chapters I to IV: General principles involved in the theories of soil mechanics (pp. 1-65) • Section B, chapters V to XI: Conditions for shear failure in ideal soils (pp. 66-234)

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• Section C, chapters XII to XV: Mechanical interaction between solid and water in soils (pp. 235-344) • Section D, chapters XVI to XIX: Elasticity problems of soil mechanics (pp. 345-479) Following deliberations regarding the tasks and aims of soil mechanics in chapter I, Terzaghi writes about the yield condition of Coulomb and Mohr (see eq. (2.90)) (and other matters) in the next three chapters plus Rankine’s earth pressure theory (see section Rankine’s stroke of genius). The focus of section B is earth pressure theory (Fig. 2.75), the theory of ground failure, the stability of embankments and anchored sheet pile walls. Section B constitutes the heart of the content of Terzaghi’s monograph, although sections C and D do contain essential concepts of theoretical soil mechanics such as consolidation theory, elastic theory analyses of foundation problems and an insight into the emerging foundation dynamics. The latter was to be given its first summarising treatment as a subdiscipline of soil mechanics by Hans Lorenz (1905-1996) in the shape of his book Grundbau-Dynamik (dynamics in foundation construction) (Lorenz, 1960). In the preface to his Theoretical Soil Mechanics, Terzaghi announces a second volume: “The properties of real soils under field conditions will be discussed in

Fig. 2.75 Approximate determination of the point of application of passive earth pressure for cohesive soil material (Terzaghi, 1943, p. 102).

2.8 Earth pressure theory in the consolidation period of geotechnical engineering

a companion volume” (Terzaghi, 1943, p. VIII). Working together with Ralph B. Peck (1912-2008), this second volume (given the provisional title Introduction to Soil Mechanics) involved not inconsiderable difficulties: “It wasn’t until 1946 that the book neared completion. By then it was no longer an introduction to soil mechanics but a compendium on engineering mechanics of soils, based on precedent, wisdom, and the useful elements of soils” (Goodman, 1999, p. 213). This second volume was given the title Soil Mechanics in Engineering Practice (Terzaghi & Peck, 1948). In that same year, MIT professor Donald W. Taylor (1900-1955) published his book Fundamentals of soil mechanics (Taylor, 1948), which was heavily criticised by Peck: “Blind application of theory can directly lead to disaster … this is the idea which nearly ruined soil mechanics and against which the best efforts of Terzaghi and a few others have only recently been able to make headway” (cited after (Goodman, 1999, p. 213)). Nevertheless, Taylor’s book became very popular. Terzaghi’s two complementary volumes were translated into German and given the titles Bodenmechanik in der Baupraxis (Terzaghi & Peck, 1951) and Theoretische Bodenmechanik (Terzaghi & Jelinek, 1954). So the equation ‘foundations + soil mechanics = geotechnical engineering’ came true, provided foundations is equated to ‘soil mechanics in practice’ and soil mechanics to ‘theoretical soil mechanics’.

2.8 Earth pressure theory in the consolidation period of geotechnical engineering It was in 1954 that Karl Keil proposed Geotechnik as a technical term and set out the object of this engineering science discipline (Keil, 1954). However, the term Geotechnik would only become gradually established in the language of the German building industry at the start of the integration period of geotechnical engineering (1975 to date). The outward sign of this was the first edition of the journal Geotechnik (Fig. 2.76) in September 1978 as the publication of the Deutsche Gesellschaft für Erd- und Grundbau (now the Deutsche Gesellschaft für Geotechnik, DGGT, German Geotechnical Society) founded in Karlsruhe on 21 September 1950 (Smoltczyk, 1992). The aim of this journal was to provide a common publishing forum for the subdisciplines of geotechnical engineering at that time, i.e. soil mechanics, rock mechanics, foundation construction and engineering geology. Rock mechanics and engineering geology were therefore added to the ‘foundations + soil mechanics = geotechnical engineering’ equation in 1978. The scientific areas of study of these fields was expanded and their theoretical principles intensified during the consolidation period of geotechnical engineering (1950-1975). Progress in both disciplinary processes had been ongoing since the end of the 1960s through the use of computers for geotechnical calculations in research and major projects. Nevertheless, not until the integration period (1975 to date) would earth pressure theory benefit from the possibilities for modelling the mechanical soil behaviour arising out of the latter development.

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Fig. 2.76 Cover of the journal Geotechnik, June 2015 issue.

2.8.1

New subdisciplines in geotechnical engineering

New scientific subdisciplines of geotechnical engineering appeared in the form of rock mechanics and geomechanics. These two subdisciplines founded by Leopold Müller (1908-1988) were crowned by the multi-volume work Der Felsbau (rock engineering), the first volume of which was published by Müller as early as 1963 (Müller, 1963). His work as a whole was acknowledged by the Österreichische Gesellschaft für Geomechanik (ÖGG, Austrian Society for Geomechanics) [ÖGG, 2008], whose publication Geomechanik und Tunnelbau – Geomechanics and Tunnelling has been published in German and English since 2008. Whereas Müller tends to stress the empirical side of rock mechanics, so the work of Walter Wittke which followed shifted the relationship between empiricism and theory in favour of theory, but without neglecting the empirical side (see (Wittke, 1984 & 2014), for example). Wittke has been involved with specific applications of the finite element method (FEM) in tunnelling since the 1970s (Kaiser, 2008, p. 78). Important preliminary work on this was provided by the pioneer of the finite element method (FEM), O. C. Zienkiewicz, and his collaborators, in the shape of their article Stress analysis of rock as a ‘no tension’ material (Zienkiewicz et al., 1968), and the creator of the discrete element method (DEM), Peter A. Cundall (Cundall, 1971). The professor of structures in Braunschweig, Heinz Duddeck, made considerable contributions to the theory of structures fundamentals for tunnelling (e.g. (Duddeck, 1976 & 1978)).

2.8 Earth pressure theory in the consolidation period of geotechnical engineering

All three of these researchers created the scientific basis for work in tunnels and caverns in the modern age; they would have a huge impact on rock mechanics in the integration period of geotechnical engineering (1975 to date). The Soviet scientist Vadim V. Sokolovsky (1912-1978), internationally renowned through his contributions to plastic theory (e.g. (Sokolovsky, 1955)), managed to transform the Winkler system of equations for the earth continuum (eqs. 2.89 and 2.94) into a canonic system of equations of the hyperbolic type through transformation of the variables and to show that the two characteristic families of slip planes correspond to the plastic limit state of equilibrium (Sokolovsky, 1960 & 1965). Fig. 2.77 shows an example of such slip plane families. Sokolovsky thus ascertained only the physical non-linearity of the earth continuum. When setting up the relationships between stresses and strains, however, the geometric non-linearity of the earth continuum still has to be taken into account. This very general approach to modern continuum mechanics was pursued by Gerd Gudehus from the University of Karlsruhe, who advanced fundamental research into soil mechanics systematically under Hans Leussink (1912-2008). Gudehus made the non-linear field theory of mechanics – e.g. of a C. A. Truesdell and a W. Noll (Truesdell & Noll, 1965) – workable for theoretical soil mechanics (Gudehus, 1968), and left his mark on a style of theory that would prove vital to the numerical analysis of mechanical soil behaviour with computers. 2.8.2

Determining earth pressure in practical theory of structures

Papers dealing with the experimental and numerical determination of earth pressure acting on retaining structures and aiming at practical calculations appeared several times a year in the journals Brücke und Straße, Die Bautechnik and Der Bauingenieur. Isolated articles doubting Coulomb’s earth pressure theory were also published (Kohler, 1956) plus those claiming to deal with the “principles of a new earth pressure theory” (Hartmann, 1968). Fig. 2.77 Slip planes for a retaining wall with inclined wall line and horizontal terrain line with constant uniformly distributed load p from y = 0 to 2.0 (Sokolovsky, 1960, p. 95).

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The methods of earth pressure theory were still being dealt with in a fashion in the fifth edition of the third volume of the four-volume work Praktische Baustatik (practical theory of structures) (Schreyer et al., 1967, pp. 106-136). In the course of the changeover from engineering schools to polytechnics in Germany in the early 1970s, theory of structures was also given a new look. So the sections on masonry arch theory and earth pressure theory also disappeared from Praktische Baustatik (now three volumes) in the 1970s. Earth pressure theory thus vanished from the scientific canon of practical theory of structures at the end of the consolidation period of geotechnical engineering. Nevertheless, determining earth pressure in a simple way remained a task of the structural engineer working on buildings until well into the integration period of geotechnical engineering. The modified Culmann E line

Edgar Schultze (1905-1986) tackled line loads using the existing methods for calculating earth pressure ((Krey 1926b, p. 90), (Jacoby, 1948, p. 176), (Müller-Breslau, 1947, pp. 72 & 86) and (Möller, 1922a, p. 92)). Although Krey had essentially solved the problem, he had not provided any directly usable equations. According to Schultze, this was the reason why “one has found a number of incorrect views or inadmissible simplifications in recent times, too” (Schultze, 1950, p. 7). He named the work of E. Jacoby (Jacoby, 1948) as an example. Schultz modified the Culmann E line (see Fig. 2.35) for the load case of earth pressure due to a line load. Fig. 2.78 shows his diagram for the case of P behind the slip plane. He specified the modified Culmann E line for the other case, too (P in front of slip plane). As a “simple and correct method for calculating the earth pressure distribution” over the wall line is lacking (Schultze, 1950, p. 7), Schultze developed just such a graphical method and checked it analytically. Finally, he generalised his method for inclined loads. Hans Lorenz and his collaborators Helmut Neumeuer and Rudolf Lichtl took a close look at the work of Schultze in the light of new research by Terzaghi [Lorenz, 1954]. They came to the conclusion that “the Culmann E line used in practice so far, or rather the method given by

Fig. 2.78 The Culmann E line for a line load P (Schultze, 1950, p. 8).

2.8 Earth pressure theory in the consolidation period of geotechnical engineering

Schultze, always supplies values that lie too far on the safe side” [Lorenz, 1954, p. 315]. Therefore, they suggested using the Boussinesq stress theory in future, but called for more research owing to the fact that this latter theory leads to earth pressure values for line loads which are much lower than those given by classical earth pressure theory. Nevertheless, practising engineers returned to the Culmann E line again and again. Helmut Schmidt from the Underground Railways Division of Hamburg Building Authority therefore extended the Culmann E line to cover cohesive soils [Schmidt, Helmut, 1966]. Using the Schmidt method, it is possible to determine graphically not only the active earth pressure Ea , but also the passive earth pressure Ep ; in his sample calculation, Schmidt used polar coordinates paper for Ep and therefore arrived at a clear presentation of his solution [Schmidt, Helmut, 1966, p. 82]. New findings regarding passive earth pressure

So far, the earth pressure problem had been treated as a planar problem – characterised by the fact that, for example, the Coulomb earth pressure wedge, as a prism, has an assumed unit length B = 1 perpendicular to the plane of projection. In the case of elements with a narrow compression area, such as anchor plates for sheet pile walls, dolphins, well and mast foundations, piles, wing walls to bridges and short sections of wall, B influences the earth pressure. The three-dimensional earth pressure can no longer be described geometrically by a unit prism; instead, adequate geometrical models must be found in order to take account of three-dimensional effects. The dissertation of Heinz Zweck therefore investigated the influence of lateral bodies of earth – the three-dimensional effect – on the passive earth pressure Ep (Zweck, 1952), the main findings of which were summarised by him in a paper (Zweck, 1953). Whereas so far it had been assumed that the three-dimensional earth resistance was made up of the Coulomb component of eq. (2.73) multiplied by B 1 (2.128) ⋅ H2 ⋅ γE ⋅ λp ⋅ B 2 and a term that represents the influence of the lateral bodies of earth proportional to the cube of the wall height H, Zweck arrived at the following result after evaluating his tests (Fig. 2.79): Ep,Coulomb =

1 (2.129) ⋅ H2 ⋅ γE ⋅ λp ⋅ B′ 2 The second component of Ep is therefore only proportional to the square of H. Zweck captured the three-dimensional effect through the equivalent width B′ , thus based the total passive earth pressure Ep,lat.bod. =

1 (2.130) ⋅ H2 ⋅ γE ⋅ λp ⋅ (B + B′ ) 2 on the prism model with total width B′′ = B + B′ . By extrapolating the model tests, Zweck arrived at an answer that states that the component of the lateral bodies of earth for the 3 m high wall must be equal to the active earth pressure Ep = Ep,Coulomb + Ep,lat.bod. =

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Fig. 2.79 Model tests on three-dimensional earth pressure with shell-shaped slip surface (Zweck, 1953, p. 191).

“on a 45.5 cm wide section of wall within an infinitely long wall” (Zweck, 1953, p. 193). Using eq. (2.129), he calculated the value Ep,lat.bod. = 15.2 t for this – a value that agreed well with the passive earth pressure component resulting from a large-scale test on an H = 3 m high and B = 5.60 m wide retaining wall, which was 16 t. Anton Weißenbach’s dissertation Der Erdwiderstand von schmalen Druckflächen (earth resistance of narrow compression areas) (Weißenbach, 1961), the main results of which were published in an article of the same name in Die Bautechnik (Weißenbach, 1962), went way beyond the preceding work on this theme – even beyond that of Zweck, whose work was subjected to a careful criticism by Weißenbach. Based on large-scale tests on soldier pile walls and experiments with models, Weißenbach achieved a generalised solution to the problem of earth resistance on narrow compression areas. This and the further development of his calculation proposal form one of the focal points of his standard work on calculation principles for excavations (Weißenbach, 1975), the second edition of which was published together with Achim Hettler (Weißenbach & Hettler, 2011). A detailed presentation of Weißenbach’s ground-breaking publication (1961) is unnecessary here because Hettler has already provided a commentary (Hettler, 2013). The dissertation of Hans Jörg Mayer-Vorfelder converts Ohde’s method for determining the passive earth pressure (Ohde, 1938a) into the “Ohde calculus of variations” (Mayer-Vorfelder, 1970). The key elements of this work, too, appeared one year later in the form of a paper (Mayer-Vorfelder, 1971). He compared the earth resistance coefficient 𝜆ph of his method with the tabulated values of Caquot/Kérisel (Caquot & Kérisel, 1967), Coulomb/Jumikis (Jumikis, 1962), Krey (Krey, 1932) and Ohde (Ohde, 1956) (Fig. 2.80). As can be seen, the earth resistance coefficients 𝜆ph calculated according to Coulomb and assuming planar slip planes deviate considerably from the other

2.8 Earth pressure theory in the consolidation period of geotechnical engineering

Fig. 2.80 Earth resistance coefficients 𝜆ph plotted against wall friction angle 𝛿 for 𝜌 = 20∘ , 30∘ and 40∘ (Mayer-Vorfelder, 1971, p. 149).

methods with curved slip planes for larger values of the wall friction angle 𝛿 and value of internal friction 𝜌. This means that the passive earth pressure is much too large, so earth resistance calculations with planar slip planes rule themselves out for reasons of safety alone. Based on Kötter’s mathematical earth pressure theory for the passive case, Mayer-Vorfelder developed another method, which he called the “equilibrium method”. Here, too, he compared his two methods with those of Ohde and Caqout/Kérisel, and recommended using only his equilibrium method and the tabular values of Caqout/Kérisel for earth resistance calculations. Mayer-Vorfelder wrote a program in ALGOL 60 for his method, the procedure for which and the calculated results for 𝜆ph can be seen in his dissertation (Mayer-Vorfelder, 1970). Without a computer, he would not have been able to compile the comprehensive tables of earth resistance coefficients 𝜆ph supplied by his iterative method. Mayer-Vorfelder saw the future task of determining earth resistance as “finding

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a method that makes use of electronic computer systems to carry out the variation in possible slip plane forms – described by polygonal lines if necessary – in such a way that it is possible to find the absolutely lowest earth resistance while maintaining all equilibrium relationships” (Mayer-Vorfelder, 1971, p. 153). In this search for equilibrium, iterative methods would assert themselves more and more during the integration period of geotechnical engineering (1975 to date).

2.9 Earth pressure theory in the integration period of geotechnical engineering Im The International Symposium on Numerical Methods for Soil and Rock Mechanics took place at the University of Karlsruhe in September 1975. Two years later, the proceedings appeared under the title of Finite Elements in Geomechanics (Gudehus, 1977) as the fifth volume of the Wiley Series in Numerical Methods in Engineering edited by R. H. Gallagher and O. C. Zienkiewicz. Concerning the aims, the editors wrote the following: “This volume is intended for readers both from research establishments and industry who have already some experience with FEM. It may also be of interest for practitioners in adjacent subjects ... As has been shown, FEM can make a vital contribution towards the integration of theory and practice” (Gudehus, 1977, Preface). FEM and other computer-based intellectual technologies were crucial to placing the engineering sciences on a uniform methodological foundation and, as a consequence, led to integration between theory and practice: science was directly effective in practice as practical engineering was given a scientific footing. It is, after all, the essential feature of the system of non-classical engineering sciences, which appeared historico-logically in the evolution of the engineering sciences with the innovation phase in research and exemplary engineering projects and then infiltrated the entire practical calculations of the engineer during the diffusion phase. These phases can be combined to form the integration period, which for geotechnical engineering began in the mid-1970s. The founding of the International Journal for Numerical and Analytical Methods in Geomechanics in 1977 (Fig. 2.81) and the magazine Computers and Geomechanics in 1985 plus the compendium of Gudehus in 1977 (Gudehus, 1977) can be regarded as publishing milestones in the innovation phase of geotechnical engineering – a period that stretches from 1975 to the late 1980s. Only after that did numerical engineering methods start to disseminate across practical geotechnical calculations. Numerical methods for assessing the deformation behaviour of geotechnical structures have already proved their worth in practice in the first decade of the 21st century: “The verification of serviceability anchored in Eurocode 7 is mostly only possible with the help of numerical analyses when complex structures are involved. And when verifying limit states, numerical calculations are becoming increasingly important” (Wolffersdorff & Schweiger, 2008, p. 501). Earth pressure theory as the basis of the numerical verification of limit states is again attracting attention in geotechnical numerical analysis, which can now look to the Empfehlungen (recommendations) of the DGGT’s Numerical Analysis Study

2.9 Earth pressure theory in the integration period of geotechnical engineering

Fig. 2.81 Credits page of the first issue of International Journal for Numerical and Analytical Methods in Geomechanics (January 1977).

Group (DGGT, 2014). The section on earth pressure by Achim Hettler in the Grundbau-Taschenbuch (foundations manual) also had the aim of “making sets of instructions available … to foundation and structural engineers” (Hettler, 2008, p. 289) – a goal that the author also realised brilliantly in the 8th edition of that work (Hettler, 2017a). Added to this is the fact that in the integration period of geotechnical engineering, knowledge of earth pressure theory and its

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historical development is still necessary for the assessment of historical retaining walls. 2.9.1

Computer-assisted earth pressure calculations

It was 1958 when the first computer-assisted earth pressure studies appeared in Géotechnique (Little & Price, 1958). Based on the simplification of the slip circle method of Fellenius (see Fig. 5-65 in (Kurrer, 2018, p. 342)) by Alan W. Bishop (1920-1988) (Bishop, 1955), this method of slices (see Fig. 2.2) was programmed by A. L. Little and V. E. Price to determine the critical slip circle of earth dams by way of iteration. Investigating the stability of embankments, Martin Ziegler carried out a critical comparison of the customary variations on the slip circle method and derived an implicit, non-linear conditional equation for safety (Ziegler, 1988). He showed in his work that the method of slices of Gudehus (Gudehus, 1981, pp. 169-173) satisfies force equilibrium. Crucial here is that the safety of embankments is determined by way of iteration and this is only possible practically with the help of computers. The determination of active and passive earth pressure is simpler. The mathematician Hermann Groß from the Chair of Rock Mechanics at the University of Karlsruhe has derived analytical formulas for calculating the active and passive earth pressures with planar slip planes in soils with friction, cohesion and a continuous uniformly distributed load on the inclined terrain line (Groß, 1981). By way of a three-stage evaluation of complicated trigonometric equations it is in the end possible to quantify the earth pressure, which, however, even back then, with programmable desktop computers already available, did not present any difficulties. Although a closed-form solution to the earth pressure is possible with the Groß method, there are many cases it cannot handle, such as non-uniform surcharges, line loads, etc. In a five-part essay, the Hamburg-based civil engineers Hugo Minnich and Gerhard Stöhr translated the graphical Culmann method into an analytical one with the aim of covering as many practical cases as possible. In the first part they postulate the general formula for passive earth pressure for the load G0 (Fig. 2.82) acting on the terrain line. Similarly, the authors present the formulas for the active earth pressure in the second part of their series of articles and from now on call their method the “G0 method” (Minnich & Stöhr, 1981b). Finally, they modify their G0 method in order to determine the active earth pressure for line loads (Minnich & Stöhr, 1982a) and cohesive soils (Minnich & Stöhr, 1982b). In the fifth part, they compile the mathematical apparatus for their G0 method and take into account the adhesion between the wall line and the soil material as well (Minnich & Stöhr, 1984). Compared with the formulas of Groß, the advantage of the G0 method is its greater applicability, but at the cost of less clarity, with the authors carrying out their sample calculations on programmable pocket calculators. Like Minnich and Stöhr, Hermann Lohmiller also considered broken terrain lines and non-uniform surcharges when determining earth pressure and wrote a program for the modified classical earth pressure theory (Lohmiller, 1995), which

2.9 Earth pressure theory in the integration period of geotechnical engineering

Fig. 2.82 Culmann drawing for passive earth pressure (Minnich & Stöhr, 1981a, p. 198).

he later extended to stratified soils (Lohmiller, 2009). Programs specifically for earth pressure calculations have been available since the early 1990s. 2.9.2

Geotechnical continuum models

In 1914 Otto Mohr introduced Die Lehre vom Erddruck (earth pressure theory) in his Abhandlungen aus dem Gebiete der Technischen Mechanik (treatises from the field of applied mechanics) with the following words: “Determining the earth pressure against a retaining wall is a statically indeterminate task in all circumstances because the number of unknown variables exceeds the number of equilibrium conditions by far. A famous physicist is said to have answered – as he was asked whether he regarded the calculation as possible – that it is perhaps possible if the genesis of the body of soil is known. Without doubt, what he wanted to insinuate was that it is necessary to include the small deformations of the wall and the body of soil, which happen during the building of the structure, in order to derive the many unknowns from the proper relationships between the deformations and the internal forces in a similar way to analysing elastic structures. Those relationships are, however, unknown, because something similar to Hooke’s law does not exist for non-elastic bodies. Therefore, trying to get closer to the solution by this means is totally hopeless. The only other way is the one that has always been used: Introduce assumptions regarding the stress states in the body of soil and try to establish their reliability or probability through experience. The biggest difficulty of this approach is obtaining reliable test results. The results of countless tests carried out so far allow virtually anything to be verified within wide limits” (Mohr, 1914, p. 236). Mohr’s epistemological pessimism with respect to the material laws of the soil was all too justified. Such material laws could only be formulated at the end of the consolidation period of geotechnical engineering (1950-1975) on the basis of non-linear continuum mechanics, implemented for research with the help of

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FEM on mainframe computers and later, in the integration period of geotechnical engineering (1975 to date), gradually used for practical applications as well. The reason why the consolidation period of theory of structures (Kurrer, 2018, pp. 22-24) began 50 years and its integration period (Kurrer, 2018, pp. 24-26) 25 years before the same periods in geotechnical engineering is due to the fact that theory of structures had already concluded its elastic theory principles in the final quarter of the 19th century in the form of the theory of statically indeterminate elastic trusses. It was not until the second half of the consolidation period of theory of structures (1925-1950) did thoughts turn to non-elastic material laws as well. Owing to the steeper gradient in the disciplinary development of soil mechanics, the difference in the timing of the different developmental periods decreased to just 25 years. The reason why this discrepancy remained for so long is due to the fact that the material laws for soils are far more complex than those of structural mechanics, irrespective of the processes that were taking place in structural mechanics to break down the dominance of the linear. When it comes to experimental earth pressure research, it had already reached a new quality with the constitution of soil mechanics in the early 1920s, its establishment and completion (1925-1950) and then its dialectic integration into geotechnical engineering by the middle of the 20th century. One example is the direct shear apparatus at the earthworks laboratory of the Prussian Testing Institute for Hydraulics and Shipbuilding, which determined the angle of internal friction 𝜌 and the cohesion c of the Mohr-Coulomb yield criterion (eq. 2.90) (Fig. 2.83). The triaxial test apparatus, which started to appear in earthworks laboratories during the establishment and classical phases of soil mechanics (1925-1950), can also be used to measure the shear parameters. The cube-shaped soil sample is “compressed” in the three coordinate directions with the principal stresses 𝜎 1 , 𝜎 2 and 𝜎 3 .

Earth backfill c Steel strap

Pressure (turned through 90° sign in Pressure lever d this version, i.e. ⊥to plane of drawing) plate a g

g

f i Rollers r Test plate Steel strap

Ball bearing

p

q

Fig. 2.83 Direct shear apparatus for determining shear parameters at the earthworks laboratory of the Prussian Testing Institute for Hydraulics and Shipbuilding (Krey, 1932, p. 10).

2.9 Earth pressure theory in the integration period of geotechnical engineering

Fig. 2.84 Vertical stress diagram from triaxial test (Fellin, 2000, p. 13).

σ1

g

σ1max

σ2

E0 σ1min

Unl oa

din

g

Loadin

ε1

In the triaxial test with a cylindrical soil sample, the sample is compressed by a stress 𝜎 1 while applying a constant horizontal radial compressive stress 𝜎 2 in the vertical direction and, for example, the strain 𝜀1 is measured. The stress–strain curve shown in Fig. 2.84 is a simple example of the hypoplastic material law for cohesionless soils, e.g. sand, which can no longer be presented in the form 𝜎 = f(𝜀). Hypoplastic material laws are rate laws and describe the change in the stress for a change in the strain. Only strain terms of the first order are possible here. The non-linear material behaviour is controlled by the way the stiffness depends on the stress. Here, the change in the stiffness upon changing between loading and unloading (Fig. 2.84) can be illustrated by using the absolute value. Wolfgang Fellin has constructed a simple hypoplastic material law for didactic reasons (Fellin, 2000, p. 13): | d𝜀 | dσ1 d𝜀 = C3 ⋅ (σ1 + σ2 ) ⋅ 1 + C4 ⋅ (σ1 − σ2 ) ⋅ || 1 || dt dt | dt |

(2.131)

using the constants C3 =

E0 2 ⋅ σ2

(2.132)

C4 =

E0 2 ⋅ σ2 ⋅ sin ρ

(2.133)

and

The terms of the deviatoric stresses in eq. (2.131) can be used to simulate failure conditions (Fellin, 2000, p. 14). Generally, the hypoplastic material law is represented by the tensor equation ( ) d𝜺 d𝝈 = f 𝝈, (2.134) dt dt which is not linear in d𝜺/dt. “The results of a geotechnical FEM calculation stand and fall with the quality of the material law used to describe the material behaviour of the soil. Soils are strongly non-linear and inelastic and exhibit a distinctive change in volume as a result of shear deformations ... Hypoplastic

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material laws can model all the above properties very well” (Fellin & Kolymbas, 2002, p. 830). Dimitrios Kolymbas (Kolymbas, 1977 & 1988a,b), Gerd Gudehus (Gudehus, 1994 & 1996) and Peter-Andreas von Wolffersdorff (Wolffersdorff, 1996) are just some of the researchers who have investigated this type of material law for coarse- and fine-grained soils. Martin Ziegler based his research into earth pressure on the work of Kolymbas (1977), one of the first hypoplastic approaches for sand (Ziegler, 1987), which was later developed further by Khalid Abdel-Rahman – taking into account scale effects for earth pressure (see (Hettler & Abdel-Rahman, 2000)). Dimitrios Kolymbas and Ivo Herle provide an excellent insight into material laws for soils in the Grundbau-Taschenbuch (Kolymbas & Herle, 2008). In their contribution they criticise that in commercial software products for FEM analyses, the choice of material laws is extremely limited, and most programs contain only the simplest material laws, e.g. the Mohr-Coulomb model, and implement their own formulations of elastoplastic material laws with isotropic hardening but with little documentation (Kolymbas & Herle, 2008, p. 284). This latter class of material laws with deviatoric and volumetric hardening (hardening-soil model) was used by the consulting engineers responsible for the extension to the power station on the Rhine at Iffezheim when creating the three-dimensional analytical model needed to design the 35 m deep oval excavation required (Fig. 2.85). The plan form of the 35 m deep main excavation with clear dimensions of 51 x 36 m was made up of two three-centred arches; it had an underwater concrete ground slab anchored into the subsoil (Raithel & Kirchner, 2011). The 45 m high shoring consisted of 1.5 m thick diaphragm walls (without anchors) and included a capping beam and compression ring, which rendered bracing and anchors unnecessary. Such a design mobilised the arching effect of the

Headwater excavation Tailwater Main excavation

excavation

Existing power station Retaining wall

Fig. 2.85 Analytical model of the main excavation for extending the power station on the Rhine at Iffezheim – construction phase with partial bracing due to plant installation (water level omitted for clarity) (source: Kempfert + Partner Geotechnik).

2.9 Earth pressure theory in the integration period of geotechnical engineering

Depth [m] below ground level

Slice 22A 0

Capping beam

–5 Groundwater level –10

Compression ring

–15 –20 Change of soil material

–25 –30 –35

0

50

Underwater concrete ground slab 300

200 250 100 150 Earth pressure [kN/m2]

Earth pressure from continuum model EaR = three-dimensional active earth pressure (R = 35 m or 15.5 m) E0B = EAB earth pressure recommendation (E0B = (EaR + E0)/2) E0 = earth pressure at rest Slice 1

5,5

.1 m

5m

.3

ca

ca

Slice 32

Slice 22A

Fig. 2.86 Comparison of the earth pressure distribution for slice 22A of the oval excavation (Raithel & Kirchner, 2011, p. 871).

two three-centred arches and achieved an economic structural system for the main excavation. The results obtained from the three-dimensional continuum model were verified by plausibility checks – especially with respect to the magnitude of the calculated earth pressures acting on the sides of the excavation. Fig. 2.86 shows the good agreement between the earth pressure distribution obtained from the three-dimensional continuum model and the numerical three-dimensional earth pressure according to the modified slice element theory (Walz & Hock, 1987 & 1988).

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This highly complex geotechnical structure was awarded the Ulrich Finsterwalder Engineering Prize in 2015 (Jesse & Rauschenbach, 2015, pp. 24-25). 2.9.3

The art of estimating

The simplification of the earth pressure calculations by Otto Franzius has already been mentioned in section 2.6.1 (Franzius, 1918). He proposed always using the value of the horizontal earth pressure according to Prony’s earth pressure formula (eq. (2.72) or (2.73)). In doing so, he related the active earth pressure Ea according to eq. (2.72) to the hydrostatic pressure W : ( ρ) 1 1 = ⋅ H2 ⋅ γE ⋅ λa = μa ⋅ W (2.135) Ea = ⋅ H2 ⋅ γE ⋅ tan2 45∘ − 2 2 2 with 1 (2.136) W = ⋅ H2 ⋅ γW 2 and ( ρ) . (2.137) μa = γE ⋅ tan2 45∘ − 2 In eq. (2.136), 𝛾 W = 1.00 t/m3 is the unit weight of water. Franzius developed eq. (2.73) for the passive earth pressure similarly: Ep = μp ⋅ W

(2.138)

with

( ρ) . (2.139) μp = γE ⋅ tan2 45∘ + 2 The Ea and Ep values were tabulated by Franzius (Franzius, 1918, p. 187/188): dry topsoil (𝛾 E = 1.40 t/m3 , 𝜌 = 40∘ ) → Ea = (1/3)⋅W and Ep = 6⋅W wet topsoil (𝛾 E = 1.65 t/m3 , 𝜌 = 30∘ ) → Ea = (1/2)⋅W and Ep = 5⋅W dry clay (𝛾 E = 1.60 t/m3 , 𝜌 = 40∘ ) → Ea = (1/3)⋅W and Ep = 7⋅W wet clay (𝛾 E = 2.00 t/m3 , 𝜌 = 20∘ ) → Ea = 1⋅W and Ep = 4⋅W dry sand (𝛾 E = 1.60 t/m3 , 𝜌 = 31∘ ) → Ea = (1/2)⋅W and Ep = 5⋅W damp sand (𝛾 E = 1.80 t/m3 , 𝜌 = 40∘ ) → Ea = (2/5)⋅W and Ep = 8⋅W wet sand (𝛾 E = 2.10 t/m3 , 𝜌 = 29∘ ) → Ea = (3/4)⋅W and Ep = 6⋅W wet gravel (𝛾 = 1.86 t/m3 , 𝜌 = 25∘ ) → E = (3/4)⋅W and E = 4.5⋅W

• • • • • • • • E a p • sand underwater after deducting uplift and horizontal hydrostatic pressure (𝛾 E - 𝛾 W = 2.10 – 1.00 = 1.10 t/m3 , 𝜌 = 25∘ ) → Ea = (1/2)⋅W and Ep = 2.5⋅W Looking at the figures in the table, we can see that “it does not depend on recalculating the earth pressures for every design, but rather on focusing on cautious yet reasonable assumptions” (Franzius, 1918, p. 187). Franzius recommends determining the earth pressure with his table for preliminary designs, whereas “more precise methods must be employed for the final design after establishing the correct constants for the backfill soil by way of tests” (Franzius, 1918, p. 187). In conversation with Dr.-Ing. Hans-Peter Andrä, he said that his father – Dr.-Ing. Wolfhart Andrä (1914-1996) – always estimated the active

2.9 Earth pressure theory in the integration period of geotechnical engineering

earth pressure using the hydrostatic pressure formula (eq. 2.136) (Andrä, 2015), i.e. EAndrä = W . He justified this with the fact that entering 𝛾 E = 2.00 t/m3 and the earth pressure coefficient 𝜆a = 0.50 into eq. (2.135) results in the hydrostatic pressure formula (eq. 2.136). According to Franzius, this estimate is also valid for the most unfavourable case of wet clay as a backfill material, indeed results in Ea ≈ W when using Prony’s formula (eq. 2.135). So Andrä senior fully satisfies the purpose of preliminary calculations. The earth pressure at rest according to eq. (2.116) should be assumed when designing non-yielding basement walls; Andrä senior was right again here, although the deviation of the earth pressure at rest for loosely bedded sand (𝛾 E = 1.80 t/m3 , 𝜌 = 30∘ ) (Broms, 2007, p. 130) is E0 =

1 1 1 ⋅ H2 ⋅ γE ⋅ (1 − sin ρ) = ⋅ H2 ⋅ 1.80 ⋅ (1 − sin 30∘ ) = ⋅ H2 ⋅ 0.90 2 2 2 (2.140)

and with his formula 1 1 (2.141) ⋅ H2 ⋅ γW = ⋅ H2 ⋅ 1.00 2 2 is only +11%. So eq. (2.141) may also be used for the earth pressure at rest according to eq. (2.116): E0 ≈ W = EAndrä . As the structural conscience of Fritz Leonhardt, Wolfhart Andrä was able to estimate the earth pressure simply, but not too simply. Despite great progress in geotechnical engineering theories, Coulomb’s earth pressure theory – with its modifications on the one hand and simplifications for estimating the earth pressure on the other – is still a safe guideline for the civil engineer searching for important states of equilibrium in geotechnical retaining structures. EAndrä = W =

2.9.4 The history of geotechnical engineering as an object of construction history The history of construction, which has been taking shape on an international scale since the late 1990s, contains only isolated references to the history of geotechnical engineering knowledge. Researchers such as Massimo Corradi (1995 & 2002), Jacob Feld (1928 & 1948), Jacques Heyman (1972), Walter Kaiser (2008), Jean Kérisel (1953 & 1985), Fritz Kötter (1893), Jean-Henri Mayniel (1808), Alec W. Skempton (1956, 1981 & 1985) and Jacques Verdeyen (1959) have written on important individual themes of the historical development of geotechnical engineering. But we are still awaiting a synthesis of this work. This chapter, which summarises the history of earth pressure theory from 1700 to the present day within the scope of the history of the theory of structures, should therefore be understood as one building block in a systematic history of geotechnical engineering knowledge in the context of a historical study of construction. The scientific discipline of construction history is still young and increasingly refers to important construction tasks in their historical context. Special

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Research Area 315 “Conservation of historically significant structures” at the University of Karlsruhe, active from mid-1985 to the end of 1999, has provided a sound foundation that is recorded in 14 yearbooks [Wenzel, F., 1987-1997] and the eight books of the Empfehlungen für die Praxis (recommendations for practice) series edited by Fritz Wenzel and Joachim Kleinmanns, where the work is structured, prepared and summarised according to themes. This series also includes the book written by Michael Goldscheider, Baugrund und historische Gründungen (subsoil and historical foundations), with an appraisal written from the heritage preservation viewpoint by Hannes Eckert (Goldscheider, 2003). In contrast to new structures, the surveying, assessment and repairing of historical foundations throws up two special aspects (Goldscheider, 2003, p. 1): • In addition to the normal demands regarding the stability, integrity and serviceability of a structure, there is also the demand for retention of the maximum historical value. • The many different historical foundations are totally different to modern ones. The differences lie in the material, in the design and detailing and in the greater exploitation of the strength of the soil. These special aspects are not only essential for historical buildings, but also for the foundations to other historical structures. For example, the historical value of the Fleisch Bridge in Nuremberg (1596-1598) with its pile foundation (Lorenz & Kaiser, 2012, p. 125) is indisputable. And the fortifications of a Vauban or the Prussian fortification structures of the 19th century (Fig. 2.87) with their retaining walls have aesthetic, historical, scientific or social values for past, present and future generations, i.e. are worthy of preservation. Therefore, the European Union, working with fortifications researchers from many different countries, are making great efforts to reassess this inheritance [Neumann, Hans-Rudolf, 2005 & 2014]. Assessing the worthiness for preservation is much more difficult in the case of historical structures of a geotechnical kind, such as dams, cuttings, embankments, excavations, tunnels, caverns, quay walls and transport infrastructure systems. Excavations are temporary structures and disappear from society’s memory if not recorded in the specialist literature. The struggle of Swedish civil engineers to discover the causes of the loss of stability of Stigberg Quay in 1916 led to an earth pressure method (Kurrer, 2018, pp. 341-343) that, in principle, is still employed today; the structure itself may have passed into history, but the method remains in its historical genesis. Therefore, as with theory of structures, historical methods of analysis in geotechnical engineering contribute to the heritage value of historical structures. Compared with buildings, in geotechnical engineering the relationship between product and method is shifted in favour of the latter. For this reason, research into the scientific canon of geotechnical engineering is a constituent part of the history of its knowledge within the scope of a historical study of construction. In particular, knowledge of geotechnical theory processes contributes to shaping how geotechnical engineering sees itself in scientific and technical terms and is hence also integral to its social responsibility. Furthermore, research into the history of geotechnical engineering knowledge has an effect on

2.9 Earth pressure theory in the integration period of geotechnical engineering

Fig. 2.87 Poster for a series of events “Praktiken und Potenziale von Bautechnikgeschichte” (practices and potential of construction history) held at the Deutsches Technikmuseum Berlin in 2014 (source: Chair of Construction History and Structural Preservation, Brandenburg University of Technology Cottbus-Senftenberg).

geotechnical engineering itself – connecting the technical with the historical. For example, in the surveying, assessing and repairing of historical retaining walls [Alsheimer, 2015], knowledge of both geotechnical experimentation and historical earth pressure theories is necessary – and here the art of estimating, as demonstrated by the example in section 2.9.3, could play a significant role. In this respect, the history of geotechnical engineering knowledge helps to raise the quality when engineers are working on historically important geotechnical

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design systems while trying to do justice to heritage requirements. Nevertheless, as geotechnical engineering and the historical study of theory of structures progress towards a historical design theory of geotechnical engineering, there is still a “hard slog” (Brecht) ahead.

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Schreyer, C., Ramm, H. and Wagner, W. (1967). Praktische Baustatik. Teil 3., 5th ed. Stuttgart: B. G. Teubner. Schroeter, A. (1940). Der Coulombsche Erddruck aus Hinterfüllung und bei Auflasten, insbesondere Kurzstreckenlasten. Das Einflußlinien-Verfahren bei Erddruck. Die Bautechnik 18 (44/45): 505–514. Schroeter, A. (1947). Die klassische Erdschubtheorie und ihre Vollendung nach 175 Jahren. Berlin: Georg Siemens Verlagsbuchhandlung. Schultze, E. (1950). Der Erddruck aus einer Linienlast. Die Bautechnik 27 (1): 7–12. Schwarz, H. (1922). Die Erddruckberechnung von Freund. Zentralblatt der Bauverwaltung 42: 598–599. Schwedes, J. and Schulze, D. (2003). Lagern von Schüttgütern. In: Handbuch der Mechanischen Verfahrenstechnik. Vol. 2, ed. by Heinrich Schubert, pp. 1137-1253. Weinheim: WILEY-VCH Verlag. Schwedler, J.W. (1891). Eine Abhandlung J. W. Schwedlers über eisernen Oberbau. Centralblatt der Bauverwaltung 11: 89–96. Senft, A. (1921). Regarding (Freund, 1920). Zentralblatt der Bauverwaltung 41: 270–271. Skempton, A. W. (1956). Alexandre Collin and his pioneer work in soil mechanics. In: Landslides in Clays, by Alexandre Collin: 1846. Trans. by W. R. Schriever, pp. xi-xxxiv. Toronto: Toronto University Press. Skempton, A.W. (1981). Landmarks in early soil mechanics. In: Proceedings of the 7th European Conference on Soil Mechanics, Vol. 5, 1–26. Brighton. Skempton, A.W. (1984). Selected Papers on Soil Mechanics. London: Thomas Telford. Skempton, A. W. (1985). A history of soil properties. In: Proceedings of the Eleventh Conference on Soil Mechanics and Foundation Engineering, Golden Jubilee Vol., pp. 95-121. Rotterdam/Boston: A. A. Balkema. Smith, A. (1776). An Inquiry into the Nature and Causes of the Wealth of Nations. London: Printed for W. Strahan & T. Cadell in the Strand. Smoltczyk, U. (1992). Deutsche Gesellschaft für Erd- und Grundbau e.V.: Entstehung und Entwicklung. Geotechnik, Vol. 15, special issue, pp. 7-13. Sokolovsky, V. V. (1955). Theorie der Plastizität. Trans. from the Russian by O. Friederici. Berlin: VEB Verlag Technik. Sokolovsky, V. V. (1960). Statics of soil media. Trans. by D. H. Jones & A. N. Schoefield. London: Butterworth. Sokolovsky, V. V. (1965). Statics of granular media. Trans. by J. K. Lusher. Oxford: Pergamon Press. Stahl, H.J. (1941). Regarding (Schroeter, 1940). Die Bautechnik 19 (23): 254–256. Statens Järnvägar (ed.), (1922). Geotekniska Kommission, 1914-1922. Slutbetänkande angivet till Kungl. Järnvägsstyrelsen den 31 maj, 1922. Stat. Järnv. Geot. Medd., No. 1, Stockholm. Staudt, K.G.C.v. (1847). Geometrie der Lage. Nuremberg: Verlag von Bauer und Raspe – Fr. Korn. Swain, G.F. (1882). Mohr’s Graphical Theory of Earth Pressure. Journal of the Franklin Institute 54 (4): 241–251. Sylvester, J. J. (1860). On the pressure of earth on revetments. Philosophical Magazine, 4th series, No. 136, suppl., pp. 489-499.

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Szabó, I. (1977). Geschichte der mechanischen Prinzipien und ihrer wichtigsten Anwendungen. Basel: Birkhäuser. Taylor, D.W. (1948). Fundamentals of Soil Mechanics. New York: John Wiley & Sons. Terzaghi, K.v. (1920). Old Earth-Pressure Theories and New Test Results. Engineering News-Record 85 (14): 632–637. Terzaghi, K.v. (1925). Erdbaumechanik auf bodenphysikalischer Grundlage. Leipzig/Vienna: Franz Deuticke. Terzaghi, K.v. (1935). Fünfzehn Jahre Baugrundforschung. Der Bauingenieur 16 (3/4): 25–31. Terzaghi, K.v. (1936). Distribution of the lateral pressure of sand on the timbering of cuts. In: Proceedings of the International Conference on Soil Mechanics and Foundation Engineering, Vol. I, 211–215. Cambridge/Mass: Havard University Press. Terzaghi, K.v. (1943). Theoretical Soil Mechanics. New York: John Wiley & Sons. Terzaghi, K.v. and Peck, R.B. (1948). Soil Mechanics in engineering practice. New York: John Wiley & Sons. Terzaghi, K. v. and Peck, R. B. (1951). Bodenmechanik in der Baupraxis. Trans. from the English by Alfred Bley. Berlin: Springer. Terzaghi, K. v. and Jelinek, R. (1954). Theoretische Bodenmechanik. Trans. from the English & ed. by Richard Jelinek. Berlin: Springer. Timoshenko, S.P. (1953). History of strength of materials. With a brief account of the history of theory of elasticity and theory of structures. New York: McGraw-Hill. Truesdell, C.A. (1956). Zur Geschichte des Begriffes “innerer Druck”. Physikalische Blätter 12 (7): 315–326. Truesdell, C. A. and Noll, W. (1965). The non-linear field theories of mechanics. Handbuch der Physik, Vol. 3, Part 3, (Ed. by S. Flügge). Berlin/New York: Springer. Türke, H. (1990). Die Statik im Erdbau, 2., rev.ed. Berlin: Ernst & Sohn. Verdeyen, J. (1959). Quelques notes sur l’histoire de la Mécanique des Sols. Brussels: Presses Universitaires de Bruxelles. Vitruvius, P. M. (1981). Zehn Bücher über Architektur. Trans. & annotated by Curt Fensterbusch, 3rd ed. Darmstadt: Wissenschaftliche Buchgesellschaft. Waltking, F.-W. (1943). Über Erweiterungsmöglichkeiten der Coulombschen Erddrucklehre. Die Bautechnik 21 (6/7): 52–54. Walz, B. and Prager, J. (1979). Lösung von Erddruckproblemen nach der Elementscheibenmethode. Die Bautechnik, Vol. 56, No. 11, pp. 375-379 & No. 12, pp. 424-427. Walz, B. and Hock, K. (1987). Berechnungen des räumlichen aktiven Erddrucks mit der modifizierten Elementscheibentheorie. Report No. 6, Bergische Universität Gesamthochschule Wuppertal, Fachbereich Bautechnik. Wuppertal. Walz, B. and Hock, K. (1988). Räumlicher Erddruck auf Senkkästen und Schächte – Darstellung eines einfachen Rechenansatzes. Bautechnik 65 (6): 199–204. Weißenbach, A. (1961). Der Erdwiderstand von schmalen Druckflächen. In: Mitteilungen der Hannoverschen Versuchsanstalt für Grund- und Wasserbau, Franzius-Institut der TH Hannover, No. 19, pp. 220-338. Hannover. Weißenbach, A. (1962). Der Erdwiderstand von schmalen Druckflächen. Die Bautechnik 39 (6): 204–211.

References

Weißenbach, A. (1975). Baugruben. Teil II. Berechnungsgrundlagen. Berlin: Ernst & Sohn. Weißenbach, A. and Hettler, A. (2011). Baugruben. Teil II. Berechnungsgrundlagen., 2nd ed. Berlin: Ernst & Sohn. Wenzel, F. (ed.) (1987-1997). Erhalten historisch bedeutsamer Bauwerke, Jahrbücher, Vol. 14. Berlin: Ernst & Sohn. Weyrauch, J.J. (1878). Zur Theorie des Erddrucks. Zeitschrift für Baukunde 1 (2): 193–208. Weyrauch, J. J. (1880). Theorie des Erddrucks auf Grund der neueren Anschauungen. Allgemeine Bauzeitung, Vol. 45, pp. 63-67 & pp. 77-85. Winkler, E. (1861). Über den Druck im Inneren von Erdmassen. Dissertation, University of Leipzig. Handwritten ms. in Leipzig University archives, sig. Phil. Fak. Prom. 404. Winkler, E. (1871a). Versuche über den Erddruck. Zeitschrift des Österreichischen Ingenieur- und Architekten-Vereines 23: 255–262. Winkler, E. (1871b). Bemerkungen zum ‘Beitrag zur Theorie des Erddrucks vom Baurath Mohr’. Zeitschrift des Architekten- und Ingenieur-Vereins zu Hannover 17: 494–496. Winkler, E. (1872). Neue Theorie des Erddruckes nebst einer Geschichte der Theorie des Erddruckes und der hierüber angestellten Versuche. Vienna: R. v. Waldheim. Winkler, E. (1885). Ueber Erddruck auf gebrochene und gekrümmte Wandflächen. Centralblatt der Bauverwaltung 5: 73–76. Wittke, W. (1984). Felsmechanik. Berlin: Springer. Wittke, W. (2014). Rock mechanics based on an anisotropic jointed rock model. Berlin: Ernst & Sohn. Wittmann, W. (1878). Geometrische Erddrucktheorie und Anwendung derselben auf die Stabilitätsbestimmung von Stützmauern und Gewölben. Zeitschrift für Baukunde 1 (1): 53–76. Wolffersdorff, P.-A.v. (1996). A hypoplastic relation for granular materials with a predefined limit state surface. Mechanics of Cohesive-Frictional Materials 1: 251–271. Wolffersdorff, P.-A.v. and Schweiger, H. F. (2008). Numerische Verfahren in der Geotechnik. In: Grundbau-Taschenbuch. Teil 1: Geotechnische Grundlagen, 7th ed., ed. by Karl Joseph Witt, pp. 501-557. Berlin: Ernst & Sohn. Woltman, R. (1794). Beyträge zur hydraulischen Architektur. Dritter Band. Göttingen: Johann Christian Dieterich. Woltman, R. (1799). Beyträge zur hydraulischen Architektur. Vierter Band. Göttingen: Johann Christian Dieterich. Ziegler, M. (1987). Berechnungen des verschiebungsabhängigen Erddrucks in Sand. Veröffentlichungen des Instituts für Bodenmechanik und Felsmechanik der Universität Karlsruhe (101). Ziegler, M. (1988). Standsicherheitsberechnungen nach dem Gleitkreisverfahren. Geotechnik 11 (2): 71–79. Zienkiewicz, O.C., Valliappan, S., and King, I.P. (1968). Stress analysis of rock as a ‘no tension’ material. Géotechnique 18 (1): 56–66. Zimmermann, H. (1896). Ueber den Erddruck auf Stützmauern. Centralblatt der Bauverwaltung, Vol. 16, pp. 150-153 & 354.

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Zweck, H. (1952). Ermittlung des Erdwiderstandes vor einer Wand bei Bildung eines Gleitprismas unter Berücksichtigung der beiden seitlichen Randkörper. Dissertation, TH Karlsruhe. Zweck, H. (1953). Erdwiderstand als räumliches Problem. Die Bautechnik 30 (7): 189–193.

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3 Methods for the determination of earth pressure 3.1 Overview and bound methods 3.1.1

Overview of the methods

In literature, different limit equilibrium methods are available for the determination of earth pressure. If the wall movement is sufficiently large, then the ultimate limit state is generally reached. Depending on the type of soil, the state of the soil and the type of wall movement, slip surfaces (Fig. 3.1b) are frequently observed, which can be theoretically determined as a good approximation by kinematic mechanism methods (see chapter 3.2). Kinematic mechanism methods include the oldest, still common procedure for determining the active earth pressure, based on Coulomb (1776) and which requires a simple wedge analysis. Assuming that failure zones occur (Fig. 3.1a), static stress field methods are suitable (see chapter 3.4). For example, Rankine’s theory (1857), also an early soil-mechanic study, belongs to the static stress field methods. The determination of passive earth pressure according to DIN 4085 is also based on a static stress field theory, namely the method by Sokolowski (1965) and Pregl (2002). In contrast to the kinematic mechanism method, which only gives the resultant of earth pressure, the static stress field method also provides the earth pressure distribution. Despite the inadequacies of the kinematic and static stress field methods - for example, slip surfaces or failure zones are not always clearly defined (Fig. 3.1c) - these methods are suitable as an approximation and are therefore used in practice to determine the limit equilibrium. If these methods reach their limits, model tests and large-scale tests are appropriate (see chapter 3.5). In the last few years, the Finite Element Method (FEM) has become established. One advantage of this is the fact that kinematic as well as static limit conditions can be correctly recorded and not only the load bearing capacity, but also the serviceability and the associated deformation can be modelled (see chapter 3.6). A completely different, not continuum-mechanical approach is chosen in the case of so-called microscopic theories; see e.g. Thornton (2000). The starting points for this are the single soil grain and the grain-to-grain contacts. For numerical calculations, programs based on the Discrete Element Method (DEM) are available. In order to minimise computing time, various Earth Pressure, First Edition. Achim Hettler and Karl-Eugen Kurrer. © 2020 Ernst & Sohn Verlag GmbH & Co. KG. Published 2020 by Ernst & Sohn Verlag GmbH & Co. KG.

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a)

b)

c)

Fig. 3.1 Types of wall movement and failure according to Ohde (1992). a) Failure zone with rotation about the toe, b) Curved slip surface and rotation about the top, c) Hybrid type with deflection of the wall.

simplifications have been introduced, such as a spherical instead of polygonal particle shape, see Cundall et al. (1996). A program family called Particle Flow Code (PFC) was developed for numerical simulation. Herten thus calculates the spatial earth pressure on shaft constructions (Herten 1999). However, this method is still very complex for practical purposes. For example, Neuberg et al. (2007) require computing times between 5 and 72 hours for a calculation run to simulate the passive earth pressure in front of soldier piles with up to 45,000 elements, depending on computer performance, geometry and number of elements. The DEM and other such methods are not considered in this book. For these subjects, refer to Gudehus (1996) and Weißenbach (1985).

3.1.2

Upper and lower bounds

The solutions of static stress field and kinematic mechanism methods can be classified as lower and upper bounds in the sense of plasticity theory, see Drucker, Greenberg, Prager (1952), Koiter (1960) and Gudehus (1972). Put simply, the lower bound method states: If a static stress field can be found within the soil that is in equilibrium with its body forces (self-weight) and external loads and nowhere violates the failure criterion, then the external loads form a lower bound for those that will cause collapse. The upper bound method states: If a failure mechanism exists, for which supposing an incremental displacement the work done the external loads and the forces of its self-weight are smaller than the work done by plastic deformations in failure zones and slip surfaces, then the external loads can be classified as the upper bound for the collapse loads, cf. also Atkinson (2007) and Powrie (2013). The bound methods strictly apply to perfectly plastic materials. Amongst other things, an important requirement is the so-called normality condition, which is generally not fulfilled for soils. Nevertheless, the static stress field and kinematic mechanism methods have been well proven in many applications, especially for earth pressure, and the error seems to be relatively small. A detailed presentation and application to soils can be found in Goldscheider (2013b).

3.2 Kinematic mechanism methods for active earth pressure

3.2 Kinematic mechanism methods for active earth pressure Kinematic mechanism methods are based on simplified, rigid failure bodies, in which the movement is concentrated at slip surfaces. The geometry of the failure mechanism is varied until the resultant of active earth pressure is a maximum or the resultant of passive earth pressure is a minimum (see chapter 3.3). In case of a vertical, rigid wall with height h, which is moving away from the earth without rotation, it is possible to observe, for example in model tests with dry, dense sand, that a straight slip surface is formed in the soil (Fig. 3.2a). For reasons of simplification, the earth pressure resultant as well as the ground surface are assumed to be horizontal. Following Coulomb’s theory, a wedge with the inclination ϑ is assumed, and the acting forces are determined (Fig. 3.2b). The earth pressure resultant E′a and the self-weight G of the wedge are in equilibrium with the reacting force Q in the slip surface (Fig. 3.2c). In the ultimate limit state, the force Q is inclined at the friction angle φ against the normal on the slip surface. The friction force T points against the direction of movement, which is here, in the active case, upward. From the equilibrium of forces, one obtains tan (ϑ − φ) 1 E′a = ⋅ γ ⋅ h2 ⋅ . (3.1) 2 tan ϑ The hypothesis of Coulomb states that the relevant inclination of the slip surface ϑ is obtained when the earth pressure resultant E′a reaches the maximum Ea , which is the active earth pressure. Fig. 3.2d shows exemplary the course of the function tan (ϑ − φ) f= (3.2) tan ϑ in eq. (3.1). In order to determine the maximum, the derivation tan (ϑ − φ) tan ϑ − cos2 (ϑ − φ) cos2 ϑ df = dϑ tan2 ϑ

(3.3) tan (ϑ - φ) tan ϑ 0.3

τ σ E′a

h ϑ

E′a Q

φ N T

b)

c)

φ = 40°

0.2 G

0.1

ϑ-φ

0.0

Q

a)

φ = 30°

Ka 40° 50° 60° 70° 80° ϑ ϑa

d)

Fig. 3.2 Example of active earth pressure according to Gudehus (1981). a) Sliding wedge, b) Forces on sliding wedge, c) Force polygon, d) Earth pressure coefficient depending on ϑ with maximum.

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is set to zero, giving the solution φ (3.4) ϑ = ϑa = 45 + 2 for the angle of the slip surface ϑa of active earth pressure. By substituting ϑ = ϑa in eq. (3.1), the resultant of the active earth pressure is obtained ( φ) 1 (3.5a) Ea = ⋅ γ ⋅ h2 ⋅ tan2 45 − 2 2 with the earth pressure coefficient ( φ) Ka = tan2 45 − . (3.5b) 2 At φ = 30∘ , this gives Ka = 1/3. At φ = 0, Ka = 1 and the soil has the same coefficient as a liquid. According to Gudehus (1981), the hypothesis of Coulomb is equivalent to the principle of minimum safety. Accordingly, the relevant slip surfaces are inclined in such a way that for a given supporting force, the safety factor with respect to this force becomes the minimum. In other words, the failure mechanism which leads to the minimum safety level is always found in nature. In case of active earth pressure, the wedge suffices as a failure mechanism, with only some few exceptions. Müller-Breslau (1906) has extended the equations for any inclinations of the wall, of the ground level and of the earth pressure resultant. He also derived the formulas, which are now customary in practice, for calculating the active earth pressure, see chapter 4. For the general case of earth pressure due to soil self-weight, surcharges and cohesion, the inclination of the slip surface ϑa depends also on the cohesion, as proved by Groß (1981). The equations can no longer be split additively into shares for soil self-weight, surcharge and cohesion, see chapter 17.2. The wedge is also suitable for determining the active earth pressure in case of surcharges and a discontinuous ground level (Fig. 3.3). As in the simple case of Fig. 3.2, the inclination ϑ of the slip surface is varied until a maximum is reached, i.e. the active earth pressure. For details see Gudehus (1996), Weißenbach (1985) and DIN 4085.

ϑ

ϑ a)

b)

Fig. 3.3 Sliding wedge with variation of the angle of slip surface for the determination of active earth pressure. a) with surcharge, b) with discontinuous ground level.

3.2 Kinematic mechanism methods for active earth pressure

Depending on the limit conditions, curved slip surfaces are also observed under active earth pressure. Ohde, for example, finds in model tests with rotation about the top the collapse mechanism shown in Fig. 3.1b (Ohde 1992, 1948). The active earth pressure with the kinematic mechanism method and curved slip surfaces can generally be determined with circles and logarithmic spirals. The logarithmic spiral has the advantage that all forces on the slip surface inclined at the angle φ go through the pole and thus do not contribute to the momentum equilibrium around the pole (Fig. 3.4a). In this respect, the distribution of the normal and shear forces along the slip surface has no influence on the result. With a known self-weight G and lever arm lG of the sliding blocks and predefined inclination δa of the earth pressure resultant E′ag with the lever arm lE , assumed on the basis of empirical values, E′ag results from E′ag = G ⋅

lG . lE

(3.6)

The slip surfaces must be varied until E′ag reaches the active earth pressure force Eag (Fig. 3.4c). Since the self-weight G can only be determined with difficulty from closed formulas, it is appropriate to divide the body into individual slices (Fig. 3.4b). For details, see Weißenbach (1985). In case of a circular collapse mechanism, the direction of the reaction forces Q in the slip surface may be determined according to Krey’s assumption of a friction circle (Krey 1926), see chapter 3.3. From a strictly theoretical point of view, circular slip surfaces are only kinematically possible for volumetric deformations with constant volume, whereas logarithmic spirals are correct for soils with dilatancy (Gudehus 1996). These differences are usually neglected in practice. Composite mechanisms with several plane slip surfaces or several circular slip surfaces are mainly used for passive earth pressure, please refer here to

Pol

IGi

Pol

ψ r1

maxE′ag

IE φ

φ

φ

r IG h

δa Za

a)

G

r2

φ

δa E′ag Q′ag φ

Zag

b)

E′ag

δa E′ag

Gi φ φ

φ

Zag

QaGi c)

Fig. 3.4 Determination of earth pressure with spiralled slip surfaces according to Weißenbach (1985). a) Forces on total sliding blocks, b) Segmentation of sliding blocks into slices, c) Determination of the most unfavourable slip surface.

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chapter 3.3. Any number of slip surfaces can be considered with the Kinematic Element Method according to Gußmann (1983). The numerical implementation and the application of the procedure are described by Gußmann and Schanz (1983), current developments by Gußmann, König and Schanz (Gußmann et al. 2016).

3.3 Kinematic mechanism methods for passive earth pressure Analogously to the active earth pressure (Fig. 3.2), the resultant of the passive earth pressure Ep can also be determined for a sliding wedge according to Coulomb in case the wall is displaced against the earth. By reversing the direction of movement, the direction of the friction force changes, and the force Q is applied as shown in Fig. 3.5a. From the force polygon in Fig. 3.5b, one obtains depending on ϑ tan(ϑ + φ) 1 ⋅ γ ⋅ h2 ⋅ . (3.7) 2 tan ϑ According to Coulomb, the inclination angle ϑ is decisive where E′p reaches a minimum. From the condition dE′p /dϑ = 0, one obtains E′p =

φ . (3.8) 2 With ϑ = ϑp , the resultant of the passive earth pressure is calculated with ( φ) 1 (3.9) Ep = ⋅ γ ⋅ h2 ⋅ tan2 45 + 2 2 giving the earth resistance coefficient ( φ) . (3.10) Kp = tan2 45 + 2 In the case of a vertical wall, horizontal ground level and horizontal earth pressure force, ( φ) 1 1 Kp = tan2 45 + = . (3.11) ( )= φ 2 K a tan2 45 − ϑ = ϑp = 45 −

2

E′p

h

G

E′p

ϑ a)

G

Q Q

ϑ+φ

φ b)

Fig. 3.5 Sliding wedge for passive earth pressure. a) Sliding wedge, b) Force polygon.

3.3 Kinematic mechanism methods for passive earth pressure

As with the active earth pressure, the analysis of the sliding wedge can also be extended to inclined walls, inclined ground levels and inclined earth pressure resultants (see chapter 6). In contrast to the active earth pressure problem, particularly at high friction angles and earth pressure inclination angles in the range of δp = -φ, unrealistically high Kp -values can result from the analysis of the sliding wedge, so that curved slip surfaces are decisive (see chapter 6). In the case of a translation of the wall, the passive earth pressure can be approximately determined by rigid multi-body mechanisms with plane slip surfaces. Fig. 3.6 shows an example with a two-block mechanism according to Gudehus (1996). For the two-block mechanism, ϑ1 , ϑ12 and ϑ2 in Fig. 3.6 must be varied until E′p reaches a minimum and thus the limit value Ep of the passive earth pressure force. The earth resistance coefficients Kpt for translation thus obtained are shown in Fig. 3.7 for different inclinations δp . Significantly higher Kp -values are obtained for the sliding wedge according to Coulomb determined for δp = -φ, as the dotted line in Fig. 3.7 shows. The procedure in the so-called Kinematic Element Method (KEM) is illustrated by the following example according to Gußmann et al. (2016). Required is the plane earth resistance for translation on a vertical wall without wall friction in a terrain inclined at φ/2 for a non-cohesive soil with φ = 30∘ . In this case, Kpgh = 4.6479 for the sliding wedge according to Coulomb and for curved slip surfaces according to Pregl/Sokolowski (see chapter 6.2) Kpgh = 4.4073. Note: The earth resistance coefficients are listed with four digits behind the decimal point to illustrate the differences. First, a failure mechanism with one element is examined (Fig. 3.8a). The corresponding kinematics are shown in Fig. 3.8b. The accuracy can be gradually increased by increasing the number of elements. For example, for four elements Kpgh = 4.4625 (Fig. 3.8e and f ) and for twelve elements Kpgh = 4.477 (Fig. 3.8i and j). For 24 elements, the result from the kinematic element method approaches Kpgh = 4.4444, which forms an upper limit according to the bound methods and approximates the Kpgh -value of Pregl/Sokolowski, which is a lower bound (chapter 3.1.2) based on the static stress field method. Refer to Goldscheider (2013a) for details on bound methods.

E′p

G2 h

a)

φ

E′p G2 ϑ1

Q1

Q12

Q2 ϑ2

φ

Q1

Q12

G1

φ

ϑ12

Q2

G2

b)

Fig. 3.6 Determination of passive earth pressure with two-block mechanisms according to Gudehus (1996). a) Sliding blocks, b) Force polygon.

151

152

3 Methods for the determination of earth pressure

Kpt

Sliding wedge according to Coulomb at δp = –φ

40

δ = –φ

Fig. 3.7 Coefficient of passive earth pressure at translation for vertical wall and horizontal surface according to Gudehus (1996), determined with two-block mechanisms.

30

–2φ/3 20

–φ/3 10 0 φ/3 2φ/3 φ

0 25°

30°

1

a)

35°

40°

45° φ

1

Kpgh = 4.6479 Displacement boundary

b)

Kpgh = 4.4752 c)

d)

Kpgh = 4.4625 e)

f)

Kpgh = 4.4507 g)

h)

Kpgh = 4.447 i)

j)

v1

Fig. 3.8 Calculation of earth resistance on a vertical wall without wall friction according to Gußmann et al. (2016). a), c), e), g), i) Failure mechanism; b), d), f ), h), j) Kinematics.

3.3 Kinematic mechanism methods for passive earth pressure

sin

φ

r1



r ϑ1

E′p G

hE a)

G

E′p

h

Q

Q b)

Fig. 3.9 Determination of the passive earth pressure with circular slip surfaces according to Gudehus (1996). a) Sliding blocks with friction circle according to Krey, b) Force polygon.

Curved slip surfaces, e.g. circular slip surfaces, are decisive for rotation of the wall about the top of the wall or another upper point (Fig. 3.9). First, the intersection point of G with E′p is determined for a given centre of slip circle, a pre-set inclination δp and an assumed height hE of the earth pressure force E′p . The direction of the resulting force Q in the slip surface is obtained by the fact that Q has to intersect at a point with the lines of action of G and E′p for equilibrium and at the same time, it is the tangent line of the friction circle with the radius r ⋅ sinφ according to Krey (1926). As before, the geometry of the slip circles has to be varied until a minimum is obtained for E′p . The Kpr -values for rotation determined in this way by Groß (1981) are shown in Fig. 3.10. If one compares the earth resistance coefficients, then it is found for positive inclination angles δp : Kp according to Coulomb = Kpt in Fig. 3.7 = Kpr in Fig. 3.10. On the other hand, for negative inclination angles δp < 0, according to Coulomb Kp > Kpt > Kpr , cf. Gudehus (1981). If one follows this result, one would have to determine the most unfavourable passive earth pressure coefficient in practice depending on the movement of Fig. 3.10 Coefficient of passive earth pressure at rotation for a vertical wall, horizontal ground level and varying earth resistance inclination δp according to Groß (1981) and Gudehus (1996).

Kpr 30 δ = –φ

20

–2/3φ

–1/3φ

10

0 25°

0 1/3φ 2/3φ φ 30°

35°

40°

45° φ

153

154

3 Methods for the determination of earth pressure

the wall. This would be very cumbersome and is also not very realistic, because the wall movements are mostly unknown, and other, arbitrary mixed forms can occur. In German practice, the influence of the kinematics of the failure body is usually neglected, and the methods are used which provide the smallest Kp -value. For example, at δp = -φ, one does not use the Kp -values which result from circular slip surfaces with the kinematic mechanism method, but the Kp -values from static stress field methods, e.g. according to Caquot/Kérisel or Pregl and Sokolowski, where the kinematics are not known (see chapters 3.4 and 6).

3.4 Static stress field methods 3.4.1

Fundamentals

While rigid bodies and small shear bands are used in the kinematic mechanism method, in static stress field methods one assumes that plastic field zones of failure are formed. If the Mohr-Coulomb limit condition is used, the vertical stresses σz , the horizontal stresses σx and the shear stresses τzx in the plane case in Cartesian coordinates must comply with the limit condition (Gudehus 1996). √ ( σ − σ )2 σ + σx z x + τ2zx = z ⋅ sin φ + c ⋅ cos φ. (3.12) 2 2 At the same time, the equilibrium conditions have to be met 𝜕σz 𝜕τ + zx = γ (3.13a) 𝜕z 𝜕x 𝜕σx 𝜕τ + zx = 0 (3.13b) 𝜕x 𝜕z taking the specific weight of the soil γ into account. Equations (3.12) and (3.13) can be converted into two equations for the mean stress σm and the direction ψ of the largest principal stress σ1 dσm + 2 ⋅ σm ⋅ tan φ ⋅ ds1 dσm − 2 ⋅ σm ⋅ tan φ ⋅ ds2

dψ cos(ψ + μ) = −γ ⋅ ds1 cos φ dψ cos(ψ − μ) =γ⋅ ds2 cos φ

longitudinal s1

(3.14a)

longitudinal s2

(3.14b)

using the abbreviation μ = π/ψ = –φ/2, see Gudehus (1996) and with s1 and s2 being the characteristics of the system. On these lines, the shear stress reaches the shear strength, i.e. the ratio of shear stress to normal stress becomes a maximum. Equations (3.14a) and (3.14b) go back to Kötter, see Kézdi (1962). In the general case, these equations can be solved with the method of characteristics of Sokolowski (1960). Pregl has followed this path and derived coefficients for the passive earth pressure, which are included in DIN 4085:2011-05. The solutions by Rankine and Caquot/Kérisel represent special cases of equation (3.14), which are based on simplifying requirements and assumptions.

3.4 Static stress field methods

The failure zones presupposed in the static stress field methods can only occur under certain limit conditions, e.g. in the Rankine case with the active earth pressure on a wall and rotation about the toe. At the same time, for example, the earth pressure inclination must be the same as the ground inclination. In most cases, small shear bands or combinations of small shear bands and failure zones occur. In this respect, the validity of static stress field methods is limited. On the other hand, kinematic mechanism methods can lead to good approximations, even if the assumed collapse mechanism is not observed (Gudehus 1996). Static stress field methods also have the restriction that from a theoretical point of view the kinematic requirements for the stress-limit condition have to be proved. In most of the solutions known from the literature, no statements are made about the kinematics. Goldscheider (2017, 2000) developed a procedure based on kinematics, in which it is possible to derive zones of failure. This is based on the theory by Spencer, which leads to a coincidence of static and kinematic characteristics, see chapter 3.4.5. 3.4.2

Rankine’s solution

Rankine (1857) acts on the following assumptions: • The soil is non-cohesive and homogeneous and has specific weight γ. • The soil is deformed in such a way that the limit condition in Eq. (3.12) is achieved everywhere and the principal stresses have the same direction in the considered area. For a vertical wall with horizontal earth pressure and horizontal ground level and a linear increasing distribution of the earth pressure with the depth, this results in: ( φ) (3.15) eah = γ ⋅ z ⋅ tan2 45 − 2 for the active case and ( φ) (3.16) eph = γ ⋅ z ⋅ tan2 45 + 2 for the passive case. As this example shows, static stress field methods also provide an earth pressure distribution in contrast to kinematic mechanism methods. The earth pressure coefficients ( φ) (3.17) Kah = tan2 45 − 2 for the active case and ( φ) (3.18) Kph = tan2 45 + 2 for the passive case are identical to the results for Coulomb’s sliding wedge, which only applies to this particular case. In general, different values are obtained. From the equations (3.15) and (3.16), one obtains the earth pressure resultant from the soil self-weight for a wall of height h Eag =

1 ⋅ γ ⋅ h2 ⋅ Kah 2

(3.19)

155

156

3 Methods for the determination of earth pressure

and 1 (3.20) ⋅ γ ⋅ h2 ⋅ Kph . 2 The additive splitting of the total earth pressure force Ea into three components for soil self-weight (index g), infinite uniformly distributed surcharge (index p) and cohesion (index c) also applies to the considered particular case with Epg =

Ea = Eag + Eap + Eac

(3.21a)

as well as Ep = Epg + Epp + Epc

(3.21b)

with identical earth pressure coefficients for the shares of soil self-weight and infinite uniformly √ distributed surcharge √ as well as the earth pressure coefficients Kac = −2 ⋅ c ⋅ Kah and Kpc = 2 ⋅ c ⋅ Kph , see also Gudehus (1996). The characteristics in the Rankine case, which are often equated with slip surfaces, are two parallel series of straight lines with the inclinations ( φ) (3.22a) ϑa = ± 45 + 2 for the active case and ( φ) ϑp = ± 45 − (3.22b) 2 for the passive case, cf. Fig. 3.11. If appropriate limit conditions are chosen, the slip surfaces can be observed in tests, see chapter 3.5.7. In the case of inclined ground level and inclined earth pressure, the equilibrium and limit conditions are only fulfilled if the earth pressure acts parallel to the ground level. In addition, it can be shown that only a vertical surcharge is compatible with a Rankine zone. For details see Gudehus (1996); Kézdi (1962). This example shows that the Rankine solution is severely restricted in its applicability. 3.4.3

Theory of Boussinesq/Résal/Caquot

In the theory of Boussinesq/Résal/Caquot it is assumed that the soil stresses increase linearly with the depth on rays from the top of the wall (Gudehus 1996; Kézdi 1962) (see Fig. 3.12). Based on this, Caquot/Kérisel published extensive eah

eph

h ϑp a)

ϑp z

b)

z

Fig. 3.11 Rankine’s solution for a vertical wall with horizontal ground level and horizontal earth pressure according to Gudehus (1996). a) Active case, b) Passive case.

3.4 Static stress field methods

Fig. 3.12 Centre of similarity at the top A of the wall. A

β

E δ

E α

tables in the year 1948, which still have inconsistencies mainly at earth pressure inclinations δp = -φ. For this reason, one should rather use the tables from Caquot/Kérisel/Absi from 1973 (Caquot et al. 1973). 3.4.4

Solution by Pregl/Sokolowski

Pregl solved the pair of equations (3.14) by means of the method of characteristics of Sokolowski (1960), and developed extensive tables and analytical equations. The cases of passive earth pressure from soil self-weight, infinite uniformly distributed surcharge and cohesion for various inclinations of earth pressure, wall and ground level are determined. In addition to the earth pressure coefficients, the shape of the slip surface can also be determined (Fig. 3.13). The results of Pregl were included in the appendices of DIN 4085:2011-05 and then directly included in the standard text of DIN 4085:2017-08 (see chapter 6). 3.4.5

Analysis of Goldscheider

Starting from the equations of Kötter-Massau and the theory of Spencer (1982), Goldscheider developed an approximation method, which permits the combination of zones with plastified stress fields and rigid body mechanisms. Due to Spencer’s requirement that static and kinematic characteristics coincide, it is possible to make statements on kinematics even in failure zones. Fig. 3.14, for example, shows a fan-shaped failure zone consisting of radiating s2 -characteristics and logarithmic spirals for s1 -characteristics. The fan-shaped zones are found in limit equilibrium analysis of slopes and foundations as Pv

h

ep

Ep

α

ϑ2

ϑ1

β

IB r1

δp ω

ν

γ,c,φ

r2 ϑ3 z

I sp

Fig. 3.13 Earth resistance and sliding blocks according to Pregl (2002).

x

157

158

3 Methods for the determination of earth pressure

Fig. 3.14 Deformation of a fan-shaped failure zone according to Goldscheider (2000).

O ψo ψo-ψo A1

O′ 6 O′1

V′A6 = V′A1

VA1

A6

90°-φ

γ2(p1) r = ro P1 S2

P2 γ2(p6)

P3

r = ro •e(ψ-ψo)tan φ

P4 P6

S1

P5

transition zones between two differently oriented Rankine zones. In Fig. 3.14, all radiating s2 -lines rotate about their respective toes P1 , P2 to P6 on the s1 -line. Fig. 3.15 shows the use of the procedure for a double-anchored wall rotated about the toe. The failure mechanism consists of the Rankine zones I, II and V, the fan-shaped zone III and the rigid sub-body IV. The solution in Fig. 3.15 is exact regarding its kinematics and has only small, but negligible errors in the statics. For details, see Goldscheider (2000). 3.4.6

Approach of Patki/Mandal/Dewaikar

Patki/Mandal/Dewaikar (Patki et al. 2015) act on the assumption of a logarithmic spiral as a failure surface (Fig. 3.16) and produce static equilibrium at the failure body. Walls in non-cohesive soil are investigated with varying inclinations of the wall, earth pressure and ground level. The position of the pole as well as the initial P

x

V

I

IV

β

x

O S

III S 2

S1

α μ

μ

I

H h

R

II

S2

Io *

σ1

Io

S1 90°-φ

Fig. 3.15 Twice anchored wall rotated about the toe with composite failure zone mechanisms, I to III and V failure zones, IV rigid zone according to Goldscheider (2000).

3.5 Tests and measurements

Final radius

Pole of the log spiral O θm

H

ϕ

θcr

i

B

D

Soil reaction (R) Normal

Tangent

Initial radius Ppγ

1./3

λ>0 δ

1.

W ϕ

θν

Tangent J

RJD Normal

Failure surface (log spiral)

Fig. 3.16 Proposed failure mechanism with original terms according to Patki et al. (2015).

radius and end radius of the logarithmic spiral are varied until a minimum for the earth resistance coefficients is obtained from soil self-weight. In the Patki et al. case, this is named Kpγ . The results accord very well with upper and lower bound solutions of other authors (Chen 1975; Kobayashi 1998; Soubra 2000), as shown in Fig. 3.17. The contribution by Patki et al. also includes extensive earth pressure tables. It should be noted that the earth resistance coefficients can increase significantly with high friction angles, wall inclination and earth pressure inclination (Fig. 3.18). However, this does not appear to be a defect in the proposed method, which as the comparison in Fig. 3.18 shows, also includes test results (pushing methods A and B).

3.5 Tests and measurements 3.5.1

Fundamentals and scaling laws

In addition to static stress field and kinematic mechanism methods as well as the Finite Element Method, small- or large-scale tests are also suitable for the determination of the active and passive earth pressure, cf. Gudehus (1996); Terzaghi (1934). Small-scale model tests are generally carried out as so-called 1g tests at 1g gravity acceleration. In the centrifuge, the pressure level can be set to that of an n-fold larger prototype by increasing to n ⋅ g in the model. This leads to a similar pressure level. Small-scale tests have the advantage that extensive parameter studies can be carried out with comparatively little effort, although difficulties can arise from the similarity requirements of the scaling laws. Centrifuge tests are more complex, but they have the advantage of model similarity in the pressure level. The most complex methods are large-scale tests, which are hardly suitable for parameter studies. Very valuable results can be obtained by means of measurements on existing constructions and components, e.g. on excavation walls. Ideally, one will consider various methods. As an example, reference is made to Gässler, who developed a design method for soil nailing based on the kinematic mechanism method and carried out extensive model tests as well as some

159

3 Methods for the determination of earth pressure

8 ϕ = 30°

6 Kpγ

Shiau et al. (2008) [Lower bound] ϕ = 25°

Shiau et al. (2008) [Upper bound] Proposed analysis

4 ϕ = 20°

2 0

1/3

2/3

1

δ/ϕ

a) 45

ϕ = 45°

40 35 30 Kpγ

160

Shiau et al. (2008) [Lower bound]

25 ϕ = 40°

20 15

ϕ = 35°

Shiau et al. (2008) [Upper bound] Proposed analysis

10 5 0 0 b)

1/3

2/3

1

δ/ϕ

Fig. 3.17 Comparison of the proposed values for Kpγ with the theoretical results according to Shiau et al. (2008) for the case of a vertical wall (λ = 0) with horizontal backfill (i = 0∘ ). a) φ = 20∘ , 25∘ and 30∘ , b) φ = 35∘ , 40∘ and 45∘ according to Patki et al. (2015).

large-scale tests for validation (Gässler 1987). In recent years, numerical methods, especially the FEM, have been given high priority, see chapter 3.6. Generally, this should also be validated by tests. When carrying out small-scale model tests, the relevant dimensionless parameters in the model (index M) and in the prototype (index P) must be the same in order to achieve model similarity, see Görtler (1975). For example, • there is geometric similarity when the ratio of the lengths to each other in the model and in the prototype are identical, • Bending stiffnesses must be the same in dimensionless form.

3.5 Tests and measurements

120

ϕ = 42°, λ = – 30°

Proposed analysis

100

Soubra (2000) Chen (1975)

Kpγ

80

Sokolovski (1960)

60

Pushing method A

40

Pushing method B

20

Pushing method B (Residual)

0 0.0

0.3

0.6

0.9

tan δ 200

ϕ = 42°, λ = – 40°

180

Kpγ

160 140

Proposed analysis

120

Soubra (2000)

100

Chen (1975)

80

Sokolovski (1960)

60

Pushing method A

40

Pushing method B

20 0 0.0

0.3

0.6

0.9

tan δ 500 ϕ = 42°, λ = – 60°

450 400 350

Proposed analysis

Kpγ

300

Soubra (2000)

250

Chen (1975)

200

Sokolovski (1960)

150

Pushing method B

100 50 0 0.0

0.6

0.3

0.9

tan δ

Fig. 3.18 Comparison of the results with other theoretical and experimental results according to Patki et al. (2015) (Terms see Fig. 3.16).

161

162

3 Methods for the determination of earth pressure

In individual cases, it may be difficult to identify the relevant parameters, and simplifications are often necessary. In many cases, for example, 1g model tests are sufficient even if the pressure level is not represented as in the model. For examples, see Gudehus (1996); Hettler and Gudehus (1985); Hettler (1981, 2010). The initial state in model tests is particularly significant. In order to achieve defined conditions in sandy soils, the flotation technique has proved its worth. In this case, pressure at rest and a more or less uniform density are achieved through continuous pouring in layers. If, on the other hand, the soil is compressed in layers, the soil is preloaded. This may lead to a relatively greater effect of high horizontal stresses at the smaller scale and the scaling laws are not complied with. Special considerations are required for moist sand and gravel. For soil with capillary cohesion cK and specific weight γ, the requirement in tests on walls with different heights h is obtained by ( ) ) ( cK cK = . (3.23) γ⋅h M γ⋅h P This scaling law cannot be complied with in a 1g model test at constant γ and cK and at different wall heights. Particularly in relatively small-scale tests, the capillary cohesion can have a huge influence on the earth pressure, as can easily be demonstrated using the classic earth pressure formulas. The discussion outlines that particular considerations are necessary for the evaluation of model tests with compressed or moist sand and gravel. In individual cases, large errors are possible when transferring to other wall heights than in the tests. For these reasons, 1g model tests with cohesive soils are also particularly difficult. In these cases, the influence of the initial stress state and the cohesion must be carefully analysed. A frequently mentioned objection to model tests concerns the influence of the progressive failure on the peak load of the bearing capacity, see Gudehus (1996). This applies in particular to small-scale tests in dense sand and gravel. If it is assumed that in the case of progressive failure in non-cohesive soils, shear bands develop, whose influence can be described by a characteristic thickness or a characteristic length with a multiple of a typical grain diameter dK , the following scaling law can be derived ( ) ( ) dK dK = . (3.24) lK M lK P In equation (3.2.4), lK denotes a typical construction-related length, e.g. the wall height h. If the influence of the parameter dK /lK on the active or passive earth pressure is to be clarified, practical difficulties are encountered. For example, in 1g model tests at different geometric scales, the influences of pressure level and progressive failure occur simultaneously, and a separation is hardly possible. In principle, this would demand, for example, centrifuge tests with a constant pressure level and a different geometric scale. However, the size of available centrifuges limits this. Numerical tests with a constant pressure level and different wall heights are also possible. But here too, problems are to be expected from a numerical point of view. As Gudehus (1996) remarked: there still seems to be no quantitatively secured theory of progressive failure. However, there are indications that the course of peak loads for passive earth pressure can be explained

3.5 Tests and measurements

depending on the wall height, e.g. solely due to the influence of the pressure level, see Hettler (1997). The same applies to foundations (Hettler and Gudehus 1988), according to which progressive fracture seems to occur after the peak. 3.5.2

Evaluation of test results and application of scaling laws

In earth pressure tests under plane strain conditions using dry sands and gravels with the specific weight γ and the friction angle φ, the resultant of the active and the passive earth pressure can be presented in dimensionless form as 2 ⋅ Ea = fa (φ) γ ⋅ h2 2 ⋅ Ep = fp (φ). γ ⋅ h2

(3.25a) (3.25b)

The functions fa and fp only depend on the friction angle and the type of wall movement. In principle, it is sufficient to carry out tests at a wall height, e.g. with a model wall of 10 cm height, for a certain type of wall movement and a soil of a certain density. Assuming a constant friction angle, the same value is obtained for the functions fa and fp for a 10 m high wall under identical conditions, which corresponds to the earth pressure coefficients Ka and Kp fa = Ka

(3.26a)

fp = Kp .

(3.26b)

This assumption corresponds to the usual procedure in practice; most model tests are evaluated on this basis. As a rule, this is sufficient for practical purposes. In fact, in many non-cohesive soils a more or less pronounced decrease of the friction angle is observed with an increasing pressure level. This means that the functions fa and fp in equation (3.25) depend on material parameters with the dimension of a stress, which is usually unknown. Formally, the dependency on the stress level can also be presented as follows without knowledge of the material parameters ( ) 2 ⋅ Ea h = g φ, (3.27a) a γ ⋅ h2 h0 ( ) 2 ⋅ Ep h = g φ, (3.27b) p γ ⋅ h2 h0 where h0 denotes an arbitrary reference height, e.g. 1 m. If a representative mean friction angle φm is used to calculate the earth pressure coefficient, which decreases with increasing wall height because of the higher pressure level, Ka increases in the active case with increasing wall height. In the passive case, on the other hand, Kp decreases (Hettler 1997). For this, reference is also made to the following examples and the tests carried out by Bartl. Formally, equation (3.25) can also be transferred to normally consolidated, cohesive soils by inserting the angle φNC . However, due to the low consistency

163

164

3 Methods for the determination of earth pressure

of normally consolidated soils, it is difficult to solve the problem with models. There is also the problem of establishing a defined initial state in the model. In the case of over-consolidated soils with friction and cohesion c, one obtains analogously to equation pair (3.25) ( ) 2 ⋅ Ea c = ha φ, (3.28a) γ ⋅ h2 γ⋅h ( ) 2 ⋅ Ep c = h φ, . (3.28b) p γ ⋅ h2 γ⋅h As can be seen from the equation pair (3.28), the parameter c/(γ ⋅ h) is no longer constant in the case of an n-fold smaller model in the 1g test and the use of the same soil. In order to achieve model similarity, one would have to reduce the cohesion with the factor n, which is usually impossible, or increase the specific weight γ to n times. The latter requirement can be fulfilled in the centrifuge. In practice, one usually assumes an additive approach for the shares from soil self-weight and cohesion. This is a simplification of equations (3.28a) and (3.28b), which is not universally valid. For this, reference is made, e.g. to the investigations of Groß with the Coulomb model (Groß 1981; Gudehus 1996). However, for practical purposes, an additive approach is often sufficient - even for the share of surcharge. If this is applied, 1g model tests can again be useful. 3.5.3 Example: active earth pressure under plane strain conditions from soil self-weight Ohde carried out extensive tests on active earth pressure (Ohde 1992), showing the influence of the type of wall movement on the size and distribution of the earth pressure. In principle, four basic types of wall movement can be distinguished (see Fig. 3.19) • • • •

Parallel displacement, Rotation about the toe, Rotation about the top, Deflection.

h

h

b)

h

s

s

s a)

h

c)

s d)

Fig. 3.19 Basic forms of wall movement according to Weißenbach (1985). a) Parallel displacement of the wall, b) Rotation about the toe, c) Rotation about the top, d) Deflection of the wall.

3.5 Tests and measurements

Eag

Eag

Eag Zag

Zag 1

Zag = 3 · h

a)

Zag

Zag = 0.50·h to 0.55·h

Zag = 0.40·h to 0.45·h

b)

c)

Fig. 3.20 Distribution of active earth pressure and stresses of slip surfaces from soil self-weight for different types of wall movement according to Ohde, compiled by Weißenbach (1985). a) Rotation about the toe, b) Rotation about the top, c) Deflection of the wall. Fig. 3.21 Distribution of active earth pressure from soil self-weight with parallel movement of the wall. a) according to Terzaghi and Lehmann, b) according to Jáky and Abouleid.

Eag

Eag

Zag = 0.40·h to 0.45 · h a)

Zag = 0.40·h to 0.45 · h b)

The corresponding distributions as well as the point of action of the earth pressure resultant are shown in Figures 3.20 and 3.21. According to Ohde’s theoretical considerations, the value of the resulting earth pressure force in the active case is: • Rotation about the toe: • Rotation about the top: • Deflection of the wall:

100% 115% to 120% 105%

Weißenbach adds and, in the case of the parallel movement, concludes that the resultant should hardly exceed 105%. The wall movements required to mobilise the active earth pressure are, e.g. between 1.5‰ and 3‰ of the wall height for medium density, see e.g. Weißenbach (1985). In practice, all possible intermediate forms of wall movement can occur. In view of the small differences in the value of the resultant and the uncertainties in the estimation of the actual wall movement, it is customary in practice to apply the total earth pressure according to its magnitude irrespective of the type of support, movement and deformation of the wall. A detailed discussion with extensive citations from the literature on this subject can be found at Weißenbach (1985). 3.5.4 Example: passive earth pressure under plane strain conditions from soil self-weight In the case of passive earth pressure, much larger movements occur than in the active case, and the differences can be considerable for the basic types of wall movement. Numerous publications including tests are known from literature, e.g. by Terzaghi (1920), Streck (1926), Franzius (1928a, 1928b), Tschebotarioff

165

166

3 Methods for the determination of earth pressure

(1953), James/Bransby (1970, 1971), Roscoe (1970), Vogt (1984) and Mao (1995). A summary and evaluation can be found in Besler (1998) as well as in Bartl (2004). In addition, Gutberlet (2008), in contrast to studies done so far, investigated the earth resistance problem for homogeneous soil, but also for stratification. From these numerous publications some of Bartl’s results (Bartl 2004) are presented below. These tests are characterised by improved technology compared to previous studies. A test box with large dimensions was used with high resolution of the measurements. The total width was 1 m and the height was 0.564 m. The actual test wall had a width of 0.40 m and was equipped with eight earth pressure measuring devices distributed over the height. Extensive 1g test series were carried out with the basic types of wall movement shown in Fig. 3.22 and different densities of the “Dresdner Sands” used. These studies were supplemented by centrifuge tests with n ⋅ g at the University of Natural Resources and Life Sciences, Vienna, in order to clarify the influence of the pressure level. Figures 3.23 to 3.25 show some typical test results. The relevant dimensionless coefficient K′ph =

2 ⋅ E′ph

(3.29)

γ ⋅ h2

is shown in each case for the horizontal component E′ph of the mobilised earth resistance depending on the displacement s related to the wall height h according to Fig. 3.22. The dependency of K′ph on the density ID is evident both in parallel displacement (Fig. 3.23) and in a rotation about the top (Fig. 3.24). With rotation about the toe (Fig. 3.25), no limit value is reached in the sense of a peak or plateau. The type of wall movement has a considerable influence both on the achieved maximum values and on the associated displacements. In this respect, the influence of the type of wall movement is not generally negligible in practical cases in contrast to the active event. Due to the considerable displacements, which can be 100 times higher than in the active case, the compatibility of the deformations with the purpose of the structure must always be checked in the case of passive earth pressure.

s

s

s

h

a)

b)

c)

Fig. 3.22 Basic types of wall movement under passive earth pressure. a) Rotation about the toe, b) Parallel displacement, c) Rotation about the top.

3.5 Tests and measurements

Relative density ID

K′ph 14.0 12.0 10.0 8.0 6.0 4.0 2.0 0.0 0.00

0.05

0.10

0.15

0.20

0.25 s/h

0.19 0.19 0.26 0.35 0.43 0.45 0.53 0.55 0.59 0.62 0.63 0.81 0.81 0.82 0.82 0.82

Fig. 3.23 Mobilisation of the earth pressure force: 1g tests, parallel displacement, aluminium wall surface, variation of the initial relative density ID according to Bartl (2004). K′ph 7.0 Relative density ID . 0.26 0.47 0.65 0.75 0.87

6.0 5.0 4.0 3.0 2.0 1.0 0.0 0.00

0.05

0.10

0.15

0.20

0.25 s/h

Fig. 3.24 Mobilisation of the earth pressure force: 1g tests, rotation about the top, aluminium wall surface, variation of the initial relative density ID according to Bartl (2004).

The normalised horizontal component Kph of the passive earth pressure Eph at the ultimate limit state 2 ⋅ Eph (3.30) Kph = γ ⋅ h2 depends on the pressure level, as shown by the parallel centrifuge tests at n ⋅ g (Fig. 3.26). On the horizontal axis, the parameter h* ⋅ ng is shown in relation to a wall height of 1 m. The wall height in the test is h* and ng is the n-fold of acceleration due to gravity. Fig. 3.26 shows both the results at 1g (symbol 1g-V) and in the centrifuge at n ⋅ g (symbol Z.-V.) for various wall coatings. Alu stands for an aluminium surface, S 220 for “abrasive paper S 220”. The pressure dependency of

167

168

3 Methods for the determination of earth pressure

K′ph Relative density ID

14.0

0.28 0.29 0.41 0.51 0.55 0.56 0.59 0.60 0.61 0.80 0.82

12.0 10.0 8.0 6.0 4.0 2.0 0.0 0.00

0.05

0.10

0.15

0.25 s/h

0.20

Fig. 3.25 Mobilisation of the earth pressure force: 1g tests, rotation about the toe, aluminium wall surface, variation of the initial relative density ID according to Bartl (2004).

Kph 14.0 12.0 10.0 8.0 6.0

Z.-V., Alu Z.-V., S220

4.0

1g-V., Alu Z.-V., Alu

2.0

Z.-V., S220

0.0 0.0

2.0

1.0

3.0

4.0

5.0

h*ng/1m [-]

Fig. 3.26 Normalised passive earth pressure force for dense sand with variation of the stress level for parallel displacement according to Bartl (2004).

the Kph -value can be explained by the pressure dependency of the friction angle in the “Dresdner Sand”, see Figure 17 in Bartl (2004). In practical applications, and for example also in DIN 4085, it is generally assumed that the displacement sp /h relative to the wall height is independent of the wall height when the passive earth pressure is reached. In fact, an increase with pressure level is observed in many cases (see Fig. 3.27). According to a model theory by Hettler, the increase can be approximated as follows sp sp0

( =

h h0

)1+β (3.31)

3.5 Tests and measurements

sp/h 0.160 Z.-V., Alu

0.140

Z.-V., S220

0.120

1g-V., Alu

0.100

Z.-V., Alu Z.-V., S220

0.080 0.060 0.040 0.020 0.000 0.0

1.0

2.0

3.0

4.0

5.0

h*ng/1m [-]

Fig. 3.27 Normalised passive limit displacement for dense sand with variation of the pressure level for parallel displacement according to Bartl (2004).

where, sp0 denotes the displacement sp belonging to the height h0 and β a soil-dependent exponent. For details, see Hettler and Gudehus (1985); Hettler (1997). 3.5.5

Example: spatial earth resistance in front of soldier piles

Due to the difficulties in the static stress field and kinematic mechanism methods, tests are particularly suitable for solving spatial problems. Weißenbach (1962) investigated the spatial earth resistance in small-scale model tests in front of beams of width b0 with embedment depth t0 . Both dry and moist sands were used and, in particular, the influence of capillary cohesion cK was taken into account. Based on his investigations, Weißenbach worked out the following description of spatial earth resistance: 1 (3.32) E∗p = ⋅ γ ⋅ ωR ⋅ t30 + 2 ⋅ cK ⋅ ωK ⋅ t20 2 with the earth pressure coefficients ωR and ωK according to Fig. 3.28. A distinction is made in the range of b0 /t0 < 0.3 and b0 /t0 ≥ 0.3. For further details, see chapter 8.2 as well as Weißenbach (1985). Equation (3.32) has been adopted in a modified form for the new DIN 4085 (see chapter 8.3). 3.5.6

Example: spatial earth resistance in front of square anchor slabs

Buchholz (1930) examined the spatial earth resistance E∗p in front of square anchor slabs of height h and width b = h in medium-dense sand. A spatial earth pressure coefficient η was derived from the tests, which depends on the depth position H of the anchor slab (Fig. 3.29). From this, E∗p can be determined by means of the equation E∗p =

1 ⋅ γ ⋅ H2 ⋅ b ⋅ η. 2

(3.33)

169

3 Methods for the determination of earth pressure

ωR 10 ° 45 =



° 40

.5°

42

37.

ϕ

170

35°

8

32.5°

6

30°

4

2

0

0.1

0.2

0.3

a) ωK 10

0.5

=4

0.7 a0

0.6

42.5°



ϕ

0.4

b0 t0

40°

8

° 37.5

6

32.5° 30°

35°

4

2

0

0.1

0.2

0.3

b)

0.4

0.5

0.6

0.7 a0

b0 t0

Fig. 3.28 Coefficients for earth resistance in front of soldier piles according to Weißenbach (1962). a) Coefficient ωR for the share of friction, b) Coefficient ωK for the share of capillary cohesion.

The calculation proposal of Buchholz is limited to the ultimate state at one density. If, in addition, the mobilisation at different densities is taken into account, the investigations by Arnold et al. (Arnold et al. 2008; Arnold 2007) can be used. 3.5.7

Further examples

Further examples, particularly of soils with cohesion, can be found in Gudehus (1996). The advantages of the 1g model technique also come into play when it comes to isolating the geometry of failure mechanisms. Fig. 3.30 shows various examples

3.5 Tests and measurements

Fig. 3.29 Calculation proposal for anchor slabs in medium-dense sand according to Buchholz (1930).

η 13

12

11

10

η

9

8

7

Sandy soil

H h

6

φ = 32.5° H h

5 1.0

a)

b)

2.0

3.0

4.0

5.0

6.0

7.0

c)

Fig. 3.30 Pictures of sliding joints of active earth pressure and earth resistance from Walz (2006). a) Parallel movement, active limit state, b) Parallel movement, passive limit state, c) Rotation around the toe at the passive limit state.

of active and passive earth pressure according to Walz (2006). By means of thin, color-coded sand layers, shear bands can be shown at large deformations. If an image pattern recognition program is used, small deformations can be made visible. For example, the PIV method (Particle Image Velocimetry) makes it possible to make a simple straight slip surface visible at an early stage, see Hauser and Walz (2006). Homogeneous Rankine zones can be proved experimentally, following a basic idea of Terzaghi and Peck, see Chapter 23 in Terzaghi and Peck (1961) with the help of modern X-ray technology, as Henning Wolf (2005) has shown. Fig. 3.31a shows a schematic diagram of the experimental set-up, Fig. 3.31b shows the device. By pulling on a movable wall with the force Fx , the displacement ux is generated. Using a rubber mat as the lower limit of the test box, a linear

171

172

3 Methods for the determination of earth pressure test material

moving wall

FX, uX x 0 fixed wall

rubber mat

displacement uX strain εXX

(a)

(b)

Fig. 3.31 Schematic diagram a) and device b) of a test to produce a homogeneous active Rankine zone according to Wolf (2005).

displacement distribution and thus a constant strain εxx is imposed on the test material, resulting in homogeneous active Rankine zones with parallel shear bands (Fig. 3.32). Depending on the average grain diameter d50 , conjugate shear bands also form, cf. Fig. 3.32c.

3.6 Finite Element Method 3.6.1

General

The use of the Finite Element Method (FEM) has grown strongly in recent years and FEM has established itself as a standard tool in some areas. One can choose

3.6 Finite Element Method Displacement direction d50 = 0.35mm

d50 = 0.89mm

d50 = 1.58mm

conjugated shear band

a)

b)

c)

Fig. 3.32 Shear bands measured by Wolf (2005) for various mean grain diameters d50 . a) d50 = 0.35 mm, b) d50 = 0.89 mm, c) d50 = 1.52 mm.

from a wide range of constitutive models and programs. Herle and Kolymbas (2007) provide an overview of the constitutive models. FEM models are covered extensively by Wolffersdorff and Schweiger (2017). Schanz reports on current developments and the Recommendations of the working group “1.6 Numerics in Geotechnics” for the case of ultimate limit state and deformation calculations (Schanz 2006). In this book, only a few special features of its use for earth pressure problems are treated. If only the ultimate limit state is of interest, i.e. Ea or Ep should be determined, a linear-elastic/ideal-plastic constitutive model with the Mohr-Coulomb limit condition is generally sufficient. The results of the FE calculations can then be presented in dimensionless form, as in the case of model tests. For soils without cohesion, equations (3.25) and (3.26) are valid. For soils with friction and cohesion, a representation according to equation (3.28) can be chosen. If wall displacements are of importance, the above-mentioned simple constitutive models are no longer sufficient. In these cases, elasto-plastic constitutive models with isotropic hardening, such as the Hardening Soil model (Schanz 1998), should be used. For very high accuracy requirements, the behaviour in case of small strains should be modelled as close to reality as possible, as e.g. in the Hardening Soil Small Strain Stiffness model, see Benz (2007) or the constitutive model used by Scharinger (2007). In particular, the initial stress state should be checked, since it can have a great influence on displacements. Hypoplastic constitutive models are suitable both for the determination of ultimate limit states and for the determination of deformation. It should be noted that the influence of the pressure level on the stiffnesses and the shear parameters are sometimes modelled differently. Ziegler (1986) uses an earlier version which is homogeneous in the first order of the stress. This means, e.g. that the friction angle is independent of the pressure level and, in the case of a dimensionless representation, earth pressure coefficients are independent of the wall height, see equations (3.25) and (3.26). If, on the other hand, one works with the newer version proposed by von Wolffersdorff, the friction angle depends on the pressure level. The results can then be formulated as in equation (3.27). This means that the earth pressure (2 ⋅ E)/(γ ⋅ h2 ) in dimensionless form and thus the earth pressure

173

174

3 Methods for the determination of earth pressure

coefficient depends on the wall height. The same also applies to displacements represented in dimensionless form. If the stiffnesses are proportional to the pressure level as according to Ziegler, the relation between wall displacements s/h and mobilised earth pressure (2 ⋅ E′ )/(γ ⋅ h2 ) can be expressed as follows ) ( s 2 ⋅ E′ . (3.34) =f⋅ h γ ⋅ h2 Otherwise, the displacements depend, analogously with equation (3.27), on the wall height h. Furthermore, the parameter h/h0 is added in equation (3.34) with a variable reference wall height h0 ) ( 2 ⋅ E′ h s , . (3.35) =g⋅ h γ ⋅ h2 h0 If, e.g. the “Hardening Soil model” or hypoplasticity according to von Wolffsdorff is used, equation (3.35) is valid. 3.6.2

Examples

Early systematic calculations with the FEM were carried out by Potts and Fourie (1986). Both rough and smooth walls were tested for the basic movement types as shown in Figures 3.19a to c in the active ultimate limit state as well as in Figures 3.22a to c in the passive ultimate limit state. The behaviour was modelled with a simple elasto-plastic constitutive model with the Mohr-Coulomb limit condition. The shear parameters were c = 0 and φ′ = 25∘ . Calculations were performed both with a non-associated flow rule and the dilatancy angle υ = 0 as well as with a fully associated flow rule and υ = φ. The wall height was constant at 5.00 m. The at-rest earth pressure coefficient in the initial state was assumed to be K0 = 2 and K0 = 0.5, respectively. Potts and Fourie found a strong dependency of the earth pressure distribution on the type of wall movement. The investigations by Nakai (1985), who also modelled walls with different types of movement, are taken from the same period, see Abdel-Rahman (1999). The first application of a hypoplastic constitutive model to the earth pressure problem was undertaken by Ziegler in 1986 (Ziegler 1986). Ziegler also examined the three basic types of movement for the active and the passive states. The wall friction angle was assumed to be zero. Fig. 3.33 shows the results for the active case. Abdel-Rahman (Abdel-Rahman 1999; Hettler and Abdel-Rahman 2000) worked with a more recent version of hypoplasticity according to von Wolffersdorff, where the friction angle is dependent on the pressure level and the stiffnesses are not proportional to the pressure level. As described in equation (3.35), the dimensionless mobilisation curves depend on the wall height h for both the active and the passive cases. As an example, Fig. 3.34 shows the development of the active earth pressure for a parallel movement and medium density for a smooth wall, Fig. 3.35 shows the passive case. As observed in centrifuge tests (see Fig. 3.27), displacements increase when reaching the peak value with increasing wall heights. In the active case, the earth pressure coefficients Kah also increase with the wall height (Fig. 3.36), whereas the Kph -values decrease (Fig. 3.37). This also agrees with results from centrifuge tests (see Fig. 3.26).

3.6 Finite Element Method

2E′h/γ·h2 0.4 Rotation about the toe Parallel displacement Rotation about the top

0.3

0.2

0.1

s/h – 0.015

– 0.010

– 0.005

0

0

Fig. 3.33 Earth pressure load dependant on the displacement with active wall movement according to Ziegler (1986).

K′ah =

2 E′ah

γ·h2

0.50 h = 0.2 m h = 0.5 m h = 1.0 m h = 2.0 m h = 6.0 m h = 10.0 m h = 20.0 m

0.46 0.42 0.38 0.34 0.30 0.26 0.22 0.18

0

0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 s/h

Fig. 3.34 Active earth pressure: Load-displacement curves for various wall heights with parallel displacement and medium density (Hettler and Abdel-Rahman 2000).

The observed dependencies on the wall height are neglected in practice. Accordingly, DIN 4085 states the displacements required to mobilise the active or passive earth pressure expressed in per cent or per mill of the wall height. As a rule, this is sufficient for practical purposes. However, when transferring the results from small-scale model tests to larger wall heights, the influence of this scale effect should be checked. The results of the numerical simulation by Abdel-Rahman also show the dependency of the earth pressure distribution and the earth pressure resultant on the

175

176

3 Methods for the determination of earth pressure

K′ph = 4.8

2 E′ph

γ·h2

4.4 4.0 3.6

h = 0.2 m h = 0.5 m h = 1.0 m h = 2.0 m h = 6.0 m h = 10.0 m h = 20.0 m

3.2 2.8 2.4 2.0 1.6 1.2 0.8 0.4 0

0.02

0

0.04

0.06

0.08

0.10

0.12 s/h

Fig. 3.35 Passive earth pressure: Load-displacement curves for various wall heights with parallel displacement and medium density (Hettler and Abdel-Rahman 2000).

Kah 0.32 dense e = 0.55 medium-dense e = 0.65 loose e = 0.75

0.30 0.28 0.26 0.24 0.22 0.20 0.18 0.16 0.14 0.12

0

2

4

6

8

10

12

14

16

18

20

Wall height (h) in [m]

Fig. 3.36 Dependency of the active earth pressure coefficient Kah on the wall height h with parallel displacement (Hettler and Abdel-Rahman 2000).

type of wall movement, as described in chapter 3.5. Fig. 3.38 shows the earth pressure coefficients Kah for the active case with high density. The maximum values result for a rotation about the top. The approach of DIN 4085 is on the safe side or just slightly unsafe for the present case. With some exceptions, the present

3.6 Finite Element Method

Kph 6.6 6.4 6.2 6.0 5.8 5.6 5.4 5.2 5.0 4.8 4.6 4.4 4.2 4.0 3.8 3.6 3.4

dense e = 0.55 medium-dense e = 0.65 loose e = 0.75

0

2

4

6

8

10

12

14

16

18

20

Wall height (h) in [m]

Fig. 3.37 Dependency of the passive earth pressure coefficient Kph on the wall height h with parallel displacement (Hettler and Abdel-Rahman 2000). Kah 0.28 0.24 0.20 0.16 0.12

Parallel About the top About the toe DIN 4085

0.08 0.04 0

0

2

4

6

8

10

12

14

16

18

20

Wall height (h) in [m]

Fig. 3.38 Dependency of the active earth pressure coefficient Kah on the wall height h and comparison with DIN 4085 for high density (Hettler and Abdel-Rahman 2000).

example confirms the usual procedure in practice where the influence of the type of wall movement on the earth pressure resultant in the active case is neglected. In contrast, the conditions for passive earth pressure are different. DIN 4085 demands in this case a reduction of the Kph -value obtained according to Pregl/Sokolowski, which is assigned to a parallel movement, e.g. for a rotation

177

178

3 Methods for the determination of earth pressure

Kph 4.8 4.4 4.0 3.6 3.2 2.8 2.4 2.0

Parallel About the top About the toe DIN 4085-Parallel DIN 4085-About the top/toe

1.6 1.2 0.8 0.4 0

0

2

4

6

8

10

12

14

16

18

20

Wall height (h) in [m]

Fig. 3.39 Dependency of the passive earth pressure coefficient Kph on the wall height h and comparison with DIN 4085 for medium density (Hettler and Abdel-Rahman 2000).

about the toe. The results in Fig. 3.39 and comparison with DIN 4085 also agree within the scope of the possible accuracy with the specifications of the standard. While the earth pressure coefficients depend significantly on the wall height, the type of earth pressure distribution appears to be independent of the wall height and more or less of the density. If the wall coordinate z is normalised with the height h and the earth pressure eah (z) with the mean earth pressure Eah /h, then the curves for different heights h and density are largely identical (see Figures 3.40 to 3.42). The distributions themselves are roughly equivalent to the proposals in DIN 4085 (see chapter 4). The jumps near the wall toe are associated with numerical difficulties. However, net dependencies have not been established (Abdel-Rahman 1999). Although it is not yet possible to model the shear band formation, the softening and the processes of progressive failure numerically in a satisfactorily way, FE calculations can give evidence of the failure behaviour. Fig. 3.43 shows the mobilised friction angle calculated from the actual principal stresses T1 , T2 and T3 with T1 > T2 > T3 ( ) T1 − T3 φm = arcsin (3.36) T1 + T3 for parallel displacement, medium density and a wall height of h = 1.00 m at the ultimate limit state for the active case. A band-shaped zone with a fully mobilised friction angle φ ≈ 35∘ is clearly visible. The failure mechanism is reminiscent of the solution by Coulomb with a straight line, see Fig. 3.30 in chapter 3.5.7. With rotation about the top, a curved, band-shaped zone (Fig. 3.44) forms, which

3.6 Finite Element Method

z/h 0.0 0.1 h = 0.2 m h = 1.0 m h = 10.0 m

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0

0,2

0,4

0,6

a)

0,8

1,0

1,2

1,4

1,6

1,8

1.6

1.8

eahh/Eah

z/h 0 0.1

e0 = 0.55 e0 = 0.65 e0 = 0.75

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0,9 1.0 b)

0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

eahh/Eah

Fig. 3.40 Parallel displacement. a) Influence of the wall height on normalised active earth pressure distribution for medium density, b) Influence of the density on normalised active earth pressure distribution for a wall height of h = 1 m (Hettler and Abdel-Rahman 2000).

resembles a rigid body mechanism with a curved slip plane in the form of a logarithmic spiral, cf. Fig. 3.1b in chapter 3.1 and Fig. 3.4 in chapter 3.2. As in the case of Rankine’s solution, the simulation of a wall with rotation about the toe results in a zone of failure, see Fig. 3.45, which was also observed in model tests, cf. Fig. 3.1a.

179

180

3 Methods for the determination of earth pressure

z/h 0.0 0.1 0.2 h = 0.2 m h = 1.0 m h = 10.0 m

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

eahh/Eah

Fig. 3.41 Influence of the wall height on normalised active earth pressure distribution for rotation about the top and medium density (Hettler and Abdel-Rahman 2000).

In addition, Fig. 3.46 shows the passive case with parallel movement. As in the case of the static stress field solution of Pregl/Sokolowski (see Fig. 3.13 in chapter 3.4.4) a zone of failure with a curved lower limit occurs, see also Fig. 3.30b in chapter 3.5.7. z/h 0.0 0.1

h = 0.2 m h = 1.0 m h = 10.0 m

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

eahh/Eah

Fig. 3.42 Influence of the wall height on normalised active earth pressure distribution for rotation about the toe and medium density (Hettler and Abdel-Rahman 2000).

3.6 Finite Element Method

Faliure zone

Fig. 3.43 Band shaped failure zone for parallel displacement at the active limit state, wall height h = 1.00 m, medium density according to Abdel-Rahman (1999).

Faliure zone

Fig. 3.44 Band shaped failure zone with rotation about the top at the active limit state, wall height h = 1.00 m, medium density according to Abdel-Rahman (1999).

Failure zone

Fig. 3.45 Rankine zone with rotation about the toe at the active limit state, wall height h = 1.00 m, high density according to Abdel-Rahman (1999).

181

182

3 Methods for the determination of earth pressure

Failure zone

Fig. 3.46 Rankine zone failure with parallel displacement at the passive limit state, wall height h = 1.00 m, medium density according to Abdel-Rahman (1999).

While Abdel-Rahman takes the earth pressure angle δ = 0 as a basis, Hegert (2016) also examines rough walls. The numerical model is based on the program Plaxis2D 2012 using the Hardening Soil constitutive model developed by Schanz (1998), which is sufficient to describe the essential effects. A detailed discussion can be found in Hegert (2016). The model size is adapted to Bartl’s test box with the aim of simulating his results. The soil parameters of the “Dresdner Sand” for the Hardening Soil model used by Bartl were determined from special triaxial tests with a sample height of 10 cm and a diameter of 10 cm. The numerical model was validated and calibrated in terms of size and mesh fineness due to Bartl’s tests. This led to an - albeit insignificant - adjustment of the soil parameters. For details see Hegert (2016). In the following, some results of the numerical calculations are presented as examples and compared with Bartl’s tests, see Figures 3.23 to 3.25. Fig. 3.47 shows the mobilisation curves of the dimensionless mobilised earth resistance K′ph = E′ph /2 ⋅ γ ⋅ h2 depending on the displacement s/h related to the wall height for different densities ID = 0.2 (loose), ID = 0.5 (medium-dense), ID = 0.8 (dense) for parallel movement of the wall. In the context of the scattering, the numerical prognosis model is suitable for describing the test results as a good approximation. This also applies to the mobilisation of the inclination angle δ of the resulting earth pressure depending on the dimensionless displacement s/h (see Fig. 3.48a to c). If the earth resistance coefficients are calculated according to the proposal of Pregl/Sokolowski (see chapter 6.2) with the friction angles from the triaxial tests for the respective density and the earth pressure inclination angle from Fig. 3.48, one obtains a good compliance of the model tests and the FE calculations. Thus, irrespective of the investigations with a hypoplastic constitutive model, the Kp -values according to Pregl/Sokolowski can be used for parallel movement of the wall, see also Fig. 3.39. The FE calculations for parallel movement also confirm the usual assumption of a triangular earth pressure distribution independent of the density (see Fig. 3.49). Evaluating the so-called failure points, one obtains failure mechanisms resembling logarithmic spirals, see Fig. 3.50. For this, see also the approaches by Pregl/Sokolowski (Fig. 3.13) and Patki et al. (Fig. 3.16). Fig. 3.51 shows the results of the FE calculations for rotation about the toe with medium density (ID = 0.5), and Fig. 3.52 for rotation about the top with loose density (ID = 0.2). While the mobilisation curves from tests and FE calculation match

3.6 Finite Element Method 14 12

K′ph [-]

10 FEM

8

Bartl 044 Bartl 064

6

Bartl 065 Bartl 067

4 2 0 0.00

a)

0.02

0.04

0.06

0.08

0.10

s/h [-] 14 12

K′ph [-]

10 FEM

8

Bartl 038 Bartl 050

6

Bartl 051 Bartl 052

4

Bartl 056 2 0 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20

b)

s/h [-] 14 12

K′ph [-]

10

FEM

8

Bartl 049

6

Bartl 055 Bartl 070

4 2 0 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.22

c)

s/h [-]

Fig. 3.47 Mobilisation of normalised earth pressure load, parallel displacement according to Hegert (2016). a) ID = 0.8 (dense), b) ID = 0.5 (medium-dense), c) ID = 0.2 (loose).

183

3 Methods for the determination of earth pressure 0.6 0.4 0.2 tan δ [-]

FEM 0.0

Bartl 044 Bartl 064

–0.2

Bartl 065 Bartl 067

–0.4 –0.6 –0.8 0.00

0.02

0.04

a)

0.06

0.08

0.10

0.12

s/h [-] 0.8 0.6 0.4 FEM Bartl 038 Bartl 050 Bartl 051 Bartl 052 Bartl 056

tan δ [-]

0.2 0.0 –0.2 –0.4 –0.6 –0.8 0.00

b)

0.04

0.08

0.12

0.16

0.20

s/h [-] 0.8 0.6 0.4 0.2

tan δ

184

FEM Bartl 049

0.0

Bartl 055

–0.2

Bartl 070

–0.4 –0.6 –0.8 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.22 0.24

c)

s/h [-]

Fig. 3.48 Mobilisation of the inclination angle of the resulting earth pressure depending on the relative wall displacement s/h with parallel displacement according to Hegert (2016). a) ID = 0.8 (dense), b) ID = 0.5 (medium-dense), c) ID = 0.2 (loose).

3.6 Finite Element Method

0.0

z/h [-]

0.2

s/h=0 s/h=0.01066 s/h=0.0177

0.4

s/h=0.03

0.6

s/h=0.0486

0.8 1.0

0

2

a)

4

6 e′ph/(γ·h)

8

10

12

s/h=0.0577 Kph· (z/h) according to DIN 4085

0.0 0.2

z/h [-]

s/h=0 0.4

s/h=0.0177 s/h=0.039

0.6

s/h=0.0567 s/h=0.0798

0.8

s/h=0.0975

1.0

0

2

b)

4 6 e′ph/(γ·h)

8

10

0.0

z/h [-]

0.2 s/h=0 s/h=0.0177 s/h=0.039 s/h=0.0567 s/h=0.0798

0.4 0.6

s/h=0.1241

0.8 1.0 c)

0

1

2 3 e′ph/(γ·h)

4

5

Fig. 3.49 Mobilisation of normalised passive earth pressure e′ph depending on the relative wall displacement s/h with parallel displacement according to Hegert (2016). a) ID = 0.8 (dense), b) ID = 0.5 (medium-dense), c) ID = 0.2 (loose).

185

186

3 Methods for the determination of earth pressure

a)

b)

c)

Fig. 3.50 Evaluation of failure points and failure mechanisms for parallel displacement according to Hegert (2016). a) ID = 0.8 (dense) and s/h = 0.03, b) ID = 0.5 (medium-dense) and s/h = 0.08, c) ID = 0.2 (loose) and s/h = 0.1.

very well in general for rotation about the top (Fig. 3.52), this is only true in the range of s/h < 0.04 for rotation about the toe (Fig. 3.51). The deviation in the range s/h > 0.04 or also in Fig. 3.47b and c can be indicative of compaction effects with larger deformations and thus higher friction angles, which are not represented by the Hardening Soil model, as well as softening after the peak. Despite this restriction, the prognosis is only slightly limited in the field of serviceability, so that these effects may be neglected for practical application. A comparison of the

3.6 Finite Element Method

12.0 10.0

K′ph [-]

8.0 FEM Bartl 039 Bartl 042 Bartl 059 Bartl 060 Bartl 061 Bartl 062

6.0 0.4 2.0

0.0 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.22 s/h [-] a) 0.0

s/h=0 s/h=0.0177 s/h=0.039 s/h=0.0532 s/h=0.0887 s/h=0.1950

0.2

z/h [-]

0.4

0.6

0.8

1.0 b)

0

1

2 e′ph/(γ·h)

3

4

5

c)

Fig. 3.51 Rotation about the toe, results of FE calculations with medium density (ID = 0.5) according to Hegert (2016). a) Mobilisation curve and comparison with test by Bartl, b) Mobilisation of normalised passive earth pressure distribution, c) Evaluation of failure points.

187

3 Methods for the determination of earth pressure

4.0 3.5 3.0 K′ph [-]

2.5

FEM

2.0

Bartl 033

1.5 1.0 0.5 0.0 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.22 a)

s/h [-] 0.0

0.2

s/h=0 s/h=0.0177 s/h=0.0390

0.4 z/h [-]

188

0.6

s/h=0.0532 s/h=0.1064 s/h=0.1950

0.8

1.0 b)

0.0

5.0 e′ph/(γ·h)

10.0

c)

Fig. 3.52 Rotation about the top, results of FE calculations with loose density (ID = 0.2) according to Hegert (2016). a) Mobilisation curve and comparison with tests by Bartl, b) Mobilisation of normalised passive earth pressure distribution, c) Evaluation of failure points.

References

earth pressure distributions in Figures 3.51b and 3.52b with the proposal of DIN 4085 shows that the approaches of the DIN represent a suitable approximation for practical use, cf. right column of Table 15.2 in chapter 15.3. The evaluation of the failure mechanism indicates a plastified zone of failure in the entire area behind the wall (Fig. 3.51c) for rotation about the toe, while rotation about the top (Fig. 3.52c) is a combination of plastified zone in the upper wall region with a shear band in the form of a logarithmic spiral as the lower limit. For further results see Hegert (2016). The listed examples show only a small section of the literature on numerical simulations of earth pressure with the FEM. In some cases, special problems are also dealt with, as in Arnold and Herle, who performed FE simulations of the spatial, passive earth pressure on pressure plates (Arnold et al. 2008). Abdel-Rahman (1999) provides an overview, although only limited.

References Abdel-Rahman, K. (1999). Numerische Simulation des Erddruckproblems in Sand auf der Grundlage der Hypoplastizität. Schriftenreihe des Lehrstuhls Baugrund-Grundbau Universität Dortmund, Heft 23. Arnold, M., Herle, I. and Gabener, H.-G. (2008). Mobilisierung des räumlichen passiven Erddrucks bei Erdüberdeckung. Geotechnik 31: 12–22. Arnold, M. (2007). Mobilisierung des räumlichen, passiven Erddrucks bei Druckplatten mit Erdüberdeckung. Mitteilungen des Instituts und der Versuchsanstalt für Geotechnik Technische Universität Darmstadt, Heft 76: 167–181. Atkinson, J. (2007). The Mechanics of Soils and Foundations, second edition, CRC Press, Taylor and Francis Group. Bartl, U. (2004). Zur Mobilisierung des passiven Erddrucks in kohäsionslosem Boden. Institut für Geotechnik Technische Universität Dresden, Mitteilungen, Heft 12. Benz, T. (2007). Small-strain stiffness of soil and in numerical consequences. Mitteilungen des Institutes für Geotechnik der Universität Stuttgart, Heft 55. Besler, D. (1998). Wirklichkeitsnahe Erfassung der Fußauflagerung und des Verformungsverhaltens von gestützten Baugrubenwänden. Schriftenreihe des Lehrstuhls Baugrund-Grundbau der Universität Dortmund, Heft 22. Buchholz, W. (1930). Erdwiderstand auf Ankerplatten. Jahrbuch Hafenbautechnische Gesellschaft, Heft 12. Caquot, A., Kérisel, J. and Absi, E. (1973). Tables de Butée et de poussée. Paris, Bruxelles, Madrid: Gauthier-Villars Éditeur. Chen, W. F. (1975). Limit analysis and soil plasticity. London: Elsevier Scientific Publishing Company. Coulomb, C. A. (1776). Essai sur une application des régles des maximis et minimis à quelques problèmes de statique relatifs à l’architecture. Mémoires présentès à l’Académie des Sciences VII, Re-Print by Editions des Sciences et Industrie 1971, Paris. Cundall, P. A., Konietzky, H. and Potyondy, D. (1996). PFC – ein neues Werkzeug für numerische Modellierungen. Bautechnik 73(8): 492–498.

189

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3 Methods for the determination of earth pressure

Drucker, D.C., Prager, W. and Greenberg, H.J. (1952). Extended limit design theorems for continuous media, Quart. Appl. Math. 9: 381–389 Franzius, O. (1928a). Neuere Einrichtungen für Versuche über Erddruck, Erdwiderstand, Bodenreibung, Fundamentreibung usw. In: Der Bauingenieur 9(7): 118-119 and 9(8): 136-137. Franzius, O. (1928b). Erddruckversuche im natürlichen Maßstabe. In: Der Bauingenieur 9(43): 787-792 and 9(44): 813-815. Gässler, G. (1987) Vernagelte Geländesprünge – Tragverhalten und Standsicherheit. Veröffentlichungen des Instituts für Bodenmechanik und Felsmechanik, Universität Karlsruhe, Heft 108. Goldscheider, M. (2013). Gültigkeitsgrenzen des statischen Kollapstheorems der Plastomechanik für Reibungsböden. Geotechnik 36(4): 243–263. Goldscheider, M. (2017). Berechnung des Erdruhedrucks bei geneigtem Gelände und geneigter Schnittfläche. Geotechnik 40 (2): 103-118. Goldscheider, M. (2000). Zum Nachweis der Geländebruchsicherheit und der erforderlichen Ankerlänge verankerter Stützwände. Bautechnik 77(9): 641–656. Görtler, H. (1975). Dimensionsanalyse. Berlin, Heidelberg, New York: Springer. Groß, H. (1981). Korrekte Berechnung des aktiven und passiven Erddrucks. Geotechnik 2: 66–69. Gudehus, G. (1981). Bodenmechanik. Stuttgart: Enke. Gudehus, G. (1996). Erddruckermittlung. Grundbau-Taschenbuch, 5. Auflage, Teil 1, Abschnitt 1.10, Berlin: Ernst & Sohn. Gudehus, G. (1972). Lower and upper bounds for stability or earth-retaining structures. Proc. 5th Europ. Conf. on Soil Mech. and found. Engng., Madrid, Vol. 1, 21-28. Gußmann, P., König, D. and Schanz, T. (2016). Die Methode der kinematischen Elemente in der Geotechnik – aktuelle Entwicklungen und Anwendungen. Geotechnik 39(1): 40-53. Gußmann, P. (1983). Die Methode der kinematischen Elemente. Mitteilungen Baugrundinstitut Stuttgart, Nr. 25, Stuttgart. Gußmann, P. and Schanz, T. (1983). KEM-Nachweise im Grundbau. Geotechnik 4(3): 127–133. Gutberlet, C. (2008). Erdwiderstand in homogenem und geschichtetem Baugrund – Experimente und Numerik. TU Darmstadt, Heft 78. Hauser, C. and Walz, B. (2006). Untersuchung der Boden-BauwerkWechselbeziehung mittels innovativer Verfahren – Einsatz von iFEM und PIV im bodenmechanischen Modellversuch. Mitteilungen des Institutes für Grundbau und Bodenmechanik TU Braunschweig, Heft 82. Hegert, H. (2016). Anwendbarkeit des Bettungsmodulverfahrens mithilfe von Mobilisierungsfunktionen zur Prognose Wandverschiebungen. Veröffentlichungen Lehrstuhl Baugrund – Grundbau der Technischen Universität Dortmund, Heft 32. Herle, I. and Kolymbas, D. (2017). Stoffgesetze für Böden. Grundbau-Taschenbuch, Teil 1, 8. Auflage, Berlin: Ernst & Sohn. Herten, M. (1999). Räumlicher Erddruck auf Schachtbauwerk in Abhängigkeit von der Wandverformung. BU-GH Wuppertal FB Bauingenieurwesen, Wuppertal.

References

Hettler, A. and Abdel-Rahman, K. (2000). Numerische Simulation des Erddruckproblems in Sand auf der Grundlage der Hypoplastizität. Bautechnik 77(1): 15–29. Hettler, A. and Gudehus, G. (1985). A pressure dependent correction for displacement result from 1g model tests. Géotechnique 35(4): 497–510. Hettler, A. and Gudehus, G. (1988). Influence of the foundation width on the bearing capacity factor. Soils and Foundations 28 (4): 1–10. Hettler, A. (1997). Maßstabseffekte beim Erddruck in Sand. Mitteillungen des Institutes für Geotechnik der TU Dresden, Heft 4. Hettler, A. (1981). Verschiebungen starrer und elastischer Gründungskörper in Sand bei monotoner und zyklischer Belastung. Veröffentlichungen Institut für Bodenmechanik und Felsmechanik der Universität Karlsruhe, Heft 90. Hettler, A. (2010). Possibilities and limitations of 1g model techniques. Proceedings of the 7th International Conference on Physical Modelling in Geotechnics, CRC Press, pp. 135-140. James, R. G. and Bransby, P. L. (1970). Experimental and theoretical investigations of a passive earth pressure problem. Géotechnique 20: 17-37. and discussion in Géotechnique (1971) 21 (2): 173-175; (4): 421-423 and Géotechnique (1972), 22 (2): 363-365. James, R. G. and Bransby, P. L. (1971). A velocity field for some passive earth pressure problems. Géotechnique 21(1): 61-83. Kézdi, A. (1962). Erddrucktheorien. Berlin, Göttingen, Heidelberg: Springer. Kobayashi, Y. (1998). Laboratory experiments with an oblique passive wall and rigid plasticity solutions. Soil and Foundations 38: 121–129. Koiter, W.T. (1960). General theorems for elastic-plastic solids. In: Progress in Solid Mechanics, Chapter IV (Ed. I.N. Sneddon and R. Hill), 166–221. Amsterdam: North Holland Publ. Comp. Krey, H. (1926). Erddruck, Erdwiderstand und Tragfähigkeit des Baugrundes. 3. Auflage, Berlin: Ernst & Sohn. Mao, P. (1995). Erdwiderstandsversuch mit drei verschiedenen Bewegungsarten. From: Franke, D. (Ed.): OHDE-Kolloquium 1993. Dresden. Mitteilungen - Institut für Geotechnik, TU Dresden, 2: 53-66. Müller-Breslau, H. (1906). Erddruck auf Stützmauern. Stuttgart: Verlag Körner, Neudruck 1947. Nakai, T. (1985). Finite Element Computations for active and passive earth pressure problems on retaining wall. Soils and Foundations 25(3): 98–112. Neuberg, C., Franke, D. and Engel, J. (2007). Modellversuch und numerische Simulation mit der Diskrete-Element-Methode zum passiven Erddruck. Bautechnik 84(6): 309–387. Ohde, J. (1992). Gesammelte Veröffentlichungen (Ed. D. Franke), Mitteilungen des Instituts für Geotechnik, Technische Universität Dresden, Heft 1. Ohde, J. (1948). Zur Erddrucklehre. Bautechnik 25(6): 121–126. Patki, M., Mandal, J. and Dewaikar, D. (2015). A simple approach based on the limit equilibrium method for evaluating passive earth pressure coefficients. Geotechnik 38(2): 120–133.

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Potts, D. M. and Fourie, A. B. (1986). A Numerical Study of the Effects of Wall Deformation on Earth Pressures. Int. Journ. For Numerical and Analytical Methods in Geomechanics, 10: 383–405. Powrie, W. (2013). Soil Mechanics – Concepts and applications, third edition, CRC Press, Taylor and Francis Group. Pregl, O. (2002). Bemessung von Stützbauwerken. Handbuch der Geotechnik, Vol. 16, Wien: Eigenverlag des Instituts für Geotechnik, Universität für Bodenkultur. Rankine, W. J. M. (1857). On the Stability of Loose Earth. Phil. Trans. Royal Soc., 147, London. Roscoe, K. H. (1970). The influence of strains in soil mechanics. In: Géotechnique 20(2): 129-170. Schanz, T. (2006). Aktuelle Entwicklungen bei Standsicherheits- und Verformungsberechnungen in der Geotechnik. Empfehlungen des Arbeitskreises 1.6 “Numerik in der Geotechnik“, Abschnitt 4. Geotechnik 29(1): 13–27. Schanz, T. (1998). Zur Modellierung des mechanischen Verhaltens von granularen Reibungsmaterialien. Institut für Geotechnik, Universität Stuttgart, Heft 45. Scharinger, F. (2007). Multilaminate Model for Soil incorporating Small Strain Stiffness. Gruppe Geotechnik Graz der Technischen Universität Graz, Heft 31. Shiau, J. S., Augarde, C. E., Lyamin, A. V. and Sloan, S. W. (2008). Finite element limit analysis of passive earth resistance in cohesionless soils. Soils and Foundations 48: 843–850. Sokolowski, V. V. (1960). Statics of soil media. London: Butterworths. Sokolowski, V. V. (1965). Statics of granular media. Oxford: Pergamon Press. Soubra, A. H. (2000). Static and seismic passive earth pressure coefficients on rigid retaining structures. Can Geotech J. 37: 463 – 478, 10.1139/t99-117. Spencer, A. J. M. (1982). Deformation of ideal granular materials. Mechanics of Solids. Ed. H. G. Hopkins and M. J. Sewell. 607–652. Pergamon Press. Streck, A. (1926). Beitrag zur Frage des passiven Erddruckes. In: Der Bauingenieur 7(1): 1-3 and 7(2): 32-37. Terzaghi, K. v. and Peck, R. (1961). Die Bodenmechanik in der Baupraxis, Springer. Terzaghi, K. v. (1920). Old Earth-Pressure Theories and New Test Results. In: Engineering New-Record 85(14): 632-637. Terzaghi, K. v. (1934). Large retaining wall tests, I: Pressure of dry Sand. Eng. News Record 112: 136–140. Thornton, C. (2000). Microscopic approach contributions to constitutive modelling. In: Constitutive Modelling of Granular Materials, (Ed. D. Kolymbas), 99–208. Berlin, Heidelberg, New York: Springer. Tschebotarioff, G. P. and Johnson, E. G. (1953). Effects of restraining boundaries on the passive resistance of sand. Princeton University. Vogt, N. (1984). Erdwiderstandsermittlung bei monotonen und wiederholten Wandbewegungen in Sand. Baugrundinstitut Stuttgart, Mitteilung 22. Walz, B. (2006). Der 1g-Modellversuch in der Bodenmechanik – Verfahren und Anwendung. Hans Lorenz Vorlesung, Veröffentlichungen des Grundbauinstitutes der TU Berlin, Heft 40. Weißenbach, A. (1985). Baugruben, Teil II, Berechnungsgrundlagen. Berlin: Ernst & Sohn, 1. Nachdruck.

References

Weißenbach, A. (1962). Der Erdwiderstand vor schmalen Druckflächen. Bautechnik 39 (6): 204–211. Wolf, H. (2005). Zur Scherfugenbänderung granularer Materialien unter Extensionsbeanspruchung, Schriftenreihe des Instituts für Grundbau und Bodenmechanik der Ruhr-Universität Bochum, (Ed. T. Triantafyllidis), Heft 37. Wolffersdorff, P.-A. v. and Schweiger, F. (2017). Numerische Verfahren in der Geotechnik. Grundbau-Taschenbuch, 8. Auflage, Teil 1, Ed. K.-J. Witt, Abschnitt 1.10, Berlin: Ernst & Sohn. Ziegler, M. (1986). Berechnung des verschiebungsabhängigen Erddrucks in Sand. Veröffentlichungen des Institutes für Bodenmechanik und Felsmechanik der Universität Karlsruhe, Heft 101.

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4 Active earth pressure under plane strain conditions 4.1 Fundamental considerations Sufficiently large wall displacement is a precondition for the assumption of active earth pressure. Depending on the type of wall movement and the density, the required displacements are between 0.5 ‰ and 5 ‰ of the wall height (see chapter 15.2). The reference value for medium-dense to dense non-cohesive soils and for stiff to semi-solid cohesive soils with a parallel movement is 1 ‰ of the wall height. This means, e.g. that a displacement of 1 cm is sufficient in order to achieve the active earth pressure for a wall of 10 m height with parallel movement. This condition is frequently met in practice. In this simple application rule, the additional dependency on wall height, which is often observed in tests, is neglected (see chapter 3.5). As shown in chapter 3, there are closed formulas for simple standard cases derived from Coulomb’s earth pressure theory and its extension to inclined earth pressure resultants, inclined walls, and inclined ground levels. One should note the sign rules for the earth pressure inclination δa , terrain inclination β and the wall inclination α shown in Fig. 4.1, which refer to the more recent versions of DIN 4085. The sign of the wall inclination has changed from the old DIN 4085 from February 1987 or the earlier articles in the Grundbau-Taschenbuch by Gudehus and other literature. Although not generally applicable (see chapter 17.2), an additive decomposition of the resultant Ea of the total earth pressure into a share of soil self-weight with index g, a share of infinite uniformly distributed surcharges with index p and a share of cohesion with index c has been established in practice. Thus, one obtains the formula for the resultant Ea Ea = Eag + Eap + Eac .

(4.1)

The earth pressure ea is equally divided. Exact formulas based on Coulomb’s earth pressure theory were derived by Groß, see Groß (1981) and chapter 17.2. For practical use, it is appropriate to divide the earth pressure with inclination δa into a horizontal and a vertical component with index h or v. In case of a vertical wall with α = 0 (see Fig. 4.2), one uses Eav = Eah



tan δa

(4.2a)

Earth Pressure, First Edition. Achim Hettler and Karl-Eugen Kurrer. © 2020 Ernst & Sohn Verlag GmbH & Co. KG. Published 2020 by Ernst & Sohn Verlag GmbH & Co. KG.

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4 Active earth pressure under plane strain conditions



Ea

DIN 4085 Edition 1987

+ δa

+α +α New

Old

Fig. 4.1 Definition of the algebraic sign according to DIN 4085:2017-08 and comparison with previous definition. Fig. 4.2 Horizontal and vertical components of earth pressure.

Ea Eav

δa Eah

and at an inclined wall with α ≠ 0 Eav = Eah



tan(δa + α).

(4.2b)

A further index is required to differentiate the characteristic value of the earth pressure from the design value in calculations. For example, eagh,k is the characteristic value of the horizontal component of the active earth pressure from soil self-weight, and the corresponding design value is eagh,d . In addition, exponents are used to distinguish between the earth pressure under plane strain and spatial condition (see chapter 7). Calculation of the active earth pressure is generally based on the theory by Coulomb, which assumes a sliding wedge. The inclination angle of the slip surface associated with Ea is designated by ϑa (Fig. 4.3). It should be pointed out that, in the case of discontinuous surcharges, the relevant active earth pressure can also occur at a different inclination of the slip surface (see chapter 4.4). The kinematic mechanism method provides only the earth pressure resultant and not its distribution. In practice, it is common to estimate • a linearly increasing distribution with the depth for earth pressure from soil self-weight and • a constant distribution for components of earth pressure from infinite uniformly distributed surcharges as well as from cohesion, see chapter 4.2.

4.2 Soil self-weight, infinite uniformly distributed surcharges and cohesion

Fig. 4.3 Active earth pressure and associated angle of slip surface ϑa .

Ea

ϑa

This assumption can be plausibly proved by deducing the resultant according to the height as proposed by Weißenbach (1985), or by relying on the static theory of Rankine, which, strictly speaking, only applies to rotation about the toe. Depending on the application, the classically determined earth pressure may be redistributed (see chapter 16.3).

4.2 Soil self-weight, infinite uniformly distributed surcharges and cohesion The horizontal earth pressure at depth z in a homogeneous soil with specific weight γ can be determined approximately by superimposing the components of soil self-weight with the earth pressure coefficient Kagh , infinite uniformly distributed surcharge p with the earth pressure coefficient Kaph and from cohesion c with the earth pressure coefficient Kach (Fig. 4.4a to c) with eah = γ • z • Kagh + p • Kaph − c • Kach leachlleaphl c)

eah d)

eah e)

Fig. 4.4 Distribution of earth pressure from combined action of soil self-weight, infinite uniformly distributed surcharge and cohesion. a) Earth pressure from soil self-weight, b) Earth pressure from infinite uniformly distributed surcharge, c) Earth pressure from cohesion, d) Superimposition of the earth pressure shares at |eaph | < |each |, e) Superimposition at |eaph | > |each |.

197

198

4 Active earth pressure under plane strain conditions

If the share from cohesion is absolutely larger than the share from an infinite uniformly distributed surcharge, the calculated tensile stresses are present up to the depth hcp (Fig. 4.4d). This case is discussed as well as the approach of minimum earth pressure in chapter 4.3. If the share from cohesion is absolutely smaller than the share from an infinite uniformly distributed surcharge (Fig. 4.4e), the earth pressure load can be obtained by integration over the wall height h Eah =

4.2.1

1 • • 2• γ h Kagh + h • p • Kaph − c • h • Kach 2

(4.4)

Vertical wall, level ground, horizontal earth pressure

The earth pressure theory of Coulomb gives an exact solution for the case of a flat vertical wall with level ground and a horizontal earth pressure force ( φ) (4.5a) Kagh = tan2 45∘ − 2 ( φ) Kaph = Kagh = tan2 45∘ − (4.5b) 2 √ ( φ) (4.5c) Kach = 2 • Kagh = 2 • tan 45∘ − 2 The corresponding inclination angle of the slip surface is ϑa = 45∘ +

4.2.2

φ 2

(4.6)

Vertical wall, level ground, inclined earth pressure

For the most common case in practice of a vertical wall with level ground and an inclined earth pressure, one can use Kagh = Kaph = [

√ 1+

and

cos2 φ sin(φ + δa ) • sin φ cos δa

]2

(4.7)

√ sin φ + tan ϑa =

tan φ tan φ + tan δa cos φ

(4.8)

see Table B.1 and Table B.2 in Appendix B. According to a proposal of Weißenbach (1985), an approximation for the share of cohesion is given by √ Kach ≈ 2 • Kagh (4.9)

4.3 Cohesion, calculated tension and minimum earth pressure

4.2.3

General case

DIN 4085 specifies the following equations for the general case: 2

Kagh

⎤ ⎡ ⎥ ⎢ ⎥ ⎢ cos(φ − α) =⎢ √ )⎥ ( ⎢ sin(φ + δa ) • sin(φ − β) ⎥ ⎥ ⎢ cos α • 1 + cos(α − β) • cos(α + δa ) ⎦ ⎣

Kaph =

cos α • cos β • K agh cos(α − β)

cos(α − β) • cos φ • cos(α + δa ) Kach = 2 • [ ] 1 + sin(φ + α + δa − β) • cos α

(4.10a)

(4.10b) (4.10c)

with the corresponding angle of slip surface from soil self-weight ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ cos(φ − α) ϑa = φ + arctan ⎢ √ ⎥ ⎢ sin(φ + δa ) • cos(α − β) ⎥ ⎢ sin(φ − α) + ⎥ sin(φ − β) • cos(α + δa ) ⎦ ⎣

(4.11)

It should be noted that, with an inclined wall or ground level and corresponding earth pressure inclination, it may be necessary to use curved or discontinuous slip surfaces. Information on the limits of application are given in the old DIN 4085:2011-05, cf. also chapter 17.2. In most cases, however, Coulomb’s theory is sufficient.

4.3 Cohesion, calculated tension and minimum earth pressure Unless an infinite uniformly distributed surcharge is taken into account or if the earth pressure from an infinite uniformly distributed surcharge is numerically smaller than the earth pressure from cohesion, the classic calculation of the earth pressure in cohesive soils results in calculated tensile stresses. In this case, it must be noted that for the determination of the classic earth pressure from soil self-weight and cohesion, a distinction must be made between walls where earth pressure redistribution and thus a compensation of calculated tensile stresses is possible and cases where no earth pressure redistribution takes place, particularly in the case of rotation about the toe. If the residual earth pressure after deduction of the cohesion component is relatively small, it is necessary to check whether the minimum earth pressure is decisive. A principal reason for this is the fact that, for cohesive soils in comparison to non-cohesive soils, a larger wall movement is required in order to reduce the at-rest earth pressure to the active earth pressure. A second reason for this is that the resultant earth pressure can reach zero due

199

200

4 Active earth pressure under plane strain conditions

to the influence of cohesion and relatively high walls can also be free of loading or only slightly loaded. The idea of the minimum earth pressure was introduced by Weißenbach. Based on measurements on excavation walls, the first draft of a recommendation on the minimum earth pressure was published in EB 4, paragraph 3 in 1970 (Weißenbach 1970), and the basic idea has remained unchanged since. The earth pressure from soil self-weight and cohesion must first be determined based on classic earth pressure theory with the shear parameters φ and c. Its resultant is to be compared with the earth pressure load from minimum earth pressure. The less favourable approach is then to be used for further calculation. Earth pressures from surcharges were always added. Over the years, there have been changes in the details; e.g. the minimum earth pressure is no longer based on a coefficient of K = 0.20, but on a substitute friction angle φErs = 40∘ . In the version of the recommendations from 1972, the earth pressure load from surcharges could be reduced by a share of cohesion. However, this share had to be reallocated to the share of soil self-weight, cf. Figure EB 7-1 from 1972 (Weißenbach 1972). Another approach is used in DIN 4085:2011-05. In addition to the comparison of the earth pressure resultants, a comparison of the earth pressure ordinates is proposed as a variant. In the interests of simplification and standardisation, the new DIN 4085:2017-08 only provides a comparison of the resultant. The question still remains unanswered in DIN 4085:2017-08 as to whether and how the surcharge is applied with the classically determined earth pressure. The Recommendations on Excavations EAB, however provide a clear procedure. 4.3.1

Determination of the classic earth pressure

As described in chapter 4.2, the shares from soil self-weight (Fig. 4.5a) and cohesion (Fig. 4.5b) are first determined and superimposed (Fig. 4.5c) but without the surcharges. In the case of walls without or with yielding support, which rotate about the toe or a lower point, the classic triangular distribution of earth pressure from soil self-weight is established and there is no earth pressure redistribution. The calculated tensile stresses shown by dashed lines in Fig. 4.5c must not be taken into account and must be set to zero till the depth is reached of hc =

c • Kach . γ • Kagh

(4.12)

Only the resultant remains 1 • • γ Kagh • (h − hc )2 . (4.13a) 2 If earth pressure redistribution is to be expected, especially in the case of slightly yielding walls, the resulting tensile stresses may be calculated against corresponding compressive stresses (Fig. 4.5d) and the resulting earth pressure results from Eah =

Eah = Eagh + Each .

(4.13b)

4.3 Cohesion, calculated tension and minimum earth pressure

hc

a)

b)

c)

d)

e)

Fig. 4.5 Determination of the active earth pressure load for continuously cohesive soil according to EAB, Chapter 3.2 (2012). a) Earth pressure from soil self-weight, b) Earth pressure due to cohesion, c) Earth pressure on unsupported excavation walls, d) Earth pressure on supported excavation walls, e) Minimum earth pressure.

4.3.2 Minimum earth pressure in comparison with the resultant of earth pressure When comparing the earth pressure resultants, the minimum earth pressure is determined from the soil self-weight with the substitute friction angle φ′ Ers,k = 40∘ while retaining the geometric variables such as wall inclination α and ground inclination β as well as the earth pressure inclination δa (Fig. 4.5e). From the coefficient K∗ = K (φ = 40∘ ) (4.14) agh

agh

Ers,k

the resultant of the minimum earth pressure is calculated by using 1 (4.15) ⋅ γ ⋅ h2 ⋅ K∗agh . 2 The larger value from equation (4.15) or from equations (4.13a) or (4.13b) is decisive. In layered soil (Fig. 4.6a), the earth pressures in the individual layers are first determined in a classic manner with the characteristic soil parameters (Fig. 4.6b). Subsequently, the minimum earth pressure is only determined in the cohesive layers (Fig. 4.6c) and a comparison of the resultants in each layer is performed. The example in Fig. 4.6 shows, as it can be seen in Fig. 4.6d, that the minimum earth pressure in the upper cohesive layer is decisive and the classically calculated earth pressure in the lower cohesive layer. The earth pressure in the central non-cohesive layer remains unchanged, see Fig. 4.6d. E∗agh =

4.3.3 Minimum earth pressure in comparison with the earth pressure ordinate The second procedure according to DIN 4085:2011-05 is based on a comparison of the classically calculated earth pressure ordinances with the minimum earth pressure. For the example in Fig. 4.7, down to the depth z∗ =

c • Kach γ • (Kagh − K∗agh )

(4.16)

201

202

4 Active earth pressure under plane strain conditions

a)

b)

c)

d)

Fig. 4.6 Determination of the total load of active earth pressure in partially cohesive soil layers according to EAB (2012). a) Soil stratification, b) Earth pressure with characteristic shear strengths, c) Earth pressure in cohesive layers with substitute friction angle, d) Minimum earth pressure.

eah

Fig. 4.7 Minimum earth pressure: comparison of the earth pressure ordinates according to DIN 4085:2011-05.

γ·z·K*agh z*

γ·z·Kagh - c·Kach

z

the minimum earth pressure is to be set at e∗agh = γ • z • K∗agh .

(4.17)

Below, one has to use the classic earth pressure determined on the basis of characteristic shear parameters. For layered soils, the procedure should be followed analogously. The revised DIN 4085, which was published in 2017, is limited to the method that compares the resultants. This simplifies the procedure. In addition, the position of the intersection point of the classic earth pressure distribution and the minimum earth pressure is associated with great uncertainties. The assumption of a minimum earth pressure is based on plausibility, and there is no exact theoretical reason. Additionally, the classic earth pressure distribution is only true for rotation about the toe. 4.3.4

Minimum earth pressure and surcharges

According to the “Recommendations on Excavations” (EAB 2012), the earth pressure from surcharges is generally to be added to the minimum earth pressure

4.4 Vertical line loads and strip loads

from soil self-weight. In exceptional cases, the earth pressure from surcharges can be reduced by means of mathematical tensile stresses from cohesion. In EB 6, paragraph 4 of the fourth edition states: “Even if for cohesive layers, the size of the earth pressure load from soil self-weight and cohesion is calculated by means of a substitute friction angle according to EB 4, paragraph 3b) (chapter 3.2), the earth pressure from vertical infinite uniformly distributed surcharges including loads up to pk = 10 kN/m2 , as well as line and strip loads is always to be determined in accordance with paragraph 3a) with the characteristic friction angle φk . As a rule, this also applies to loads of pk > 10 kN/m2 . In exceptional cases with cohesion of c′k > 30 kN/m2 , earth pressures from surcharges determined in this way may be adequately offset against calculated tensile stresses from soil self-weight and cohesion, provided that more detailed investigations are carried out and that sufficient local experience is available.” DIN 4085:2011-05 leaves the question open just like the 2017 revision. In this respect, the “Recommendations on Excavations” contain a stricter interpretation and a higher degree of safety.

4.4 Vertical line loads and strip loads 4.4.1

Introduction

If line or strip loads are present on the ground surface, the maximum of the active earth pressure can also be obtained at a different angle than ϑ = ϑa , which is determined by the soil self-weight, depending on the size and position of the loads (Fig. 4.8). In this case, according to Weißenbach (1977) and the “Recommendations on Excavations” (EAB 2012), one obtains a forced slip surface which is inclined by ϑ = ϑz . For this reason, further earth pressure formulas are required for any angle ϑ. In some cases, however, ϑ = ϑa may be used. DIN 4085 assumes for loads, which act on the active earth pressure wedge and whose absolute value is not higher than 10 % of the soil self-weight, that the inclination of the earth pressure slip surface from soil self-weight does not significantly alter and the calculation formulas in chapter 4.2 are valid. In addition, the share of earth pressure from surcharges is required for ϑ = ϑa . Line load p

Strip load p′

steep forced slip surface

flat forced slip surface ϑ = ϑz ϑ = ϑa

ϑ = ϑa

ϑ = ϑz

Fig. 4.8 Possible forced slip surfaces with line or strip loads.

203

204

4 Active earth pressure under plane strain conditions

Fig. 4.9 Transformation of limited surface loads to strip or line loads.

Ir I av

45°

The “Recommendations on Excavations” (EAB) are based on a different criterion. According to EAB, forced slip surfaces have to be examined only in the case of unsupported walls, which can rotate about the toe. In the case of supported walls, it is assumed that a forced slip surface cannot be formed and therefore does not have to be examined. If the line or strip loads are limited in their length and have a distance to the wall, the locally limited load may be equally distributed over the length (Fig. 4.9) lr = l + 2 • a V .

(4.18)

Along the length lr , the earth pressure is then determined as for an unlimited line or strip load. 4.4.2

Standard slip surface from soil self-weight

In the event that the inclination of the slip surface ϑa from soil self-weight does not change significantly due to surcharges, the earth pressure coefficient for soil self-weight must be calculated according to equation (4.10a) and for cohesion according to equation (4.10c). The additional horizontal earth pressure Eaph from a line load p is obtained for a horizontal ground, a vertical wall and an earth pressure force inclined at δa with Eaph = p



sin(ϑa − φ) • cos δa . cos(ϑa − δa − φ)

(4.19)

For a strip load p′ with the width bp′ , one has to replace p by p′ ⋅ bp and Eaph by Eap′ h in equation (4.19). With Kaph =

sin(ϑa − φ) • cos δa cos(ϑa − δa − φ)

(4.20)

one obtains Eaph = p ⋅ Kaph .

(4.21)

Weißenbach (1985) developed a table for the Kaph -values which is included as Table B.3 in Appendix B. If the wall is inclined, and the ground is horizontal, the following is valid Eaph = p



sin(ϑa − φ) • cos(α + δa ) cos(ϑa − α − δa − φ)

(4.22)

see DIN 4085:2017-08. The distribution of earth pressure from line or strip loads cannot exactly be determined with the classic theories. This explains why different approaches are proposed in the literature. Below is a detailed discussion of the general proposals

4.4 Vertical line loads and strip loads

ap

bp′

p

hpo φ hp

ap′ bp′ p′

eap′h,o

hp′

ϑa

ϑa eap′h,u

h

h

p′

hp′o φ

eaph

eap′h,o

hp′ h

ϑa eap′h,u

ϑa

ϑa

a)

b)

ϑa c)

Fig. 4.10 Distribution of active earth pressure stresses from line or strip loads for ϑ = ϑa according to Weißenbach (1985). a) Line load, b) Strip load starting at the wall, c) Strip load with distance to wall.

by Weißenbach (1985), which have been customarily applied for years, but which are, strictly speaking, only valid for rotation of the wall about the toe. This applies, e.g. to unsupported walls. In these cases, a triangular shape may be assumed for line loads p with a distance ap from the wall and a rectangle shape for strip loads p′ of the width bp′ without a space from the wall, see Fig. 4.10a and b. The zone of influence of the loads on the wall is limited by straight lines inclined at φ and ϑa , respectively. For a line load p with a distance ap from the wall, the distance between the wall head and the maximum value of the earth pressure stresses is hp0 = ap

tan φ.



(4.23)

The height of the load shape is hp = ap



tan ϑa − hpo .

(4.24)

The maximum ordinate of the earth pressure stresses from the line load p is obtained by eaph =

2 • Eaph hp

.

(4.25)

For a strip load p′ with the width bp′ and the distance ap′ from the wall (Fig. 4.10c), the upper boundary line of the earth pressure diagram is obtained by hp′ o = ap′



tan φ.

(4.26)

The height of the pressure diagram is hp′ = (ap′ + bp′ ) • tan ϑa − hp′ o . The earth pressure diagram is a trapezoid with the ordinates ( ) Eap′ h a p′ • eap′ h,o = 1+ h p′ ap′ + bp′

(4.27)

(4.28)

205

206

4 Active earth pressure under plane strain conditions

q′k ≤ φ′

ϑa

a)

b)

Fig. 4.11 Pressure diagrams for the earth pressure from vertical live loads for moderately flexibly supported walls according to EAB (2012). a) Retaining wall, live load and local distribution, b) Examples of simple pressure diagrams.

and eap′ h,u =

Eap′ h h p′



( 1−

a p′ ap′ + bp′

) .

(4.29)

In the case of a strip load without a space from the wall with ap′ = 0, one obtains from equations (4.28) and (4.29) eap′ h,o = eap′ h,u

(4.30)

and thus, a rectangle diagram (Fig. 4.10b). In the case of walls supported little yielding, the form of the load distribution can be freely selected to a large extent. Fig. 4.11 shows some suggestions according to the “Recommendations on Excavations” (EAB). 4.4.3

Analysis of any slip surface angle

If the above-mentioned special cases do not apply, the maximum of the active earth pressure must be sought by varying the angle of the slip surface. For this purpose, e.g. the graphic method by Culmann (see chapter 4.7) or the method proposed by Minnich and Stöhr (1981, 1982) is useful, which can also be implemented numerically. If one assumes the situation shown in Fig. 4.12a with a vertical wall and horizontal terrain, the force polygon rotates by 90∘ - φ according to the Culmann method (Fig. 4.12b and c), one obtains in dependency of ϑ, the horizontal component of the earth pressure resultant from soil self-weight, strip load p′ with the width bp′ and line loads p using the sine rule E′ah = (G + p + p′ • bp′ )



sin(ϑ − φ) • cos δa . cos(ϑ − φ − δa )

(4.31)

Analogously, one obtains for the cohesion component (Fig. 4.13) E′ach = −c ⋅ h •

cos φ • cos δa . cos(ϑ − φ − δa ) • sin ϑ

(4.32)

For the cases shown in Fig. 4.8, one first calculates the earth pressure in a classic manner according to chapter 4.2 and compares it with the results obtained

4.4 Vertical line loads and strip loads bp′

bp′

ap′

p

ap′

p′

p′

p

p′

90 - ϑ+ φ+ δa G

δa Qa

Ea

G

φ

ϑ-φ

ϑ

Total earth pressure Ea

G

90 - δa ϑ

Total load G + p

a)

90 - δa Total load G + p′·bp′

φ

φ

ϑ

ϑ-φ

90 - ϑ+ φ+ δa Total earth pressure Ea

b)

c)

Fig. 4.12 Analysis of forced slip surfaces according to Weißenbach (1985). a) Forces on sliding wedge, b) Culmann structure for line loads, c) Culmann structure for strip loads. ϑ

h

K

δa

90 - φ

K

φ

90 - ϑ + φ 90 - φ ϑ-φ Q′ac

90 - ϑ + φ δa

E′ac ϑ-φ

Q′ac

ϑ a)

ϑ

ϑ - δa δa

E′ac

90 - ϑ + φ + δa

E′acv E′ach

b)

Fig. 4.13 Consideration of cohesion for the determination of earth pressure from line and strip loads according to Weißenbach (1985). a) Forces on sliding wedge, b) Force polygon.

by inserting ϑz into equations (4.31) and (4.32). The larger value is decisive. Irrespective of this, the issue of the minimum earth pressure has to be clarified in the case of cohesive soils. Contrary to the case of ϑ = ϑa , Weißenbach (1985) assumes that the influence of the load is noticeable down to the wall foot in the case of forced slip surfaces (Fig. 4.14). With a line load p, the distance between the wall head and the beginning of the pressure diagram is obtained by hpo = ap

tan φ



(4.33)

with hp = h − hpo the maximum ordinate of the pressure diagram is obtained from 2 • Eaph eaph = . hp

(4.34)

(4.35)

For a strip load p′ with the width bp′ and the distance ap′ from the wall, the distance between the start of the pressure diagram is correspondingly hp′ o = ap′



tan φ.

(4.36)

207

208

4 Active earth pressure under plane strain conditions

ap hpo

ap′

p

φ

h - hpo

p′

φ

hp′o

eaph

bp′ eap′h,o

h - hp′o

ϑz

ϑz

a)

eap′h,u

b)

Fig. 4.14 Distribution of active earth pressure from line and strip loads for ϑ = ϑz according to Weißenbach (1985). a) Line loads, b) Strip loads.

Analogously, the height of the trapezoid is h p′ = h − h p′ o .

(4.37)

The ordinates of the pressure diagram are obtained according to the procedure for ϑ = ϑa ( ) Eap′ h a p′ • eap′ h,o = 1+ (4.38) h p′ ap′ + bp′ and eap′ h,u =

Eap′ h h p′



( 1−

a p′ ap′ + bp′

) .

(4.39)

In general, the resulting earth pressure load can be calculated from the following equation in case of a variable surface inclination β and wall inclination α as well as a vertical load V and a horizontal force H (Fig. 4.15) (Goldscheider 2015). Ea =

(G + V − C • sin ϑ) • sin(ϑ − φ) + (H − C • cos ϑ) • cos(ϑ − φ) cos(ϑ − φ − δ − α) (4.40)

4.5 Horizontal line and strip loads The earth pressure load EaHh from horizontal line or strip loads H, which act within the earth pressure wedge from soil self-weight and have a negligible influence on the inclination ϑa of the sliding wedge from soil self-weight, can be determined with the equation cos(ϑa − φ) • cos δa . (4.41) EaHh = H • cos(ϑa − φ − δa ) Equation (4.41) applies to vertical walls and horizontal terrain, and can be extended to inclined walls and rising terrain for any slip surface inclination, see equation (4.40).

4.6 Layered soil

Ea

V β

δ+α H

C

α

ϑ-φ

G

Q G

φ

δ

Ea δ+α

90°+φ-ϑ

Q ϑ

90°+φ-ϑ

C ϑ

ϑ H

V

Fig. 4.15 Earth pressure with the impact of additional forces H and V on the sliding wedge according to Goldscheider (2015).

For rotation about the toe, the distribution of the earth pressure may be selected as in the case of vertical line or strip loads, see chapter 4.4. If potential forced slip surfaces have to be examined, the inclination of the slip surface ϑa in equation (4.41) has to be replaced by ϑ. If the wall head is fixed, individual case studies are necessary. Consideration must be given to the load application, the load spread as well as the yielding and the stiffness of the supporting structures.

4.6 Layered soil In the case of layered soil, there are no exact theoretical solutions. For practical purposes, it has been found to be sufficient to apply equation (4.3) in layers and instead of γ ⋅ z, set the vertical stress σzi = Σγi ⋅ hi

(4.42)

in layer i with the specific weight γi and thickness hi . This results in eah = Σγi ⋅ hi • Kagh + p • Kaph − c • Kach .



(4.43)

At the layer boundaries, one obtains from the surcharge γi ⋅ hi two earth pressure ordinates, determined with earth pressure coefficients of the associated layer (Fig. 4.16a and b). The procedure assumes that the most unfavourable slip surface occurs in each layer (Fig. 4.16c). In fact, a rather continuous slip surface will develop for kinematic reasons (Fig. 4.16d), for which the maximum value of the total earth pressure load is obtained. According to DIN 4085, equation (4.43) can also be used with layers which are inclined parallel to the surface. If the inclination of the layer and the surface are different, it is recommended to determine the earth pressure coefficients based on the slope inclination β. Gudehus (1981) proposes consideration of stratification in order to work with weighted mean values for the specific weight, friction and cohesion. The shares from cohesion in the respective layer are proportional to the respective length of

209

210

4 Active earth pressure under plane strain conditions

a)

b)

c)

d)

Fig. 4.16 Determination of earth pressure with layered soil according to Weißenbach (1985). a) Soil stratification, b) Earth pressure distribution, c) Mathematical slip surface, d) Actual slip surface.

γ1

h1 γ2

h

φ1

c1

φ2

c2

h2 ϑ

ϑ a)

b)

ϑ c)

Fig. 4.17 Determination of weighted average of a) unit weight, b) cohesion and c) the friction angle according to Gudehus (1981).

the slip surface and the friction forces are proportional to the depth. For the case shown in Fig. 4.17, the average values of γ, c and φ are obtained by [ ( )2 ] ( )2 h2 h2 + γ2 • γ = γ1 • 1 − (4.44a) h h [ ( )] h2 h c = c1 • 1 − (4.44b) + c2 • 2 h h [ ( )2 ] ( )2 h1 h1 . (4.44c) φ ≈ φ1 • + φ2 • 1 − h h It should be mentioned that Gudehus proposed a different approximation for the determination of φ a few years later (Gudehus 1996). Examples and further approximations can be found in the supplement to DIN 4085 which was published in December 2018 (see section 17.6).

4.7 Discontinuous ground level In the case of a discontinuous ground level, the maximum of the earth pressure load can be determined graphically according to Culmann by varying the inclination of the slip surface. In doing so, different slip surfaces are examined, see

4.7 Discontinuous ground level 2

3

1 cis ive slip su rf De

F3

Ea

rth

pr

F2

es

su

re

lin

e

L 4

5

6

7

ac e

B

F7

F6

ine

el

op

sl ral

tu

na

F5

G7

F4 H Ea Eah

G6 G4

G5

G3

F1 φ + δa

G2 G1

A ϑa

φ

Fig. 4.18 Determination of earth pressure according to the Culmann procedure.

ϑ-φ E′ag G 90-φ φ

δa E′ag

a)

ϑ

Q′ag ϑ-φ

Q′ag

G

90-δa E′ag δa

b)

90-φ-δa φ

Q′ag

E′agv

φ+ δa 90-φ-δa

ϑ-φ

90 -δa

Q′ag ϑ-φ

G

c)

E′ E′ag agh

E′agv

φ

φ ϑ

E′agh

δa

ϑ

d)

Fig. 4.19 Explanation of the Culmann structure for the determination of earth pressure from soil self-weight according to Weißenbach (1985). a) Forces on sliding wedge, b) Force polygon, c) Force polygon in Culmann structure, d) Determination of earth pressure components.

Fig. 4.18. Starting from the forces on the sliding blocks (Fig. 4.19a), the force polygon in Fig. 4.19b is turned clockwise by 90∘ - φ (Fig. 4.19c). The line of action of the self-weight G coincides with the natural slope line, which is inclined at φ against the horizontal axis. The resultant E′ag of the earth pressure is directed parallel to the so-called earth-pressure line, which is inclined at φ + δa against the vertical axis. The reaction force Q′ag is in the slip surface. For the slip surfaces 1, 2, etc., the force polygon is drawn, and the earth pressure loads associated with the individual slip surfaces (see points F1 to F7 in Fig. 4.18) are combined to a curve. The maximum can be determined graphically by drawing a parallel to the natural slope line, which is tangential to the earth pressure curve in point H. As shown in Fig. 4.18, the decisive slip surface and the resultant Ea of the active earth pressure can be constructed at the contact point. The Culmann procedure is very versatile. Line loads and strip loads as well as cohesion (Weißenbach 1985) or seepage flow (DIN 4085:2017-08) can be considered. With this method, the determination of the active earth pressure load can be explained clearly. However, it is generally too complicated for practical use. An

211

212

4 Active earth pressure under plane strain conditions

b1

b2

eaghB eaghA

b3

γ· Kah3· h

h′3

h′3 + h2 - h1 h2

h3

K

β1

γ· Kah1· h

K

h1

h′2

F

β2

hB hA F β3

A hC

C a)

γ· Kah2· h

B

C b)

eaghC

Fig. 4.20 Determination of earth pressure with discontinuous ground surface according to Jenne (1960), see Weißenbach (1985). a) Wall and ground level, b) Distribution of earth pressure.

analytical solution, which can be implemented numerically, has been proposed by Minnich and Stöhr (1981). The Culmann procedure only provides the earth pressure forces. A proposal by Jenne (1960) is suitable for the approximate determination of the earth pressure distribution (see Fig. 4.20). In this case, the inclinations β1 , β2 or β3 are assumed one after the other, and the respectively associated earth pressure line is determined. The line eagh1 = γ ⋅ Kah1 ⋅ h applies up to the intersection with line eagh2 = γ ⋅ Kah2 ⋅ h, which, in turn, reaches the intersection with eagh3 . The resultant is obtained from the shaded area in Fig. 4.20b.

4.8 Discontinuous wall surfaces For multiply discontinuous wall surfaces as shown in Fig. 4.21, the earth pressure coefficient with the corresponding wall inclination may be determined in each individual section. Special rules are provided for a recessed wall. For this purpose, one should refer to DIN 4085:2017-08 and chapter 11 (L- and T-cantilever retaining walls).

4.9 Distribution of active earth pressure The classic triangular distribution of the active earth pressure from soil self-weight is, strictly speaking, only valid for rotation of the wall about the toe, see chapters 3.5 and 3.6. Depending on the structure and yielding of a wall, other distributions are also possible (Weißenbach 1977). For this, see chapter 16.3. DIN 4085:2017-08 specifies idealised forms of distribution for the basic types of wall movement. For this subject, one should refer to Table 15.1 in chapter 15.2.

References

α1 A eagh1(α1) eagh2(α2) α2 = 0

eagh3(α3)

α3

Fig. 4.21 Approach for earth pressure to discontinuous wall surface.

References EAB (2012). Recommendations on Excavations (EAB), 5th ed. Berlin: Ernst & Sohn. Goldscheider, M. (2015). Darstellung von Wasserdrücken im Boden mit strömendem Grundwasser. Geotechnik 38 (2): 85–95. Groß, H. (1981). Korrekte Berechnung des aktiven und passiven Erddrucks. Geotechnik 2: 66–69. Gudehus, G. (1996). Erddruckermittlung. Grundbau-Taschenbuch, 5. Auflage, Teil 1, Abschnitt 1.10. Berlin: Ernst & Sohn. Gudehus, G. (1981). Bodenmechanik. Stuttgart: Enke. Jenne, G. (1960). Praktische Ermittlung des Erddrucklastbildes. Bautechnik 37: 233–237. Minnich, H. and Stöhr, G. (1981). Analytische Lösung des zeichnerischen Culmann-Verfahrens zur Ermittlung des aktiven Erddrucks nach der “G0 -Methode”. Bautechnik 58 (8): 261–270. Minnich, H. and Stöhr, G. (1982). Analytische Lösung des zeichnerischen Culmann-Verfahrens zur Ermittlung des aktiven Erddrucks für Linienlasten nach der "G0 -Methode". Bautechnik 59 (1): 8–12. Weißenbach, A. (1970). Empfehlungen des Arbeitskreises Baugruben der Deutschen Gesellschaft für Erd- und Grundbau. Bautechnik 47 (7): 223-233. Weißenbach, A. (1972). Empfehlungen des Arbeitskreises Baugruben der Deutschen Gesellschaft für Erd- und Grundbau. Bautechnik 49 (6): 192-204 und (7): 225-239. Weißenbach, A. (1985). Baugruben, Teil II, Berechnungsgrundlagen. Berlin: Ernst & Sohn, 1. Nachdruck. Weißenbach, A. (1977). Baugruben, Teil III, Berechnungsverfahren. Berlin: Ernst & Sohn.

213

215

5 At-rest earth pressure 5.1 Soil self-weight and infinite uniformly distributed surcharges Strictly speaking, the at-rest earth pressure is only defined for flat terrain and normally consolidated cohesive soils or non-cohesive soils that are bedded horizontally in layers and without compaction or are formed by sedimentation. If a vertical wall is installed without disturbance, the at-rest pressure acts on this wall as long as no movement occurs. In practice, however, there are cases where these requirements do not apply, such as with immovable walls in sloping terrain. Approximate formulas for different situations have been developed for the user, although no great accuracy can be expected. Due to geological processes or the effect of former construction projects, the initial stress state in the soil may have greatly changed. There are practically no possibilities for the measurement of the stresses. 5.1.1

Horizontal ground

In the ideal case, the ratio of the horizontal stress σh to the vertical stress σv is constant over the depth, and the following relationship is valid σh = K0g ⋅ σv

(5.1)

where K0g is the coefficient of at-rest earth pressure from soil self-weight. In order to determine K0g , e.g. triaxial tests are suitable, where the vertical effective stress is increased and at the same time lateral spreading is prevented by means of a corresponding control of the lateral pressure. For non-cohesive soils, the following approach has become established in practice K0g = K0 = 1 − sin φ

(5.2)

which results from the simplification of an equation derived from Jáky (1944). The approach is confirmed by existing tests and also recommended in DIN 4085:2017-08. In the case of over-consolidated cohesive soils, cohesion is not taken into account and the effective friction angle φ’ is set for φ. As shown by the investigations of Pelz (2011), this simplification is sufficient for practical purposes. Earth Pressure, First Edition. Achim Hettler and Karl-Eugen Kurrer. © 2020 Ernst & Sohn Verlag GmbH & Co. KG. Published 2020 by Ernst & Sohn Verlag GmbH & Co. KG.

216

5 At-rest earth pressure

In the case of a vertical wall and horizontal ground, the at-rest earth pressure e0gh from the soil self-weight is calculated at depth z with e0gh = K0 ⋅ γ ⋅ z

(5.3)

including K0 according to equation (5.2). The earth pressure load for a wall of height h is 1 (5.4) ⋅ K0 ⋅ γ ⋅ h2 . 2 If infinite uniformly distributed surcharges p are added, their share is assumed to be constantly distributed in the at-rest earth pressure with E0gh =

e0ph = K0 ⋅ p.

(5.5)

From this, the resultant for a wall of height h is obtained as E0ph = K0 ⋅ p ⋅ h.

(5.6)

When the wall is inclined, the at-rest earth pressure is calculated under the condition that the stress field in the soil is the same as in the case of a vertical wall, i.e. the wall does not cause any change of stresses. Thus the earth pressure inclination angle to the wall is the result of the calculation and cannot be preset. According to Goldscheider (2017), the at-rest earth pressure e0 acting on an inclined wall is composed of a normal component e0,n and a shear component e0,t (Fig. 5.1). These components are, as a function of depth, as follows [ ] 1 + K0g 1 − K0g e0,n = γ ⋅ z ⋅ − ⋅ cos 2α (5.7) 2 2 1 − K0g e0,t = γ ⋅ z ⋅ ⋅ sin 2α. (5.8) 2 The earth pressure inclination angle is obtained from the ratio of e0,t and e0,n ( ) e0,t δ0 = arctan . (5.9) e0,n If the wall inclination angle is α > 0, the earth pressure inclination angle is also δ0 > 0; if the wall inclination angle α < 0, then δ0 < 0. Separate considerations are necessary if the calculated inclination angle δ0 according to equation (5.9) is higher or absolutely higher than the wall friction angle. This is possible in the case of smooth walls. In these cases, one can revert to a suggestion by Weißenbach (1985) or DIN 4085. In the case of layered soils or groundwater, the factor γ ⋅ z in equations (5.7) and (5.8) has to be replaced by the corresponding sum of all layers from the ground level to the considered depth z.

α>0

δ0

Fig. 5.1 At-rest earth pressure on an inclined wall with horizontal ground.

e0 e0,t Z e0,n

5.1 Soil self-weight and infinite uniformly distributed surcharges

In the case of strong geological preload, e.g. through Ice Age glaciers, continuous horizontal prestress is to be expected, which is higher than the at-rest earth pressure according to equation (5.2). According to Gudehus (1996), the increased coefficient of at-rest earth pressure K0c can be estimated with the following equation ( )m σv ′ (5.10) K0c = K0 ⋅ σ z′ where K0 denotes the coefficient of at-rest earth pressure for unpreloaded terrain according to equation (5.2), σv′ is the previous maximum, and σz′ is the current effective vertical stress, cf. also Schmidt (1966) and Gudehus (1996). The exponent m is between 0.4 for loose and 0.7 for dense sand, and 0.4 for light and 0.5 for distinctly plastic clay. A value of m = 0.5 is recommended in DIN EN 1997-1 (EC 7-1). It should be noted that the at-rest earth pressure cannot exceed the passive earth pressure. In particular for non-cohesive soils, the question is still not clarified to what extent an increased at-rest earth pressure can be reduced to a smaller value, e.g. by construction phases. 5.1.2

Inclined ground

For terrain which is inclined at an angle β, a Rankine state occurs in the limit case of β = φ, in which the earth pressure acts parallel to the ground surface. At β = φ and δa = φ and for a vertical cross-section Kag = K0g = Kpg = cos φ

(5.11)

i.e., active and passive earth pressures, and thus the intermediate at-rest earth pressure, are the same. For the horizontal share, one obtains K0gh (β = φ) = cos2 φ

(5.12)

which is used in practice as a horizontal coefficient of the at-rest earth pressure. The result is the horizontal stress on a fixed vertical wall e0gh = cos2 φ ⋅ γ ⋅ z.

(5.13)

The vertical share is e0gv = tan φ ⋅ cos2 φ ⋅ γ ⋅ z.

(5.14)

Franke (1974) proposes the interpolation formula for a slope inclined at β < φ K0g (β) = 1 − sin φ + (cos φ + sin φ − 1) ⋅

β . φ

(5.15)

It is assumed that the at-rest earth pressure acts parallel to the ground surface. Weißenbach (1985) interpolates in the same way the earth pressure coefficient for the horizontal component: K0gh (β) = K0 + (K0gh (β = φ) − K0 ) ⋅

β β = 1 − sin φ + (cos2 φ − 1 + sin φ) ⋅ . φ φ (5.16)

217

218

5 At-rest earth pressure

Fig. 5.2 Explanation of the angle ψ, the coordinate z and the directions of the components e0,n and e0,t in equations (5.19) to (5.23).

β ψ β

α

Z δ0

e0 e0n e0t

For the limit values β = 0 and β = φ, the results from equations (5.15) and (5.16) are identical; they differ from each other for intermediate values. DIN 4085:2011-05 also provides an equation for the general case with inclined terrain, inclined wall and inclined earth pressure. An objection can be raised against this approximation that, strictly speaking, the earth pressure inclination angle cannot be preset for an inclined wall. A mechanically correct proposal for the stresses in the soil has been developed by Goldscheider (2017) on the assumption that the direction of the earth pressure is parallel to the slope inclination, and that the earth pressure coefficient in a rotated Cartesian co-ordinate system parallel or vertical to the slope surface is interpolated according to Franke’s (1974) suggestion in equations (5.15) and (5.17). Using Mohr’s circle of stress, the normal component e0,n and e0,t (Fig. 5.2) can be determined with ( ) ( ( )) |β| |β| β φ ∘ K0η = K0g + (5.17) ⋅ (K0η − K0g ) ⋅ 1 − κ ⋅ sin ⋅ 180 φ φ and K0g according to equation (5.2) and φ

K0η :=

1 + sin2 φ cos2 φ

(5.18)

to β

e0,n = γ ⋅ z ⋅ [K0η ⋅ cos2 β ⋅ cos2 ψ + cos2 β ⋅ sin2 ψ − cos β ⋅ sin β ⋅ sin 2ψ] (5.19) [ e0,t = γ ⋅ z ⋅ −

]

β

cos2 β ⋅ (K0η − 1) 2

⋅ sin 2ψ − cos β ⋅ sin β ⋅ cos 2ψ

(5.20)

where ψ:=β − α

(5.21)

and z is the vertical depth of the considered wall point under the ground surface (Fig. 5.2). The scaling factor κ controls the form of the interpolation and can be adapted to the results of more accurate deformation calculations. The value κ = 0.3 corresponds with slight deviations to the linear interpolation by Franke in equation (5.15) (Goldscheider 2017).

5.2 Concentrated loads, line loads and strip loads

In the special case of a vertical wall with α = 0, it is ψ := β, and instead of equations (5.19) and (5.20), these formulas are valid β

e0,n = γ ⋅ z ⋅ cos2 β ⋅ [K0η ⋅ cos2 β + sin2 β − tan β ⋅ sin 2β]

(5.22)

β

e0,t = γ ⋅ z ⋅ cos2 β ⋅ [−(K0η − 1) ⋅ sin β ⋅ cos β − tan β ⋅ cos 2β].

(5.23)

It should be noted that when angles β, α and ψ are inserted into equations (5.19) and (5.20) or β into equations (5.22) and (5.23), the signs according to the definitions in equation (5.21) are valid. A negative algebraic sign of e0,t according to equations (5.20) or (5.23) means that this component acts on the soil behind the wall, as in Fig. 5.2. Further details see Goldscheider (2017). A similar approach is chosen by Gudehus, for details see chapter 4.2.1.2 in Gudehus (1996).

5.2 Concentrated loads, line loads and strip loads Earth pressures from concentrated, line and strip loads on inflexible walls are usually estimated based on the theory of the elastic half-space. The equations derived by Fröhlich (1934) were evaluated by Weißenbach (1985) and further developed for the practical calculation. Fröhlich introduces a so-called concentration index υ for the equations. If the stiffness modulus is constant over the depth, it has to be set at υ = 3. This is approximately the case for prestressed soils. In all other cases, υ = 4 may be assumed. Theoretically, this corresponds to a linearly increasing stiffness modulus. Exemplary, the procedure for a line load p at υ = 3, i.e. a constant stiffness modulus, is explained. The line load has the distance ap from the wall. The horizontal earth pressure e0ph and the vertical share e0pv on the wall in the depth z (Fig. 5.3) are sought. ap

p

p

ψ z* ψzs

ψz Z

ap

p ψ z* ψzs

ψz

e0pv

ψz

h

h

e0ph z

ap a)

z max e0ph

b)

max e0pv c)

Fig. 5.3 Distribution of stresses from a line load in the elastic half-space according to Weißenbach (Weißenbach and Hettler 2011). a) Load spread, b) Horizontal stresses e0ph , c) Vertical stresses e0ph .

219

220

5 At-rest earth pressure

By using tan ψz =

ap

(5.24)

z

one obtains for υ = 3 2 π 2 = π

eaph = eapv

⋅ ⋅ ⋅ ⋅

p ⋅ sin3 ψz ⋅ cos ψz ap p ⋅ sin2 ψz ⋅ cos2 ψz . ap

(5.25) (5.26)

As can be seen in Fig. 5.3b and c, both the horizontal and vertical pressure increase initially, and at the angle ψ*z a maximum is reached p ap p = 0.159 ⋅ ap

maxe0ph = 0.207 ⋅

for ψz∗ = 60∘

(5.27)

maxe0pv

for ψz∗ = 45∘ .

(5.28)

By integration over the wall height h up to the angle ap tan ψzs = h

(5.29)

the resulting earth pressure load on a wall with the height h can be calculated as p ⋅ cos2 ψzs π ) p (π 1 = ⋅ − arcψzs − ⋅ sin 2ψzs . π 2 2

E0ph =

(5.30)

E0pv

(5.31)

In the limit case of an infinitely high wall with h ad infinitum, tan ψ zs moves to zero, and the greatest possible earth pressure forces are obtained max E0ph = 0.318 ⋅ p

(5.32)

max E0pv = 0.500 ⋅ p.

(5.33)

The corresponding equations for υ = 4 as well as concentrated and strip loads are compiled completely by Weißenbach (1985). As the numerical evaluation and the figures show, radiation of concentrated, line and strip loads in the soil results in both horizontal and vertical stresses in the assumed vertical plane. This also applies to the transition from the strip load to an infinite uniformly distributed surcharge. The solution incorporates shear forces on the wall, which are neglected in practice. Instead, equation (5.5) is used in these cases and e0pv = 0 is set. As a rule, in excavation walls, an adjacent structure will already be present before the wall is constructed. In this case, the already described at-rest earth

5.2 Concentrated loads, line loads and strip loads

1m 2m σ = 10 kN/m2 or 20 kN/m2

5m

γ = 18 kN/m3 φ‘ = 30° c = 0 kN/m2 E50 = Eoed = 15000 kN/m2 Eur = 45000 kN/m2

5m Y X

Fig. 5.4 Numerical model and material parameters.

pressure due to the structure load has already been reached. If, on the other hand, a non-displaceable wall is first constructed and then a surcharge load is applied, the earth pressure determined according to the elasticity theory is to be doubled due to the reflection principle. The justification results from the necessity of achieving zero displacement as the limit condition in the cross section directly behind the wall. This is not the case with the half-space solution. It assumes that the originally vertical plane deforms in the soil. A critique in the determination of the resultant of the at-rest earth pressure from surcharges is that, as a rule, smaller values emerge than in the active case, thus reversed as compared to the shares from soil self-weight. However, this is mechanically correct, as the following example shows. The program PLAXIS was used to calculate the development of earth pressure resultants depending on the displacement for a wall with a height of 5.00 m and an embedment depth of 5.00 m with rotation about the toe. The model and the material parameters of the Hardening Soil model are shown in Fig. 5.4. The calculation procedure was as follows: • • • •

Apply the strip load of 10 kN/m2 or 20 kN/m2 to the half-space, Activate the retaining wall, Soil excavation of 5.00 m with simultaneous fixing of the wall, Gradual rotation of the wall up to a head displacement of 2.50 cm

Fig. 5.5 shows the results for an interface element with Rinter = 1.0, Fig. 5.6 with Rinter = 0.5. In both cases, the share from the soil self-weight decreases, while the earth pressure loads due to surcharges increase. Remark that the loads are calculated from the top of the wall to the bottom of the excavation. In special cases, when the surcharges become very large compared to the soil self-weight, the sum of the earth pressure loads from soil self-weight and surcharges for the at-rest condition can become smaller than in the active case. In order to avoid under-dimensioning, the earth pressure from surcharges then has to be set on the safe side, see DIN 4085:2017-08.

221

5 At-rest earth pressure

Earth pressure force (kN/m)

140 120

Soil self-weight

100

Surcharge 10kPa

80 Soil self-weight + surcharge 10kPa

60

Surcharge 20kPa

40 Soil self-weight + surcharge 20kPa

20 0

0

2 1 1.5 Displacement (cm)

0.5

a)

2.5

3

160

Variation of earth pressure (%)

140 120 100

Soil self-weight

80

Surcharge 10kPa

60

Surcharge 20kPa

40 20 0

0.5

0

1

b)

1.5 2 Displacement (cm)

2.5

3

Fig. 5.5 Dependency of the earth pressure resultant wall displacement, Rinter = 1 according to Hölter (2016). a) Absolute values, b) Relative variation. 140 Earth pressure force (kN/m)

222

120 100

Soil self-weight Surcharge 10kPa

80

Soil self-weight + surcharge 10kPa Surcharge 20kPa

60 40

Soil self-weight + surcharge 20kPa

20 0

0

0.5

a)

Fig. 5.6 (Continued)

1

1.5 2 2.5 Displacement (cm)

3

3.5

References

Variation of earth pressure (%)

160 140 120 100 Soil self-weight

80

Surcharge 10kPa

60

Surcharge 20kPa

40 20 0

0

b)

0.5

1

1.5 2 2.5 Displacement (cm)

3

3.5

Fig. 5.6 (Cont’d) Dependency of the earth pressure resultant wall displacement, Rinter = 0.5 according to Hölter (2016). a) Absolute values, b) Relative variation.

References Franke, E. (1974). Ruhedruck in kohäsionslosen Böden. Bautechnik 51 (1): 18–24. Fröhlich, O. K. (1934). Druckverteilung im Baugrund. Wien: Springer. Goldscheider, M. (2017). Berechnung des Erdruhedrucks bei geneigtem Gelände und geneigter Schnittfläche. Geotechnik 40 (2): 103-118. Gudehus, G. (1996). Erddruckermittlung. Grundbau-Taschenbuch, 5. Auflage, Teil 1, Abschnitt 1. 10, Berlin: Ernst & Sohn. Hölter, R. (2016). Berechnungen zum Erddruck mit Auflast. Lehrstuhl Baugrund-Grundbau, unveröffentlicht. Jáky, J. (1944). A nyugalmi nyomas Aenyezöje (Die Ruhedruckziffer). Magyar Mern. Ep. Egyl., Közlönye (Mitteilungen des Vereins der ung. Ing. und Arch.) 22: 355. Pelz, G. (2011). Die Berücksichtigung einer Vorbelastung bei der Mobilisierung des passiven Erddrucks für körnige Böden. TUM Zentrum Geotechnik, Heft 48. Schmidt, B. (1966). Discussion of Earth Pressure at Rest Related to Stress History. Canad. Geot. Journal 3 (38): 239–242. Weißenbach, A. and Hettler, A. (2011). Baugruben – Berechnungsverfahren. 2. Auflage, Berlin: Ernst & Sohn. Weißenbach, A. (1985). Baugruben, Teil II, Berechnungsgrundlagen, Berlin, Ernst & Sohn, 1. Nachdruck.

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225

6 Passive earth pressure under plane strain conditions 6.1 Fundamental considerations All earth pressure calculations represent approximations, and this applies above all to passive earth pressure. The size and distribution of passive earth pressure are subject to wide limits, depending on the model and type of wall movement, which can easily go beyond the usual safety factors. Therefore, a high degree of diligence is required when determining the earth resistance. In particular, the displacements required to mobilise earth resistance must be observed. If, for example, a share of the earth resistance is to be taken into account as a supporting force for horizontally loaded foundations (Fig. 6.1a), then compatibility with the rising structure has to be ensured. In the case of nearly inflexible diaphragm walls, which are designed with an increased active earth pressure (Fig. 6.1b), the safety factor of the earth resistance on the toe of the wall may have to be adopted in order to consider wall displacements. The earth pressure inclination angle and the analysis model have a great influence on the earth resistance coefficient. This raises the question of the analysis model and, because of the ease of use, the scope of Coulomb’s theory. According to DIN 4085:2011-05 and the “Recommendations on Excavations” (EAB 2012), this is permissible for example for pile walls and sheet pile walls, which can be classified as rough walls with a wall friction angle of δ = φ to calculate according to Coulomb up to φ = 35∘ with the reduced absolute earth pressure inclination angle of δp = 2/3 ⋅ φ in order to correct for model error with straight slip surfaces. The Kp -values of Streck/Weißenbach (1985) can also be used up to δp = −0.9 ⋅ φ. At δp = −φ, the values of Caquot/Kérisel/Absi (Caquot et al. 1973) can be considered. It has to be kept in mind that the earth pressure coefficients published in 1948 by Caquot/Kérisel contain inconsistencies. For example, Fig. 6.2 shows a comparison of Kpgh -values with different models by Winkler (2003), which confirms the previous procedure. Refer also to the detailed investigations by Vrettos et al. (2017). Section 6.5.1 of DIN 4085:2011-05 states the following: “in general, earth pressure coefficients are to be used, which are based on curved slip surfaces or combined failure mechanisms.” In practice, this passage has often been understood as a ban on the application of Coulomb’s theory - except in special cases. The subsequent version from 2017 explicitly admits Coulomb’s wedge Earth Pressure, First Edition. Achim Hettler and Karl-Eugen Kurrer. © 2020 Ernst & Sohn Verlag GmbH & Co. KG. Published 2020 by Ernst & Sohn Verlag GmbH & Co. KG.

226

6 Passive earth pressure under plane strain conditions

a)

b)

Fig. 6.1 Verification of the possible degree of mobilisation of passive earth pressure in regard to compatibility. a) with horizontal displacements of the rising structure, b) with toe of the wall deformation at the design with increased active earth pressure.

Fig. 6.2 Comparison of Kpgh for α = 0, β = 0, φ = 35∘ according to Winkler (2003).

theory when comparability is given; see chapters 5.1 and 7.1 of the standard. The model error is relatively small if the specifications of the EAB (2012) are complied with. For a vertical wall and flat ground surface in the worst case of φ = 35∘ and δp = −35∘ , one obtains according to chapter 6.2 a value of Kpgh = 9.03 according to Pregl/Sokolowski (Pregl 2002). For Coulomb’s theory with δp = −2/3 ⋅ φ = −23.33∘ , this results in Kpgh = 9.17, which is basically the same value with a deviation of approx. 1.5%. Even at φ = 35∘ and δp = −2/3 ⋅ φ, the error is at most 25% using Kpgh = 7.26 according to Pregl. Thus comparability is given if the specifications of the EAB are complied with.

6.2 Soil self-weight, infinite uniformly distributed surcharges

6.2 Soil self-weight, infinite uniformly distributed surcharges and cohesion with parallel movement The solution for straight slip surfaces according to Coulomb and the elaboration based on the static stress field methods of characteristics by Pregl will be discussed in more detail below. First, the case of a vertical wall with horizontal terrain and variable earth pressure inclination will be covered. Further methods are described, e. g. in DIN 4085, in Weißenbach (1985) or in Gudehus (1996), as well as their cited references, see also chapter 3. As in the active case, passive earth pressure is approximately divided into three components: soil self-weight, surcharge and cohesion. The resultant is Ep = Epg + Epp + Epc .

(6.1)

The horizontal component results from Ep Eph = Ep ⋅ cos δp .

(6.2)

Hence, the vertical component is Epv = Eph ⋅ tan δp .

(6.3)

The horizontal component of the passive earth pressure can be expressed analogously to the active case in the form eph = γ ⋅ z ⋅ Kpgh + p ⋅ Kpph + c ⋅ Kpch .

(6.4)

It is common, and also shown by comparison with Finite Element calculations (chapter 3.6), to equate the results for straight slip surfaces as well as the results by Pregl/Sokolowski with the solution for a parallel movement of the wall (Winkler 2003), although, no statements are made about kinematics for example in the method of characteristics by Sokolowski (see chapter 3.4).

6.2.1

Straight slip surfaces

For the case of a vertical wall with α = 0 and a horizontal ground level with β = 0, one obtains Kpgh = Kph = [

√ 1−

cos2 φ

]2

(6.5)

− tan φ.

(6.6)

sin(φ − δp ) ⋅ sin φ cos δp

and for the inclination of the slip surface √

tan ϑp = cos φ ⋅

1 tan φ − tan δp tan φ

227

228

6 Passive earth pressure under plane strain conditions

For a smooth wall with δp = 0, the solutions are ( φ ) 1 + sin φ = Kpgh = Kph = tan2 45∘ + 2 1 − sin φ φ ϑp = 45∘ − . 2

(6.7) (6.8)

According to equation (6.4), the passive earth pressure at depth z is obtained as epgh = γ ⋅ z ⋅ Kph

(6.9)

and the resultant on a wall of height h is Epgh =

1 ⋅ γ ⋅ h2 ⋅ Kph . 2

(6.10)

The share from infinite uniformly distributed surcharges is determined by Kpph = Kpgh = Kph

(6.11)

epph = p ⋅ Kph .

(6.12)

from

The resulting passive earth pressure load on a wall of height h is Epph = p ⋅ h ⋅ Kpgh .

(6.13)

The cohesion component (Weißenbach 1985) should be approximated with the earth resistance coefficient √ √ Kpch = 2 ⋅ Kph ⋅ cos δp . (6.14) As a simplification with Kpch = 2 ⋅

√ cos δp ≈ 1, this gives

√ Kph = (Kpgh − 1) ⋅ cot φ.

(6.15)

With equation (6.15), the horizontal component of the passive earth pressure of cohesion is √ epch = 2 ⋅ c ⋅ Kph (6.16) and its resultant on a wall of height h is √ Epch = 2 ⋅ c ⋅ h ⋅ Kph .

(6.17)

An overview of the Kph -values according to equation (6.5) is provided in the tables compiled by Weißenbach, see Table B.4 in Appendix B.

6.2 Soil self-weight, infinite uniformly distributed surcharges

6.2.2

Pregl/Sokolowski

According to Pregl, it is necessary to strictly differentiate between the earth pressure coefficients Kpgh , Kpph and Kpch . For the case of a vertical wall with a horizontal ground surface, the formulas given in DIN 4085:2017-08 are simplified for a) soil self-weight δp ≤ 0

Kpgh = cos δp ⋅

1 + sin φ ⋅ (1 − 0.53 ⋅ δp )0.26+5.96⋅φ 1 − sin φ

(6.18a)

δp > 0

Kpgh = cos δp ⋅

1 + sin φ ⋅ (1 + 0.41 ⋅ δp )−7.13 1 − sin φ

(6.18b)

b) infinite uniformly distributed surcharges δp ≤ 0

Kpph = cos δp ⋅

1 + sin φ ⋅ (1 − 1.33 ⋅ δp )0.08+2.37⋅φ 1 − sin φ

(6.19a)

δp > 0

Kpph = cos δp ⋅

1 + sin φ ⋅ (1 − 0.72 ⋅ δp )2.81 1 − sin φ

(6.19b)

c) cohesion

(

δp ≤ 0

Kpch = cos δp ⋅

δp > 0

Kpch = cos δp ⋅

(

) 1 + sin φ − 1 ⋅ cot φ ⋅ (1 − 1.33 ⋅ δp )0.08+2.37⋅φ (6.20a) 1 − sin φ

) 1 + sin φ − 1 ⋅ cot φ ⋅ (1 + 4.46 ⋅ δp ⋅ tan φ)−1.14+0.57⋅φ . 1 − sin φ (6.20b)

It should be noted that δp and φ are always to be used in radians, where no trigonometric functions are specified. For example, for φ = 35∘ and δp = −φ/2 it is [ ]0.26+5.96⋅ π ∘ ⋅35∘ 180 1 + sin 35∘ π ∘) Kpgh = cos(−17.5∘ ) ⋅ ⋅ 1 − 0.53 ⋅ ⋅ (−17.5 ∘ ∘ 1 − sin 35 180 = 6.32. A compilation of the Kph -values calculated from equations (6.18) to (6.20) can be found in tables B.5a to B.5c in Appendix B. In the general case for α ≠ 0 and β ≠ 0, according to Pregl: soil self-weight: Kpgh = cos(δp + α) ⋅ Kpgh,0 ⋅ ipg ⋅ gpg ⋅ tpg

(6.21a)

infinite uniformly distributed surcharges: Kpph = cos(δp + α) ⋅ Kpph,0 ⋅ ipp ⋅ gpp ⋅ tpp

(6.21b)

cohesion: Kpch = cos(δp + α) ⋅ Kpch,0 ⋅ ipc ⋅ gpc ⋅ tpc

(6.21c)

229

230

6 Passive earth pressure under plane strain conditions

Table 6.1 Coefficients according to Pregl/Sokolowski 𝛅p

ipg

ipp

ipc

≤0

(1 – 0.53 ⋅ δp )(0.26 + 5.96⋅φ)

(1 – 1.33 ⋅ δp )(0.08 + 2.37⋅φ)

= ipp

>0

(1 + 0.41 ⋅ δp )−7.13

(1 – 0.72 ⋅ δp )2.81

(1 + 4.46 ⋅ δp ⋅ tan φ)(−1.14 + 0.57⋅φ)

𝛃

gpg

gpp

gpc

≤0

(1 + 0.73 ⋅ β)2.89

(1 + 1.16 ⋅ β)1.57

(1 + 0.001 ⋅ β ⋅ tan φ)(205.4 + 2232⋅φ)

>0

(1 + 0.35 ⋅ β)

(1 + 3.84 ⋅ β)

e2⋅β⋅tan φ

(0.42 + 8.15⋅φ)

0.98⋅φ

𝛂

tpg

tpp

≤0

(1 + 0.72⋅ α ⋅ tan φ)(−3.51 + 1.03⋅φ)

>0

(1 – 0.0012 ⋅ α ⋅ tan φ)(2910 − 1958⋅φ)

e−2⋅α⋅tan φ cos α

tpc

= tpp

including the basic values Kpgh,0 , Kpph,0 and Kpch,0 for α = β = δp = 0 according to equations (6.7), (6.11) as well as (6.15) and the coefficients ip , gp , tp to take into account that δp ≠ 0, β ≠ 0 and α ≠ 0, cf. Table 6.1. 6.2.3

Comparison

Fig. 6.3 shows a comparison of the Kpgh -values according to Pregl/Sokolowski with the Kph -values determined on the basis of Coulomb’s wedge theory. As already explained in chapter 6.1, the differences in the range of −φ/2 ≤ δp ≤ 0 are relatively small. The deviations at φ = 30∘ and φ = 35∘ in the range of 0 ≤ δp ≤ +φ are striking, but are no longer present at φ = 40∘ .

a)

Fig. 6.3 (Continued)

6.3 Rotation about the top or the toe

b)

c)

Fig. 6.3 (Cont’d) Comparison of the Kpgh -values depending on the earth pressure inclination angle δp , calculated with Coulomb’s wedge theory and curved slip surfaces according to Pregl/Sokolowski for a vertical wall (α = 0) and horizontal ground surface (β = 0). a) φ = 30∘ , b) φ = 35∘ , c) φ = 40∘ .

6.3 Rotation about the top or the toe All investigations show that the passive earth pressure in both size and distribution strongly depends on the type of wall movement (see chapter 3). In many practical cases, combined wall movements occur. DIN 4085:2017-08 states on this topic:

231

232

6 Passive earth pressure under plane strain conditions

Fig. 6.4 Example for a limited rotation of the wall about the toe.

“If, as in most cases, a combination of wall movement from toe rotation and parallel displacement, or from top rotation and parallel displacement occurs, and the foot of the wall is displaced by a certain amount which produces the passive earth pressure when the displacement is parallel, one can assume that the size and distribution of the earth pressure load should be the same as for pure parallel displacement of the wall, and a reduction can be avoided.” This means that the passive earth pressure can often be used for parallel displacement and no reduction is required. If there is pure rotation about the toe or the top, as in Fig. 6.4, the following reductions are recommended: Epgh,about the toe ≤ 0, 50 to 0, 67 Epgh,Parallel

(6.22)

Epgh,about the toe ≤ 0, 67 Epgh,Parallel .

(6.23)

For details, see DIN 4085:2017-08, chapter 7.1.

6.4 Distribution of passive earth pressure The investigations described in chapter 3 show a strong dependency of the distribution of passive earth pressure on the type of wall movement. DIN 4085:2017-08 proposes the simplified distributions shown in Fig. 6.5. With parallel displacement, a triangular distribution is used. The resultant Epgh, Parallel acts at third height

a)

b)

c)

Fig. 6.5 Simplified distribution of passive earth pressure according to DIN 4085. a) Rotation about the toe, b) Parallel displacement, c) Rotation about the top.

References

(Fig. 6.5b). A double triangle with the position of the resultant at half the height of the wall is suggested for rotation about the toe (Fig. 6.5a). In the case of rotation about the top, the distribution is approximately parabolic with the resultant acting at 0.2 times the wall height h. For details on the distribution of the vertical shares see DIN 4085.

References Caquot, A., Kérisel, J. and Absi, E. (1973). Tables de Butée et de poussée. Paris, Bruxelles, Madrid: Gauthier-Villars Éditeur. EAB (2012). Recommendations on Excavations. 5th ed., Berlin: Ernst & Sohn. Gudehus, G. (1996). Erddruckermittlung. Grundbau-Taschenbuch, 5. Auflage, Teil 1, Abschnitt 1.10. Berlin: Ernst & Sohn. Pregl, O. (2002). Bemessung von Stützbauweken. Handbuch der Geotechnik, Bd. 16, Wien: Eigenverlag des Instituts für Geotechnik, Universität für Bodenkultur. Vrettos, C., Becker, A. and Merz, K. (2017). Vergleichsberechnungen Erddruck Angabe von Berechnungsformeln. Bericht für PraxisRegelnBau. Weißenbach, A. (1985). Baugruben, Teil II, Berechnungsgrundlagen, Berlin: Ernst & Sohn, 1. Nachdruck. Winkler, A. (2003). Ermittlung des passiven Erddrucks mit Beiwerten. Bautechnik 80: 81–89.

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7 Spatial active earth pressure 7.1 Fundamental considerations For relatively narrow walls with width l compared to the depth z, for example slurry-supported diaphragm wall, the active earth pressure is reduced compared to the case under plane strain conditions. The reason for this is a re-distribution of the earth pressure from the yielding earth wall to less yielding lateral areas, which is often referred to as arch action. As in the case under plane strain conditions, the spatial active earth pressure e(r) ah can be split into shares from soil self-weight e(r) , infinite uniformly distributed agh and cohesion e(r) : surcharge e(r) aph ach e(r) = e(r) + e(r) + e(r) . ah agh aph ach

(7.1)

If the spatial shares of earth pressure are applied to the case under plane strain conditions, equation (7.1) can be transcribed using the coefficients μagh , μaph and μach e(r) = μagh ⋅ eagh + μaph ⋅ eaph + μach ⋅ each ah

(7.2)

with eagh , eaph and each according to chapter 4. The shape coefficients at depth z depend on the ratio z/l and on the friction angle φ. Numerous theoretical investigations are available to determine the coefficients. For example, Piaskowski and Kowalewski (1965) assume a monolithic rigid block (Fig. 7.1) similar to Coulomb’s wedge theory. In the so-called shoulder theory, shear stresses τxy are applied to the lateral surfaces of a sliding wedge (Fig. 7.2). The modified element slab theory by Walz and Hock (1987) represents a further development of Terzaghi’s silo theory for spatial earth pressure issues (see chapter 12). This approach can also be used for circular problems with cylindrical walls (see chapter 7.2) and for layered soils. An overview can be found in Lorenz and Walz (1982). In the case of a vertical, smooth wall with α = 0 and δa = 0 as well as a horizontal ground surface with β = 0, DIN 4085:2017-08 uses the approximation formulas derived from D. Franke for the DIN with the data by Piaskowski/Kowalewski (1965) (φ ⋅ z) 2 (7.3) μagh = 1 − ⋅ arctan π 2⋅l Earth Pressure, First Edition. Achim Hettler and Karl-Eugen Kurrer. © 2020 Ernst & Sohn Verlag GmbH & Co. KG. Published 2020 by Ernst & Sohn Verlag GmbH & Co. KG.

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7 Spatial active earth pressure

Fig. 7.1 Model of sliding rigid blocks according to Piaskowski and Kowalewski (1965).

Fig. 7.2 Calculation of earth pressure according to the so-called shoulder theory according to Prater respectively (Lorenz and Walz 1982; Stocker and Walz 2001).

μaph = μagh .

(7.4)

Depending on the depth and size of the friction angle, the reduction factor μagh to determine the spatial earth pressure lies between 0.2 and 0.3 (Fig. 7.3). As already in DIN 4085 from 1987, the share from cohesion is set on the safe side at μach = 1.

(7.5) E(r) ah

The resulting spatial earth pressure force can be calculated with the following formula

per m width on a wall of height h

(res) (res) E(r) = μ(res) ⋅ Eagh + μaph ⋅ Eaph + μach ⋅ Each ah agh

(7.6)

by using ⎞ ⎡⎛ ⎤ ( ) ⎟ ⎢⎜ ⎥ φ ⋅ h 2 2 ⋅ l 1 (res) ⎟ ⋅ arctan ⎥ = 1 − ⋅ ⎢⎜1 + ( − μagh ) 2 π ⎢⎜ 2⋅l φ ⋅ h⎥ φ⋅h ⎟ ⎟ ⎢⎜ ⎥ ⎣⎝ ⎦ ⎠ 2⋅l ] [ ( ( ) )2 φ⋅h φ⋅h 2 2⋅l (res) = 1 − ⋅ arctan ⋅ ln 1 + + μaph π 2⋅l φ⋅h⋅π 2⋅l (res) μach = 1.

(7.7)

(7.8) (7.9)

7.2 Cylindrical surfaces

Fig. 7.3 Reduction ratio μagh for the determination of spatial active earth pressure according to DIN 4085.

In all equations, φ is to be entered in radians. The reduction factors in equations (7.3) to (7.5) and (7.7) to (7.9) can also approximately be used in accordance with DIN 4085:2017-08 for cases α, β, δa ≠ 0. As the investigations by tom Wörden (2010) and by tom Wörden and Achmus (2013) show, the proposal of DIN 4085 is on the safe side. The reduction factor 2D λ = E3D a ∕Ea as the ratio of the resultants in the spatial and in the plane strain case according to DIN 4085 forms an upper bound, independent of the ratio n = wall height/wall width (see Fig. 7.4). According to tom Wörden and Achmus, the approach is approximately λ = 1 − 0.88 ⋅ (1 − 0.7n ).

(7.10)

7.2 Cylindrical surfaces For wells, shafts or slurry suspended boreholes with cylindrical surfaces, there can be axially symmetrical limit stress fields for the radial stress σr , the vertical stress σz , the shear stress τrz and the cyclic stress σθ established for sufficiently large deformations. Applying the static stress field method (see chapter 3.4), two equilibrium conditions and the Mohr-Coulomb failure envelope are available for the four unknown stress components. Using the so-called Haar/v. Karmann condition, the problem is statically determined, cf. Gudehus (1996). A complete description of the problem is given by Kézdi (1962), where the approximation solution according to Beresanzew can also be found, which assumes straight slip lines. For non-cohesive soil with a surcharge q, the solution for a cylinder with radius r0 depending on the coordinate r is ( φ) [ ( )λ−1 ] ( )λ tan 45∘ − ( φ) r r 2 ⋅ 1− (7.11) σr = γ ⋅ r ⋅ ⋅ tan2 45∘ − +q ⋅ λ−1 rb rb 2

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7 Spatial active earth pressure

Fig. 7.4 Comparison of shape coefficients according to tom Wörden and Achmus (2013).

with the abbreviations

) ( π φ λ = 2 ⋅ tan φ ⋅ tan + 4 2 ) ( π φ rb = r0 + z ⋅ tan . − 4 2

(7.12) (7.13)

Fig. 7.5 shows the horizontal stresses in dimensionless form with the coordinates σr /(r0 ⋅ γ) and z/r0 for the case without cohesion and without surcharge at friction angles φ between 20∘ and 35∘ . Beresanzew’s simplified theory delivers relatively small earth pressures. A precondition is yielding wall constructions. In the case of slightly yielding systems, for example, the active ultimate limit state will scarcely be reached and higher stresses are to be expected. In these cases, the EAB (2012) recommends the modified element slab theory by Walz and Hock (1987, 1988) as the upper

7.3 Retaining wall across the slope

Fig. 7.5 Horizontal stresses on the wall of a cylinder with r = r0 in sand at the active limit state according to the simplified theory of Beresanzew (Kézdi 1962).

limit value for the determination of the earth pressure force. As the example in Fig. 7.6 shows, considerable differences can result. In contrast to the theory of Beresanzew, the ratio h/d of the shaft depth h to the shaft diameter d is still used in the modified element slice theory.

7.3 Retaining wall across the slope In the case of bridge abutments and culverts through dams, wing walls are often arranged at a right angle to the dam axis. In the slope zone, the terrain is inclined in the wall plane (Fig. 7.7a and b). The earth pressure theories presented in chapter 4 for the plane strain case are not valid because they are a spatial problem. For this question, Rendulic proposed an approximation (Rendulic 1938) which was further developed by Schiel (1948). The version revised by Franke (1989, 1981) was included in DIN 4085:2011-05. The successor version contains only one note. , the active earth For the determination of the spatial active earth pressure e(B) agh pressure eagh is first calculated as in the plane strain case with α = β = δa = 0 and increased by the factor ξ = ξ ⋅ eagh . e(B) agh

(7.14)

If β ≤ 0.8, it is possible, according to Goldscheider’s internal working paper for the committee of DIN 4085, to use as a simplification ξ-values with the distribution indicated in Fig. 7.7c over the slope section and the maximum value

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7 Spatial active earth pressure

Fig. 7.6 Comparison of earth pressures on cylindrical surfaces at φ = 35∘ and δa = 0, determined with various procedures according to Walz and Hock (1988). Fig. 7.7 Retaining wall with wing wall across the dam axis.

a)

b)

c)

7.3 Retaining wall across the slope

b)

c) a)

d)

Fig. 7.8 Retaining wall at an angle to the fall line of the slope.

of ξ = 1.15. The break point in the distribution of the ξ-value can be determined by ϑaB = 45∘ (see Fig. 7.7b). In the case of retaining walls arranged at an angle to the fall line (Fig. 7.8a to d), the earth pressure can be calculated analogously, but eagh has to be determined as in the case of a plane slope with a slope inclination tan βn of the terrain profile at a right angle to the retaining wall plane (Fig. 7.8d, cross-section C-C). For more detailed investigations in the case of β > 0.8 ⋅ φ, the factor ξ and the limit angle ϑaB may be determined using the following formulas: ξ=

κ ⋅ cos2 βB 1 − sin φ

(7.15)

with slope angle βB according to Fig. 7.7b or βT according to Fig. 7.8 and √ (7.16) κ = 1 − 1 − (1 + tan2 βB ) ⋅ cos2 φ as well as the numerical values of ξ depending on φ and β/φ according to Fig. 7.9 and tan ϑaB =

tan φ ⋅ tan βB + tan φ. 1−κ

(7.17)

For cohesive soils with βB > φ, it is possible to set βB = φ as an approximation.

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7 Spatial active earth pressure

0

5

10

15

20

25

30

35

40

1.7

1.7

1.6

1.6

1.5

1.5 βB/φ = 1

1.4

1.4

ξ 1.3

1.3 1.2

1.2

0.8

0.6 0.4

1.1

1.1

0.2

1.0

1.0 0

5

10

15

20 φ in °

25

30

35

40

Fig. 7.9 Factor ξ according to eq. (7.15) for active earth pressure with ground inclined in the wall plane for α = δa = 0.

References EAB (2012). Recommendations on Excavations. 5th ed., Berlin: Ernst & Sohn. Franke, D. (1989). Beiträge zur praktischen Erddruckberechnung. Technische Universität Dresden, Habilitationsschrift (Promotion B), Dresden. Franke, D. (1981). Erddruck auf Querflügelmauern. Bauleitung – Bautechnik 12: 85–88. Gudehus, G (1996). Erddruckermittlung. Grundbau-Taschenbuch, 5. Auflage, Teil 1, Abschnitt 1.10, Berlin: Ernst & Sohn. Kézdi, A. (1962). Erddrucktheorien. Berlin, Göttingen, Heidelberg: Springer. Lorenz, H. and Walz, B. (1982). Ortswände. Grundbau-Taschenbuch, Teil 2, Abschnitt 2.14, 3. Auflage, Berlin: Ernst & Sohn. Piaskowski, A. and Kowalewski, Z (1965). Application of Thixotropic Clay suspensions of Vertical Sides of Deep Trenches without Strutting. Proc 6. ISSMFE Montreal, Vol. II, 526–529. Rendulic, L. (1938). Der Erddruck im Straßenbau und Brückenbau. Forschungsarbeiten aus dem Straßenwesen, Band 10, Berlin: Volk und Reich Verlag. Schiel, F. (1948). Erddruck auf Querflügel. Abhandlungen über Bodenmechanik und Grundbau, Forschungsgesellschaft für das Straßenwesen. Berlin, Bielefeld, Detmold: Erich-Schmidt-Verlag. Stocker, M. and Walz, B. (2001). Pfahlwände, Schlitzwände, Dichtwände. Grundbau-Taschenbuch Teil 3, Abschnitt 3. 5, Berlin: Ernst & Sohn. tom Wörden, F. and Achmus, M. (2013). Numerical modeling of three-dimensional active earth pressure acting on rigid walls. Computers and Geotechnics 51: 83–90.

References

tom Wörden, F. (2010). Untersuchungen zum räumlichen aktiven Erddruck auf starre vertikale Bauteile im nichtbindigen Boden. Institut für Grundbau, Bodenmechanik und Energiewasserbau (IGBE), Leibniz Universität Hannover, Heft 68. Walz, B. and Hock, K. (1987). Berechnung des räumlichen aktiven Erddrucks mit der modifizierten Elementscheibentheorie. Forschungs- und Arbeitsberichte aus den Bereichen Grundbau, Bodenmechanik und Unterirdisches Bauen an der Bergischen Universität Wuppertal, Bericht Nr. 6. Walz, B. and Hock, K. (1988). Berechnung des räumlichen Erddrucks auf die Wandungen von schachtartigen Baugruben. Taschenbuch für den Tunnelbau. Essen: Glückauf Verlag.

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8 Spatial passive earth pressure 8.1 Overview From a practical point of view, the question of a spatial effect at the passive earth pressure mainly arises in the following cases: • Anchor slabs, • Abutment walls for pipe jacking, • Bearing capacity in front of soldier piles. The spatial earth resistance in front of individual anchor slabs can be estimated according to Buchholz if the requirements for the density of the soil apply, cf. equation (3.33) in chapter 3.5. A general procedure was developed in accordance with Weißenbach by Arnold and Herle (Arnold et al. 2008). If the failure mechanisms of parallel anchor slabs overlap, one can calculate with an equivalent anchor wall (Fig. 8.1). The earth resistance is set in front of the imaginary wall, which reaches up to ground level, and the active earth pressure is set behind the wall (Fig. 8.2). One precondition is, however, that the embedding factor H/h is not more than 5.5 and a critical distance is not exceeded. For details, see sheet pile wall manuals of steel manufacturers, e.g. Hoesch (1986). In the case of abutment walls for pipe jacking, the Kph -values for the plane strain case or the methods described in chapters 8.2 and 8.3 can be used. The well-proven procedure of Weißenbach is available for spatial passive earth pressure in front of soldier piles (see chapter 8.2). The proposal from DIN 4085:2017-08 was derived from this procedure (see chapter 8.3).

8.2 Passive earth pressure in front of soldier piles according to Weißenbach Starting from the investigations described in chapter 3.5, Weißenbach (1985) developed a procedure for the determination of the spatial earth resistance in front of soldier piles with width bt , embedment depth t and pile spacing at . The resulting spatial earth resistance E* ph is calculated from E∗ph =

1 • • 1 γ ωR • t3 + 2 • c • ωK • t2 = • γ • ωph • at • t2 . 2 2

(8.1)

Earth Pressure, First Edition. Achim Hettler and Karl-Eugen Kurrer. © 2020 Ernst & Sohn Verlag GmbH & Co. KG. Published 2020 by Ernst & Sohn Verlag GmbH & Co. KG.

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8 Spatial passive earth pressure

Fig. 8.1 Position of an equivalent anchor wall according to the Hoesch manual for sheet pile walls (Hoesch 1986). a) Vertical cross-section, b) Ground plan.

a)

b)

The coefficients ωR and ωK depend on the ratio ft =

bt pile spacing = . t embedment depth

For ft ≤ 0.30:

√ ft √ (1.20 + 0.90 • tan φ) • ft .

(8.2)

ωR = 1.826 ⋅ KR • (0.30 + 0.60 • tan φ) •

(8.3)

ωK = 1.826 ⋅ KK

(8.4)



Equations (8.1) to (8.4) can be converted into the proposals from DIN 4085, as shown below as an example, for a share from cohesion for bt /t ≤ 0.3. If the spatial earth resistance is applied to one metre of wall width, this gives using Eph,c = 2 • c • t • KK for the plane earth resistance from cohesion the shape coefficient √ t μc = 1.826 • (1.20 + 0.90 • tan φ) • bt

(8.5)

(8.6)

as factor for the increase of earth resistance for the case of plane strain conditions. If, as suggested in EAB (2012), only 50% of the cohesion is taken, one obtains after conversion √ t . (8.7) μc = 1.1 • (1.0 + 0.75 • tan φ) • bt

8.2 Passive earth pressure in front of soldier piles according to Weißenbach

Fig. 8.2 Slip surfaces and forces on an anchor wall according to the Hoesch manual for sheet pile walls (Hoesch 1986).

This corresponds to DIN 4085:2017-08 if the embedment depth t of the soldier pile is replaced by the wall height h. For the share of t/b ≥ 0.3, a correction to the original proposal by Weißenbach was necessary in order to include the plane strain earth resistance as a limiting case, even if the reduction is down to 50% of the cohesion for b/t ad infinitum. On the other hand, the original formulas of Weißenbach apply to capillary cohesion. In this case, cohesion is not diminished. The coefficients ωR and ωK are also given as tables (Weißenbach 1985) depending on the ratio ft = bt /t and the friction angle φ. With the equivalent earth resistance coefficient ωph =

2 • E∗ph γ • t 2 • at

(8.8)

the calculation methods derived for continuous sheet pile walls can also be applied to soldier pile walls, provided the failure bodies in front of the individual soldier piles do not overlap and have an appropriate spacing. In case of overlapping, the equivalent earth resistance coefficient is taken from the following equation ωph =

√ bt at − bt 4⋅c • • K • K Kph (δp ≠ 0). (8.9) ph (δp ≠ 0) + ph (δp = 0) + at at γ•t

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8 Spatial passive earth pressure

This assumes the idea of a continuous wall with different earth pressure inclination angles in front of the soldier piles and in the area between the soldier piles. For non-cohesive soils, the last part of equation (8.9) does not apply. Setting δp = 0 gives ωph = Kph (δp = 0) in this case. The earth pressure inclination angle has to be limited at a restricted vertical movement by wall friction to δ∗p = − (φ − 2.5∘ ) δ∗ = − 27.5∘

for soils with φ ≤ 30∘ for soils with φ ≥ 30∘

p

Details of the calculation procedure can be found in Weißenbach (1985).

8.3 Procedure according to DIN 4085 for limited wall sections As in the active case (see chapter 7.1), the resulting spatial passive earth pressure E(r) per metre of wall length can be expressed in the form ph (res) • • E • E = μ(res) Epph + μ(res) E(r) pgh + μpph pch ph pgh pch

(8.10)

with shape coefficients μ(res) for the individual components of the plane strain ph resultant earth resistance Eph per metre wall length from soil self-weight, infinite uniformly distributed surcharges and cohesion in chapter 6.2. The shape coefficients depend on the ratio b/h of the wall width b to the wall height h and on the friction angle φ. The formulas given in DIN 4085:2017-08 were derived from the recommendations of Weißenbach (1985). A distinction has to be made between walls with b/h < 0.3 and b/h ≥ 0.3. For relatively narrow walls with b/h < 0.3, the wall section cuts through the soil body, the soil is laterally displaced, and the earth surface is lifted. At b/h ≥ 0.3, Weißenbach observed failure bodies. The equations given in DIN 4085 lead to the following shape coefficients: • Soil self-weight μ(res) pgh



= 0.55 • (1 + 2 • tan φ) •

μ(res) = 1 + 0.6 pgh



h b



h b

tan φ

b < 0.3 h b for ≥ 0.3 h

for

(8.11a) (8.11b)

• Infinite uniformly distributed surcharges (res) = μpgh μ(res) pph

(8.12)

i.e. the shape coefficients from the soil self-weight and infinite uniformly distributed surcharges are identical.

References

• Cohesion μ(res) pch

√ = 1.1 (1 + 0.75 tan φ) •



= 1 + 0.3 μ(res) pch





h b

h • (1 + 1.5 • tan φ) b

b < 0.3 h b for ≥ 0.3. h

for

(8.13a) (8.13b)

Separate regulations are to be observed when capillary cohesion is used, see “Recommendations on Excavations” (EAB 2012). Equations (8.10) to (8.13) apply in the case that the failure bodies do not overlap. At a relatively small spacing, e.g. between soldier piles, it has to be checked whether the earth resistance for a continuous wall leads to smaller values and thus becomes decisive (see chapter 8.2). The average resulting earth resistance through per metre for a continuous wall results from Eph a−b b + EIIph • (8.14) a a with the following symbols: EIph resultant earth resistance under plane strain conditions according to chapter 6.2 between the individual wall sections with width b, the height h and the axial spacing a EIIph resultant earth resistance under plane strain conditions according to chapter 6.2 in front of the individual wall sections. The requirement for the setting of the inclination angle δp at EIph and EIIph has to be taken into account; for details, see DIN 4085:2017-08. If one sets a = at and b = bt for soldier piles, then equation (8.14) can be transferred to equation (8.9), taking into account the assumptions for the inclination angle δp . (through)

Eph

= EIph



References Arnold, M., Herle, I. and Gabener, H.-G. (2008). Mobilisierung des räumlichen passiven Erddrucks bei Erdüberdeckung. Geotechnik 31: 12–22. EAB (2012). Recommendations on Excavations. 5th ed., Berlin: Ernst & Sohn. Hoesch-Stahl-AG (1986). Spundwandhandbuch - Berechnung. Dortmund: Eigenverlag. Weißenbach, A. (1985). Baugruben, Teil II, Berechnungsgrundlagen, 1. Nachdruck Berlin: Ernst & Sohn.

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9 Influence of groundwater on earth pressure 9.1 Groundwater at rest When the groundwater is at rest, the formulas for the determination of earth pressure remain the same. Instead of the specific weight γ of the partially saturated soil, the specific weight γ’ for buoyancy has to be applied in the soil layers below groundwater.

9.2 Flowing groundwater In the case of flowing groundwater, the water pressure changes as a result of potential reduction and flow forces compared to groundwater at rest. This also changes the effective stresses in the soil and the earth pressure. Basically, the question can be solved by determining the flow net for the groundwater flow. Fig. 9.1 shows an example with vertical flow according to the Recommendations of the Committee for Waterfront Structures Harbours and Waterways (EAU 2012). The water pressures calculated by using the flow net are set on the sliding wedge (Fig. 9.2) and the inclination angle ϑ of the slip surface is varied until an extreme value occurs; in the case of active earth pressure, it is a maximum. For details, see EAU (2012), chapter 2.12.1.

Fig. 9.1 Example for a groundwater flow net in homogenous soil with vertical flow according to EAU (2012). Earth Pressure, First Edition. Achim Hettler and Karl-Eugen Kurrer. © 2020 Ernst & Sohn Verlag GmbH & Co. KG. Published 2020 by Ernst & Sohn Verlag GmbH & Co. KG.

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9 Influence of groundwater on earth pressure

Fig. 9.2 Consideration of groundwater with vertical flow for the determination of active earth pressure according to EAU (2012).

DIN 4085:2017-08 covers the case of horizontal flow (Fig. 9.3). The active earth pressure force Ea results for any inclination of the slip surface in Ea =

with G W1 W2 C ϑ α φ δa

(G − W1 ⋅ cos ϑ − W2 ⋅ sin α − C ⋅ sin ϑ) ⋅ sin(ϑ − φ) cos (ϑ − φ − δa − α) (W1 ⋅ sin ϑ − W2 ⋅ cos α − C ⋅ cos ϑ) ⋅ cos(ϑ − φ) + cos (ϑ − φ − δa − α)

(9.1)

total weight of the earth wedge water pressure on the slip surface water pressure between wall and earth wedge cohesion force in the slip surface inclination of the slip surface wall inclination effective friction angle inclination angle of active earth pressure

Fig. 9.3 Forces on an active sliding wedge behind a vertical retaining wall with horizontal flow according to DIN 4085:2017-08. a) Cross-section through the wall with sliding wedge and water pressure forces W1 and W2 , total weight G, cohesion force C, reaction force Q and earth pressure force Ea , b) Force polygon.

9.3 Water pressure in tension crack

Fig. 9.4 Flow around a wall with predominant vertical direction.

The determination of the earth pressure force Ep in the passive case can be carried out in the same way with a changed direction for the friction force and the cohesion force in the slip surface. As an alternative to the approach of water pressures or the resulting water pressures vertically to all bounding surfaces of the soil body, as shown in Fig. 9.3, the flow forces in conjunction with the buoyancy force may also be used for the determination of earth pressures in a structurally equivalent manner. For details, see Goldscheider (2015). Both methods are generally very complex, but approximate solutions are permitted in many cases. According to EAU, for predominantly vertical flow as in Fig. 9.4 and homogeneous soil, the influence of the flow forces on the earth pressure may be taken into account by increasing the specific weight γ’ in case of a flow from the top downwards or by a reduction for a flow from the bottom upwards. In Fig. 9.4, this leads to an increase of the active earth pressure and to a reduction of the earth resistance in front of the wall. See EAU, chapter 2.12.3 for details.

9.3 Water pressure in tension crack In addition to the water pressures resulting from flowing groundwater in Fig. 9.2, forces from water pressures in tension cracks at the ground surface can also be considered (Fig. 9.5). For Central European climatic and soil conditions, this is rather a theoretical case.

Fig. 9.5 Consideration of forces from water pressures in tension cracks, here W2 and W3 , for the determination of active earth pressure according to Pregl (2002).

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References EAU (2012). Recommendations of the Committee for Waterfront Structures Harbours and Waterways. 11th ed., Berlin: Ernst & Sohn. Goldscheider, M. (2015). Darstellung von Wasserdrücken im Boden mit strömendem Grundwasser. Geotechnik 38 (2): 85–95. Pregl, O. (2002). Bemessung von Stützbauwerken. Handbuch der Geotechnik, Band 16, Wien: Eigenverlag des Instituts für Geotechnik, Universität für Bodenkultur.

255

10 Compaction effects on earth pressure When fill is applied in layers in a large-scale manner without compaction, then the at-rest earth pressure settles to the horizontal stress, see Fig. 10.1a. σh = γ ⋅ z ⋅ K0

(10.1)

During a pass of a compaction machine over the newly filled layer, the horizontal stress increases, see dashed line in Fig. 10.1a. As a result of the interlocking impact caused by plastic deformations, an increased horizontal stress is obtained after the pass, see line 0-1-2 in Fig. 10.1a. Down to the depth zp , a linear curve is obtained with a coefficient KR . For physical reasons, the earth pressure can only reach the passive earth pressure with KR ≤ Kph . If each layer is compacted on its filled surface A to D, the permanent horizontal stress shown in Fig. 10.1b occurs. From this, the curve of the earth pressure from compaction shown in Fig. 10.1c can be derived as a simplification. As a rule, KR = Kph is set. The depth zp essentially depends on the compaction machine, as shown for example by the investigations of Broms (1971) or Spotka (1977). When working spaces are backfilled, the width must be included in the consideration. The narrower the working space, the higher will be the additional earth pressure from compaction on inflexible walls. If the backfill borders a wall, the yielding of the wall is also important. In case of sufficient deformation, the active earth pressure is generally set instead of the at-rest earth pressure, and the additional earth pressure from compaction is less than in the case of a large-scale filling or in case of an inflexible wall. The stated aspects are all taken into account in DIN 4085:2017-08. In addition, the measurements of Petersen and Schmidt (1980) are considered, who recommend an earth pressure from compaction of 40 kN/m2 for small working spaces and 25 kN/m2 for large working spaces. With an inflexible wall, the passive earth pressure has to be assumed down to the depth (Fig. 10.2). evh zP = . (10.2) γ ⋅ Kph (δp = 0) Depending on the width B of the working space, the earth pressure from compaction can be taken as evh = 25 kN/m2 to 40 kN/m2 (Table 10.1). For light

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10 Compaction effects on earth pressure

a)

b)

c)

Fig. 10.1 Effect of compaction and distribution of horizontal stresses on an inflexible wall with large-area fill according to Broms (1971). a) during and after the vehicle passes the top layer, b) with layered compaction, c) proposal for calculation. Fig. 10.2 Approach of earth pressure from compaction on an inflexible wall.

Table 10.1 Statements of earth pressure from compaction to be assumed according to DIN 4085 for intensive and light compaction Flexibility of the wall

Intensive compaction Width B of the space to be filled ≤ 1.00 m ≥ 2.50 m

Yielding

evh = 25 kN/m2

Non-yielding

2

evh = 40 kN/m

Light compaction with vibrating plate with an operating mass of ≤ 250 kg

evh = 15 kN/m2

za = 2.00 m 2

evh = 25 kN/m

za = 2.00 m

evh = 15 kN/m2

for interim values of B, one can interpolate linearly

compaction, this is evh = 15 kN/m2 . The intersection point E with the line of at-rest earth pressure is at the depth e zE = vh . (10.3) γ ⋅ K0 For yielding walls, the curve of the earth pressure from compaction as shown in Fig. 10.3 is recommended. Both the depth of action za of the compaction as well as evh are independent of the width B of the space to be filled (Table 10.1).

References

Fig. 10.3 Approach of earth pressure from compaction on a yielding wall.

If there is a state between at-rest earth pressure and active earth pressure due to the flexibility of the wall, one has to interpolate. This can be the case, for example, for external basement walls. The regulations for surcharge loads should also be observed. Further details can be found in chapters 17.4 and 17.6 and DIN 4085:2017-08.

References Broms, B. B. (1971). Lateral pressure due to compaction of cohesionsless soils. Proc. 4th Int. Conf. Soil Mech. Found. Eng., 373-384, Budapest. Petersen, G. and Schmidt, H. (1980). Bodendruckmessungen an einem Tunnelbauwerk in abgeböschter Baugrube. Der Bauingenieur 55: 109–114. Spotka, H. (1977). Einfluss der Bodenverdichtung mittels Oberflächen-Rüttelgeräten auf den Erddruck einer Stützwand bei Sand. Mitteilungen des Baugrundinstituts der Universität Stuttgart, Mitteilung 22.

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11 L- and T-cantilever retaining walls For the verification of the external stability of L- and T-cantilever retaining walls, DIN 4085 permits the simplified assumption of a vertical equivalent wall according to Fig. 11.1. The resultant earth pressure must be applied parallel to the ground surface. As an alternative, DIN 4085:2017-08 refers to a method with two slip surfaces, which was proposed by Mörsch in 1925, see chapter 2. This approach is considerably more complex and leads in many cases to comparable results, see chapter 5.91 in supplement 1 to DIN 4085 from 1987. According to DIN 4085:2011-05, two slip surfaces are formed as shown in Fig. 11.2, if the platform behind the wall is long enough. The earth pressure is applied to the surface BCD and the share from the soil body between the wall and the first slip surface in the area BC is taken into account in the self-weight of the wall. The angle αmax between the first slip surface and the vertical is obtained from αmax = ϑag − φ

(11.1)

where ϑag is calculated under the assumption that α = 0 and δa = β. The earth pressure on area BC is determined under the assumption that α = αmax and δa = φ. If the first slip surface cuts the cantilever retaining wall at point B in case of a short foundation platform (Fig. 11.3), the earth pressure must be applied to area ABCD. As before, the weight of the soil in the area between the wall and BC has to be added to the self-weight of the wall. The earth pressure in area BC in Fig. 11.3 is to be determined as shown in Fig. 11.2. At sections AB and CD, the inclination angle of the wall αWall has to be set for α and the respective earth pressure inclination angle δa must be applied for δ. DIN 4085 from 1987 limits the method with a vertical back wall in Fig. 11.1 to cases with homogeneous soil and straight ground surface without limited surcharge loads. Otherwise, the first slip surface should be used as a back wall, although there is no explanation. However, in the case of stratification or limited surcharge loads, for example, one cannot expect high precision with this method with two slip surfaces according to Fig. 11.2 or Fig. 11.3, so that this approach is of a theoretical nature with little relevance in practice.

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Fig. 11.1 L- and T-cantilever retaining wall. a) Vertical equivalent wall, b) Horizontal component of earth pressure on cross-section ED.

Fig. 11.2 L- and T-cantilever retaining wall with long platform. a) Slip surfaces, b) Horizontal component of earth pressure on cross-section BCD.

If the displacement and tilting of the wall necessary to mobilise the active earth pressure cannot occur in non-yielding rock, it is recommended to apply the at-rest earth pressure with an inclination of δ = β on a vertical equivalent wall, cf. DIN 4085, supplement 1 from February 1987. There are different recommendations for the internal structural design of the vertical wall and the earth pressure approach directly behind the wall. DIN 4085 from 1987 assumes an active earth pressure parallel to the surface and a redistribution into a trapezium, in which the earth pressure ordinates behave as 1 to ′ 2 (Fig. 11.4a). On the other hand, an increased active earth pressure with E ah = 0.50 ⋅ Eah + 0.50 ⋅ E0h is proposed in subsequent standards for the structural design of the vertical walls of L- and T-cantilever retaining walls, as in the case of almost inflexible supporting structures with a triangular distribution (Fig. 11.4b). For this, one should also refer to the tests by Arnold (2001) and the discussion by

References

Fig. 11.3 Cantilever retaining wall with short platform. a) Slip surfaces, b) Horizontal component of earth pressure on cross-section ABCD.

Fig. 11.4 Approach of earth pressure for the structural design of the vertical wall. a) Redistributed active earth pressure according to DIN 4085:1987-02, b) Increased active earth pressure according to DIN 4085:2011-05.

Schmidt (2006). If the backfill material is compressed, the earth pressure due to compaction also has to be taken into account, see chapter 10. It is customary to apply the additional pressure due to compaction only for the internal structural design of the vertical wall, but not for the external design of the equivalent wall, see chapters 17.4 and 17.6.

References Arnold, M. (2001). Modellversuche zum Erddruck auf Winkelstützwände. Ohde Kolloquium 2001. Institut für Geotechnik, Technische Universität Dresden, Mitteilungen Heft 9, Dresden. Schmidt, H.-H. (2006). Grundlagen der Geotechnik. 3. Auflage, Teubner.

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12 Silo pressure If there is soil between two parallel rough walls and there is the possibility of a relative displacement between the soil and the wall, part of the soil self-weight is transferred to the walls through shear stresses τ and the vertical stresses σz are relieved (Fig. 12.1). This also reduces the horizontal pressure σx on the walls. Terzaghi considers this problem to be an arching effect (Terzaghi 1954). On the assumption that the active limit state prevails everywhere in the soil, an exact solution was developed by Kötter, whose derivation is provided by Kézdi (1962). This solution can be assigned to the static stress field methods as defined in chapter 3.4. For practical purposes, however, it is sufficient to rely on the approximate solution of Terzaghi (1936), which is based on the silo theory by Janssen (1895) and supposes the following assumptions: • The vertical stress σz is constant at depth z. • The ratio λ of horizontal stresses σx to vertical stresses σz is constant everywhere, therefore σx = λ ⋅ σz . The inclination angle δx on the wall is constant. This results in τ/σx = tan δ. Equilibrium at the slice segment of width b (Fig. 12.2) leads to the linear differential equation of the first order dσz 2 ⋅ λ (12.1) + ⋅ tan δ ⋅ σz = γ dz b with the solution [ ( )] b⋅γ 2⋅z σz = ⋅ 1 − exp −λ ⋅ ⋅ tan δ (12.2a) 2 ⋅ λ ⋅ tan δ b [ ( )] b⋅γ 2⋅z σx = ⋅ 1 − exp −λ ⋅ ⋅ tan δ (12.2b) 2 ⋅ tan δ b ( )] b⋅γ [ 2⋅z τ= ⋅ 1 − exp −λ ⋅ ⋅ tan δ . (12.2c) 2 b For practical use, an estimation of the horizontal earth pressure on the wall may be made by eh = σx and for the vertical share ev = τ. In the case of inflexible walls, one may use the state of at-rest earth pressure and λ = K0gh and, for yielding walls, the active state and λ = Kagh , see DIN 4085:2017-08. The inclination angle δ can be set according to the roughness of the wall. The theory can also be extended to surcharges and adhesion to the wall (Terzaghi 1954). Earth Pressure, First Edition. Achim Hettler and Karl-Eugen Kurrer. © 2020 Ernst & Sohn Verlag GmbH & Co. KG. Published 2020 by Ernst & Sohn Verlag GmbH & Co. KG.

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σz

Fig. 12.1 Distribution of stresses in a non-cohesive material situated between parallel, vertical walls and relatively displaced downwards (Kézdi 1962).

z γ·z σz

τ

γ·z

σz

Fig. 12.2 Equilibrium on slice segment.

When this method is applied to silos, a distinction has to be made between filling and emptying operations. In addition, the geometry of the silo also has to be observed, especially at the discharge hopper. An overview of various theoretical approaches is provided by Hampe (1987). Keiter (2004) discusses various standards and especially the problem of stress peaks at the transition to the silo discharge hopper, the so-called switch.

References Hampe, E. (1987). Silos. Band 1. Grundlagen, 1. Auflage. Berlin: VEB Verlag für Bauwesen. Janssen, H. A. (1895). Versuche über Getreidedruck in Silozellen. Zeitschrift des Vereins Deutscher Ingenieure 39: 1045–1049. Keiter, T. (2004). Schüttgutlasten bei imperfekten Silowänden. Mitteilungen aus dem Bereich Mechanik und Statik Nr. 10/2004, Universität Dortmund. Kézdi, A. (1962). Erddrucktheorien. Berlin, Göttingen, Heidelberg: Springer. Terzaghi, K. v. (1936). Stress Distribution in dry and saturated Sand above a yielding Trap Door. Int. Conf. Soil Mech. I, Cambridge. Terzaghi, K. v. (1954). Theoretische Bodenmechanik. Berlin, Göttingen, Heidelderg: Springer.

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13 Dynamic loading In the case of strong earthquakes, for example, dynamic actions from inertial forces also have to be considered. As a rule, this results in increased loading for structural design compared to the purely static situation. In this context, the possibilities of displacement of the supporting structure, the stiffness of the construction and the dynamic properties of the backfill material, including the liquefaction potential of the soil, all have to be considered. Klapperich and Savidis (1982) discuss extensive model tests concerned with this question. In the theoretical analysis, the earth pressure from dynamic actions is split into an elastic and a plastic component. A quasi-static analysis is proposed for practical design. In this case, the two dynamic earth pressure components are again combined to a global earth pressure coefficient, which is dependent on the excitation intensity and the bedding density. DIN 4085:2017-08 follows a similar idea. If liquefaction of the backfill material can be excluded, the seismic earth pressure force can be set quasi-statically based on plane slip surfaces. For this purpose, earthquake coefficients are used in horizontal and vertical directions kh = ah /g and kv = av /g, in which ah and av denote the corresponding components of the computational value of the earthquake acceleration and g the acceleration due to gravity. The earth pressure coefficient as a result from the soil self-weight and seismic loading may be estimated with Ka,g+dyn,h =

cos2 (φ − χ − α)• cos(α + δa )•(1 ± kv ) √ [ ]2 sin(χ + δa )• sin(φ − χ − β) 2 cos χ•cos α• cos(δa + α + χ)• 1 + cos(δa + α + χ)• cos(β − α) (13.1)

using ( χ = arctan

kh 1 ± kv

) .

(13.2)

Regarding the corresponding terms for passive earth pressure, DIN 4085:2017-08 refers to DIN EN 1998-5. For further details on dynamic loading, see Vrettos (2017). It is recommended to engage a specialist for soil dynamics. Earth Pressure, First Edition. Achim Hettler and Karl-Eugen Kurrer. © 2020 Ernst & Sohn Verlag GmbH & Co. KG. Published 2020 by Ernst & Sohn Verlag GmbH & Co. KG.

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References Klapperich, H. and Savidis, S. A. (1982). Experimentelle Untersuchungen zum dynamischen Erddruck. Geotechnik 5: 139–146. Vrettos, C. (2017). Geotechnisches Erdbebeningenieurwesen. In GrundbauTaschenbuch: (Hrsg., K. J. Witt), Teil 2, 8. Auflage. Berlin: Ernst & Sohn.

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14 Particular cases 14.1 Repeated quasi-static loading In the case of repeated, quasi-static loading, the frequency of the cycles is so small that acceleration effects are negligible. On the basis of elasto-plastic mechanics, there are two basic ways to distinguish how a system can react under repeated loading. The shakedown theory, see Koiter (1960) and Martin (1985), distinguishes between incremental collapse and shakedown. An incremental collapse, i.e. a stepwise failure with a linear increase in plastic displacements depending on the number of cycles, is not known in the case of earth pressure issues, if one ignores the trivial case that the static collapse load is repeatedly reached. As a rule, a shakedown, i.e. a slow-down is observed (see Hettler 1981). A logarithmic increase of displacement with the number of cycles is often found, with the system reacting purely elastically according to the shakedown theory of elasto-plastic mechanics. In practice, repeated loading occurs to the abutments of integral bridges, which expand and contract as a result of temperature fluctuations. The same applies to lock structures, to which there is also repeated loading from filling and emptying operations in addition to the action of temperature. As shown by the investigations into locks by Vogt (1984), movements are in the per mill range. For example, head movements of approximately 40 mm were observed at a wall height of 19.45 m in the lock in Eibach, which is about 2.1 ‰. Even lower were the percentage movements at the barrage in Iffezheim. Both the structure measurements as well as the model tests on a wall of 4 m height show a rapid slow-down. For model tests with forced head displacements of 2 mm, 4 mm and 8 mm, just one loading procedure was sufficient to reach asymptotic behaviour of the earth pressure forces (see Fig. 14.1). A similar result was also found in the numerical simulation of an integral bridge using an elasto-plastic scaling law with isotropic strain hardening (Winter 2006). On the other hand, a strong increase of the coefficient K′p for the mobilised passive earth pressure was observed in 1g model tests by England et al. (2000). For example, rotation about the toe with a head deflection of ± 2.5 ‰ related to the wall height resulted in an increase of K′p = 1.4 in the first cycle to K′p = 2.5 in the 120th cycle. Finally, a limit value of K′p ≈ 2.7 was reached (Fig. 14.2). Earth Pressure, First Edition. Achim Hettler and Karl-Eugen Kurrer. © 2020 Ernst & Sohn Verlag GmbH & Co. KG. Published 2020 by Ernst & Sohn Verlag GmbH & Co. KG.

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a)

b)

c)

Fig. 14.1 Development of earth pressure with repeated wall displacement according to Vogt (1984). a) Head displacement 2 mm, 50 cycles, b) Head displacement 4 mm, 50 cycles, c) Head displacement 8 mm, 20 cycles.

Fig. 14.2 Increase of the earth pressure coefficient Kp′ with the number of cycles in 1g model tests with repeated head deflection (England et al. 2000).

An increase of earth pressures with the number of cycles was also found in centrifuge tests carried out by Springman et al. (1994). The practical design of integral structures is based on the approach by Vogt, using an upper value for the earth pressure in the summer setting which can be estimated according to the guidelines for the design and construction of civil engineering structures (BMVBS 2017). The lower value for the winter setting is set at 50 % of the active earth pressure.

14.3 Lateral pressure on piles

14.2 Pipelines The determination of earth pressure on either covered or jacked pipes is subject to considerable uncertainty. In principle, three cases can be distinguished (Fig. 14.3). If it were possible to install a pipe in the soil without disturbance and the pipe behaved exactly like the surroundings, the surcharge stress σz would be exactly σz = γ ⋅ τ at the crown of the pipe at depth z (Fig. 14.3a). This case, however, is only of theoretical significance. In fact, the installation method plays a major role. For example, if a pipe is installed in a narrow trench, and the pipe and the fill settle, an arch effect is created, as in the case of a silo, and there is relief at the crown with σz < γ ⋅ τ (Fig. 14.3b). Conversely, a rigid pipe on hard rock with yielding backfill can lead to an increased vertical load on the pipe (Fig. 14.3c). In addition to the described effects and the installation method, the lateral bedding also plays an important role in pipe design. In individual cases, it is recommended to make various unfavourable assumptions for design purposes. For further details, see Kézdi (1962), Gudehus (1996) and Stein (2003) as well as current standards, in particular the bulletin DWA-A 127 “Structural design of waste water pipes and sewers” of the German Association for Water, Drainage and Waste (DWA 2008). For district heating pipelines, temperature-induced loading can cause not only movements in the longitudinal direction, but also transversely. A detailed overview is provided by Achmus and Weidlich (2016).

Fig. 14.3 The three basic cases of loading on an embedded pipe according to Kézdi (1962). a) Pipe deforms similarly to surrounding, b) yielding pipe in narrow trench, c) Non-yielding pipe on non-yielding base.

14.3 Lateral pressure on piles For filling or excavations next to piled foundations in soft cohesive soils, considerable horizontal loading can act on piles (Fig. 14.4). The lateral pressure on the piles can theoretically be understood as a transverse flow around the pile axis. A proposal by Winter (1979) can be used for practical design. For details, see

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14 Particular cases

a)

b)

Fig. 14.4 Examples of reasons for lateral pressure on piles (EA-Pfähle 2012). a) resulting from filling, b) resulting from an excavation.

Gudehus (1996) and the Recommendations on Piling (EA-Pfähle 2012). Newer approaches are described by Kempfert and Moormann (2017) as well as Kempfert and Gebreselassie (2006).

14.4 Creep pressure If retaining walls, single piles or pile walls are built in creeping slopes with a foundation on solid rock, a creep pressure builds up over time. Due to the build-up of creep movement, earth pressure can increase up to the flow pressure or to the passive earth pressure. No proven methods are yet available for estimating such a time-dependent pressure increase, cf. Gudehus (1996). Approximate approaches for individual cases have been developed by Haefeli (1944) as well as by Brandl and Dalmatiner (1988), see also Pregl (2002).

14.5 Swelling pressure Swelling pressure can occur in cohesive backfills, which tend to increase in volume, e.g. with water ingress. Especially in tunnel construction, swelling has led to considerable damage. In the gypsum Keuper of Baden-Wurttemberg and Switzerland as well as in the marl soils of the North Alpine molasse, they can mainly be traced back to the swelling of the mineral anhydrite and of the clay minerals corrensite, montmorillonite, and certain illites (see Fecker and Reik 1987). For example, the conversion of anhydrite into gypsum on addition of water is associated with a volume increase of approx. 61%. For further information on swelling pressure, see Prinz and Strauß (2011).

14.6 Heavily fissured rock Depending on the joint inclination β, the internal jointing structure of rock can have a great influence on earth pressure. Fig. 14.5 shows an example of

14.6 Heavily fissured rock

β1

β2

β1 < β2

θ1

a)

θ2

b)

Fig. 14.5 Influence of joint orientation on earth pressure according to Brandl (1992).

two differently formed sliding blocks. Although the two retaining walls cut into the same rock formation, a higher load is applied in Fig. 14.5a due to the anisotropy of the rock behaviour and this results in a more massive retaining wall compared to Fig. 14.5b on the opposite side of the cutting. In order to be able to appropriately assess the internal jointing structure, the joint density, the type of joint filling and the anisotropic behaviour, broad experience is required. An overview of classification systems for rock is provided e.g. by Prinz and Strauß (2011). For the computational determination of the earth pressure, the data in the chapter “Excavation pits in unstable rock” of the EAB (2012) can also be used.

a)

b)

Fig. 14.6 Procedure for the determination of the magnitude and inclination of the active earth pressure force in a dam according to Rendulic (1938). a) Analysed sliding blocks left and right of cross-section a–a, b) Failure envelopes with reflection according to Engesser.

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14.7 Active earth pressure within dams In an infinitely long slope, the earth pressure is inclined parallel to the ground surface. With a discontinuous surface, such as with dams, the absolute value and the direction of the earth pressure can be determined for any vertical section using a proposal by Rendulic (1938) including the Engesser method. As shown in Fig. 14.6, sliding wedges with different inclinations of the slip surface are investigated on the left and right of the analysed section and the corresponding failure envelopes are drawn according to Engesser (Fig. 14.6b). By reflecting the failure envelopes, the desired earth pressure is achieved with the condition Eagr = Eagl . For details see DIN 4085:2017-08. If the method is applied at different sections, the curve of shear stress on a dam formation can be determined. For details see Rendulic (1938).

References Achmus, M. and Weidlich, L. (2016). Interaktion zwischen Fernwärmeleitungen und dem umgebenden Boden. Bautechnik 93 (9): 663-671. Brandl, H. and Dalmatiner, J. (1988). Brunnenfundierungen von Bauwerken in Hängen. Bundesministerium für wirtschaftliche Angelegenheiten, Straßenforschung Wien, Heft 352. Brandl, H. (1992). Retaining structures for rock masses. In: Engineering in rock masses. (Ed. F. G. Bell), 530–572. Oxford: Butterworth-Heinemann. Bundesministerium für Verkehr, Bau und Stadtentwicklung (2017). Richtlinien für den Entwurf und die Ausbildung von Ingenieurbauten (RE-ING), Teil 2: Brücken, Abschnitt 5, Integrale Bauwerke, Stand: 2017/12. Deutsche Vereinigung für Wasserwirtschaft, Abwasser und Abfall e. V. (DWA) (2008). Arbeitsblatt DWA-A 127: Statische Berechnung von Abwasserleitungen und -kanälen. 3. Auflage, Eigenverlag. EAB (2012). Recommendations on Excavations (EAB), 5th ed., Berlin: Ernst & Sohn. EA-Pfähle (2012). Recommendations on Piling (EA-Pfähle). 2nd ed., Berlin: Ernst & Sohn. England, G. L., Tsang, N. and Bush, D. (2000). Integral bridges: A Fundamental approach to the time-temperature loading problem. Imperial College, Thomas Telford. Fecker, E. and Reik, H. (1987). Baugeologie. Stuttgart: Enke. Gudehus, G. (1996). Erddruckermittlung. Grundbau-Taschenbuch, 5. Auflage, Teil 1, Abschnitt 1.10. Berlin: Ernst & Sohn. Haefeli, P.-D. (1944). Zur Erd- und Kriechdruck-Theorie. Schweizerische Bauzeitung 124 (20): 256–260 and (21): 267–271. Hettler, A. (1981). Verschiebungen starrer und elastischer Gründungskörper in Sand bei monotoner und zyklischer Belastung. Veröffentlichungen Institut für Bodenmechanik und Felsmechanik der Universität Karlsruhe, Heft 90. Kempfert, H.-G. and Gebreselassie, B. (2006). Excavations and Foundations in Soft Soils, Berlin, Heidelberg: Springer.

References

Kempfert, H.-G. and Moormann, C. (2017). Pfähle. Grundbau-Taschenbuch, (Ed. K. J. Witt), 8. Auflage, Teil 3, Berlin: Ernst & Sohn. Kézdi, A. (1962). Erddrucktheorien. Berlin, Göttingen, Heidelberg: Springer. Koiter, W. T. (1960). General Theorems for Elastic-Plastic solids. In: Prog. in Solid. Mech. (Ed. I. N. Sneddon and R. Hill). Amsterdam: North Holland Publ. Comp. Martin, J. B. (1985). Plasticity. Cambridge, London: MIT-Press. Pregl, O. (2002). Bemessung von Stützbauwerken. Handbuch der Geotechnik, Band 16, Wien: Eigenverlag des Instituts für Geotechnik, Universität für Bodenkultur. Prinz, H. and Strauß, R. (2011). Abriss der Ingenieurgeologie. 5. Auflage, Elsevier, Spektrum Akademischer Verlag. Rendulic, L. (1938). Der Erddruck im Straßenbau und Brückenbau. Forschungsarbeiten aus dem Straßenwesen, Band 10, Berlin: Volk und Reich Verlag. Springman, S. M., Ng, C. W. W. and Ellis, E. A. (1994). Full Height Bridge Abutment in Soil Undergoing Lateral Movement ANS & A Final Report to Transport Research Laboratory (Contract No. 7890). 52 pages. Stein, D. (2003). Grabenloser Leitungsbau. Berlin: Ernst & Sohn. Vogt, N. (1984). Erdwiderstandsermittlung bei monotonen und wiederholten Wandbewegungen in Sand. Baugrundinstitut Stuttgart, Mitteilung, 22. Winter, H. (1979). Fließen von Tonböden. Eine mathematische Theorie und ihre Anwendung auf den Fließwiderstand von Pfählen. Veröffentlichungen Institut für Bodenmechanik und Felsmechanik der Universität Karlsruhe, Heft 82. Winter, H. (2006). Modellierung der Boden-Bauwerk-Interaktion bei integralen Brücken mit dem Programm Sofistik. Lehrstuhl Baugrund-Grundbau und Lehrstuhl für Betonbau der Technischen Universität Dortmund, Dortmund.

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15 Mobilisation of earth pressure 15.1 Overview In the case of earth pressure calculations, it is usually assumed that the at-rest earth pressure can be assessed with its resultant E0 in the initial state. If the wall is moved away from the soil, movements sa in the per mill range generally suffice to mobilise the active earth pressure Ea . If the wall is displaced against the soil, considerably larger displacements sp are required until the passive earth pressure Ep is achieved (Fig. 15.1). As shown by tests, cf. Weißenbach (1985), medium-dense to dense non-cohesive soils as well as over-consolidated and stiff to very stiff cohesive soils are similar, which means that the data for non-cohesive soils can also be applied to cohesive soils. The same applies with restrictions for loose non-cohesive soils as well as normally consolidated and soft cohesive soils. In the case of dense non-cohesive soils, the mobilised resultant earth pressure ′ E increases again after reaching the minimum value Ea at sa and decreases again after reaching the maximum value Ep at sp . The reason for this is the softening process in the soil, combined with a progressive failure. In addition to the density for non-cohesive soils and the consistency for cohesive soils, the displacements sa and sp also depend on the type of wall movement. For this reason, the type of wall movement must always be indicated for sa or sp . In order to simplify matters, one considers the three basic movement types which are parallel movement, rotation about the top and rotation about the toe.

15.2 Limit values of displacement on reaching the active earth pressure Information on the required displacements sa are given in chapter 3.5. An overview of various investigations on this subject can be found in Weißenbach (1985). Reference values for an implementation are provided in DIN 4085:2017-08, Appendix C, Table C1 (see Table 15.1). In the right-hand column, simplified earth pressure distributions are also listed. The resultant earth pressure may be set equally for all types of wall movement (see also chapter 3).

Earth Pressure, First Edition. Achim Hettler and Karl-Eugen Kurrer. © 2020 Ernst & Sohn Verlag GmbH & Co. KG. Published 2020 by Ernst & Sohn Verlag GmbH & Co. KG.

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15 Mobilisation of earth pressure

Fig. 15.1 Qualitative relationship between the magnitude of the mobilised earth pressure force E′ depending on the wall displacement s with parallel movement.

15.3 Limit values of displacement on reaching the passive earth pressure Analogously to Table 15.1 for the active case, Table 15.2 provides reference values for the displacements sp necessary for the mobilisation of the passive earth pressure or the earth resistance. Unlike the active earth pressure, the passive earth pressure and its resultant depend strongly on the type of wall movement. The reference value is the earth resistance for a parallel movement. As explained in chapter 6, the earth resistance obtained with simple sliding wedges according to Coulomb and the earth resistance with curved slip surfaces according to Pregl/Sokolowski is assigned to the earth resistance for parallel movement.

15.4 Mobilisation functions A distinction must be made between mobilisation functions for the resultant E′a of the mobilised active earth pressure and E′p of the passive earth pressure and subgrade reaction approaches for the local active earth pressure e′a and the passive earth pressure e′p at depth z. Strictly speaking, E′a , E′p , e′a , e′p as well as the displacements s all depend on the type of wall movement. There are numerous proposals in the literature, in particular for the mobilised passive earth pressure. An overview is provided, e.g. by Bartl (2004) or Besler (1998). Some approaches are presented below. 15.4.1

Mobilised active earth pressure

In contrast to passive earth pressure, only a few studies deal with the active case. According to Weißenbach (1985), an estimation of the displacement s can be made for design values between the at-rest earth pressure and the active earth pressure with the respective resultants E0h and Eah . s = 0.05sa to 0.15sa for E′ah = 0.75 ⋅ E0h + 0.25 ⋅ Eah

(15.1a)

15.4 Mobilisation functions

Table 15.1 Reference values of wall movements required for the mobilisation of the active earth pressure force and simple load distributions of earth pressure from the soil self-weight for different types of wall movement in non-cohesive soil, vertical wall and horizontal ground level according to DIN 4085:2017-08. Type of wall movement

Earth pressure force Eagh Relative wall movement sa /h Loose Dense

0.004 to 0.005

0.001 to 0.002

0.002 to 0.003

0.0005 to 0.001

0.008 to 0.01

0.002 to 0.005

0.004 to 0.005

0.001 to 0.002

Simplified earth pressure distribution

a) Rotation about the toe

b) Parallel movement

c) Rotation about the top

d) Deflection

s = 0.15sa to 0.30sa for E′ah = 0.50 ⋅ E0h + 0.50 ⋅ Eah

(15.1b)

s = 0.30sa to 0.50sa for E′ah = 0.25 ⋅ E0h + 0.75 ⋅ Eah

(15.1c)

The limit displacements sa on reaching the full active earth pressure can be taken from Table 15.1. If, for example, with parallel movement, medium-dense sand and sa = 1 ‰ of the wall height are assumed, one achieves with the approach frequently used for nearly inflexible walls E′ah = 0.50 ⋅ E0h + 0.50 ⋅ Eah for a 10 m high wall, sa ≈ 1 mm and s ≈ 0.15 to 0.3 mm, i.e. relatively small deformations. With respect to relatively inflexible diaphragm wall constructions, the displacements induced during the production of a diaphragm wall are sufficient for a stress release to E′ah = 0.75 ⋅ E0h + 0.25 ⋅ Eah . Maintenance of the at-rest earth pressure does not seem to be realistic.

277

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15 Mobilisation of earth pressure

Table 15.2 Reference values of wall movements required for the mobilisation of the passive earth pressure force and simple load distributions of earth pressure from the soil self-weight for different types of wall movement in non-cohesive soil, vertical wall and horizontal ground level according to DIN 4085:2017-08. Type of wall movement

a) Rotation about the toe

Relative wall movement sp /h depending on the relative density D for D > 0.3

Earth pressure force Epgh simplified distribution of passive earth pressure and approximation of the magnitude of the earth pressure force∗

sp /h = −0.08 ⋅ D + 0.12

1 2

The specified equation applies approximately when the absolute value of δp is δp ≤ φ/2 within the negative range and provides mean values. Deviations of up to ± 20% should be considered. Within the scatter area, the values increase slightly with the wall height. When the absolute value of δp is δp ≤ φ/2 in the negative range, larger amounts may occur for sp /h.

⋅ Ebpgh ≤ Eapgh ≤

2 3

⋅ Ebpgh

Eapgv = Eapgh ⋅ tan δap,mean δap,mean = Ebpgh =

1 2

3 4

⋅ δap,min

⋅ γ ⋅ h2 ⋅ Kbpgh

b) Parallel movement δbp,mean = δbp,min sp /h = −0.05 ⋅ D + 0.09

c) Rotation about the top

The specified equation provides mean values. The variation for this type of wall movement is approximately ± 20 %. Within the scatter area, values increase with the wall height.

Ecpgh ≈

2 3

⋅ Ebpgh

δcp,mean = δcp,min

∗ δp,min is the largest absolute value of the inclination angle of earth pressure on the wall under consideration

The local displacement s (z) at depth z can be determined using an approach from Vogt (1984). s(z) z ′ Kah = K0h − (K0h − Kah ) ⋅ (15.2) s(z) b+ z

15.4 Mobilisation functions

with K′ah =

e′ah γ⋅z

.

(15.3)

For the parameter b, this is given at b = 0.011 for loose density and b = 0.003 for medium to high density. The approach is particularly simple because neither the type of the wall movement nor the amount of the displacement sa is taken into account. The entire active earth pressure with K′ ah = Kah is reached asymptotically only with infinite displacements. Nevertheless, the approach is likely to meet the requirements for an estimation in many cases, e.g. in the case of integral structures (see chapter 14.1).

15.4.2

Mobilised passive earth pressure

Bartl (2004) proposes the following equation for the resultant E′ph of the mobilised passive earth pressure or earth resistance [ ( )b ]c s ′ + E0h . (15.4) Eph = (Eph − E0h ) ⋅ 1 − 1 − sp The parameters b and c were determined based on tests for different types of wall movement (see Table 15.3). The limit displacement sp on reaching the full earth resistance can be found in Table 15.2. Equation (15.4) was originally proposed by Nendza (1983) with b = 2 and c = 0.5 and generalised by Franke (1989). The approach by Besler (1998) takes the influence of density into account, as well as groundwater and wall friction on the displacement and, as a further plot-point on the curve, the displacement sp,50 with a degree of mobilisation of 50% of the earth resistance. For a wall height h and a specific weight of soil γ, the result is E′ph

γ ⋅ h2 = 2

⎤ ⎡ ⎢ B ⎥ ⋅ ⎢A + s ⎥ C+ ⎥ ⎢ sp ⎦ ⎣

(15.5)

Table 15.3 Exponents for the mobilisation approach of Bartl (2004). Exponents of the mobilisation function Type of wall movement

b

c

Rotation about the toe

1.07

0.7

Parallel displacement

1.45

Rotation about the top

1.72

s is the actual wall displacement and sp is the displacement to generate Ep according to Table 15.2.

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15 Mobilisation of earth pressure

with A = Kph + C ⋅ (Kph − K0 )

(15.6a)

B = −(C + C2 ) ⋅ (Kph − K0 )

(15.6b)

as well as Kph ⋅

C=

sp,50

sp . sp,50 2 ⋅ (Kph − K0 ) ⋅ + 2 ⋅ K0 − Kph sp

(15.6c)

The displacements sp,50 and sp can be determined using the following equations sp,50 = f1 ⋅ h

(15.7a)

sp = f2 ⋅ h.

(15.7b)

For the functions f1 and f2 , a factorisation is proposed to consider density, wall friction and type of wall movement with the factors fD for the influence of the density, fδ for the influence of a negative wall friction and fB for the basic case. For fB , a distinction is made between fB,G for the serviceability limit state, and fB,B at the ultimate limit state. Furthermore, a factor fs is added for the serviceability limit state and the ultimate limit state. A distinction is also made between the cases “above groundwater” and “below groundwater” when the soil is under uplift. As a simplification, the influence of capillary cohesion on the displacements is not considered compared to the approach by Besler. Following Besler’s suggestion, the result is f1 = fD ⋅ fδ,G ⋅ fs,G ⋅ fB,G

(15.8a)

for a degree of mobilisation of 50% with fB = fB,G , fs = fs,G as well as fδ = fδ,B and f2 = fD ⋅ fδ,B ⋅ fs,B ⋅ fB,B

(15.8b)

for the ultimate limit state with fB = fB,B , fs = fs,B as well as fδ = fδ,B . The values determined by Besler are listed in Table 15.4 to Table 15.7. When determining the factors fs in Table 15.6, Besler relied on model tests. Complementary theoretical considerations suggest that an increase of the displacements under water should be avoided. Therefore, the values originally provided by Besler are shown in brackets in Table 15.6. Instead, it is recommended to use fs = 1.00 under water. For details see Weißenbach and Hettler (2011) as well as Hettler, Triantafyllidis and Weißenbach (Hettler et al. 2018). For more information on the subgrade reaction approach see Hettler and Besler (2001) as well as Hettler and Hegert (2017). Equation (15.5) can also be formulated as a subgrade reaction approach e′ph

⎤ ⎡ ⎢ B ⎥ = γ ⋅ z ⋅ ⎢A + s ⎥ C+ ⎥ ⎢ sp ⎦ ⎣

(15.9)

15.4 Mobilisation functions

Table 15.4 Factor fD to consider the influence of the density. Relative density

loose

medium-dense

dense

very dense

1.47

1.28

1.03

0.75

Table 15.5 Factor fδ to consider the influence of negative wall friction. Degree of mobilisation 50 %: 1/2 Eph

Failure state: Eph

1.57

2.44

Table 15.6 Factor fs to consider the influence of groundwater. Groundwater

Failure state: Eph

Degree of mobilisation 50 %: 1/2 Eph

above

1.00

1.00

below

1.00 (1.58)

1.00 (2.21)

Table 15.7 Factor fB for the determination of the relative wall displacement in the basic case. Type of wall movement

Failure state: Eph Degree of mobilisation 50 %:

1/2

Eph

Rotation about the toe

Parallel displacement

Rotation about the top

4.70 [%]

3.87 [%]

4.50 [%]

1.08 [%]

0.50 [%]

1.21 [%]

with the same parameters as in equation (15.5) for the mobilisation of the earth resistance. In the case of supported walls, it is generally sufficient to use the parameters for parallel displacement, as shown by numerous studies (Hettler et al. 2006). The approach in equation (15.9) can also be extended to layered soils and soils with cohesion. In addition, the influence of a preload, e.g. from excavation, can be considered (Hettler and Maier 2004). The proposal by Vogt in equation (15.2) can also be applied to the passive case K′ph

s(z) z = K0h + (Kph − K0h ) ⋅ s(z) a+ z

(15.10)

using K′ph =

e′ph γ⋅z

.

(15.11)

Vogt (1984) obtained from tests a = 0.11 for loose sand and a = 0.03 for medium-dense to dense sand.

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Pelz extends the approach to preloaded cohesive soils (Pelz 2011). In this case, a lateral pressure coefficient dependent on the over-consolidation ratio OCR is introduced and the mobilisation is divided into shares of friction and cohesion. 15.4.3

Spatial mobilised passive earth pressure

Weißenbach (1962, 1985) derives empirical relations from small scale model tests to determine the displacements s(r) with a degree of mobilisation of 50% and p50 the displacement s(r) p on reaching the spatial earth resistance. In the case of rigid model beams with a ratio of width b to embedment depth t of b/t < 0.30, the following results apply s(r) p = 32 ⋅

s(r) p,50

t2 1 ⋅ √0 fD b √

= 1.4 ⋅ fD ⋅

and for b/t ≥ 0.30

(15.12) t30

b

(15.13)

√ t30

(15.14)

= 2.6 ⋅ fD ⋅ t0 . s(r) p,50

(15.15)

s(r) p = 59 ⋅

1 ⋅ fD

The formulas are not dimensionally homogeneous; the displacements result in mm when b and t are used in m. The function fD reflects the influence of the relative density D with fD = 1 + 0.5 ⋅ D.

(15.16)

Based on American large-scale tests and their own Finite Element calculations, Jung (1970) and Jung and Vrettos (2016) developed an iterative procedure for the estimation of the modulus of subgrade reaction Ksh in front of soldier piles in medium-dense sand. The displacement-dependent modulus of subgrade reaction Ksh as the quotient between the mobilised passive earth pressure e′ph and the displacement s is dependent on the width b, the embedment depth t and the depth z ( ) ( ) ( ) s b z (15.17) Ksh = γ ⋅ f1 ⋅ f2 ⋅ f3 s0 b0 t with f1 (s/s0 ) as the base function and standardisation constants γ = 18.5 kN/m3 , s0 = 0.001 m, b0 = 0.3 m, as well as the approximation equations derived from Finite Element calculations: ( ) ( ) s s f1 = 850 ⋅ exp −0.065 ⋅ (15.18) s0 s0 ( ) b b f2 (15.19) = 0 b0 b

References

and

( ) z z = 2.2 ⋅ . (15.20) t t For the first iteration step, Jung and Vrettos recommend the estimate f3

b z Ksh = 1730 ⋅ 0 ⋅ . γ b t

(15.21)

Arnold and Herle extend the approach by Bartl in equation (15.4) to describe the mobilisation of the spatial passive earth pressure in front of short earth-covered walls and anchor slabs. For details, see Arnold et al. (2008).

References Arnold, M., Herle, I. and Gabener, H.-G. (2008). Mobilisierung des räumlichen passiven Erddrucks bei Erdüberdeckung. Geotechnik 31: 12–22. Bartl, U. (2004). Zur Mobilisierung des passiven Erddrucks in kohäsionslosem Boden. Institut für Geotechnik Technische Universität Dresden, Mitteilungen, Heft 12. Besler, D. (1998). Wirklichkeitsnahe Erfassung der Fußauflagerung und des Verformungsverhaltens von gestützten Baugrubenwänden. Schriftenreihe des Lehrstuhls Baugrund-Grundbau der Universität Dortmund, Heft 22. Franke, D. (1989). Beiträge zur praktischen Erddruckberechnung. Technische Universität Dresden, Habilitationsschrift (Promotion B), Dresden. Hettler, A. and Besler, D. (2001). Zur Bettung von gestützten Baugrubenwänden in Sand. Bautechnik 78 (2): 89–100. Hettler, A. and Hegert, H. (2017). Baugruben: Ermittlung des Bettungsmoduls auf der Grundlage von Mobilisierungsfunktionen. Bautechnik 94 (5): 263-273. Hettler, A. and Maier, T. (2004). Verschiebungen des Bodenauflagers bei Baugruben auf der Grundlage der Mobilisierungsfunktion von Besler. Bautechnik 81 (5): 323–336. Hettler, A., Mumme, B. and Vega Ortiz, S. (2006). Berechnung von Baugrubenwänden mit verschiedenen Methoden: Trägermodell, nichtlineare Bettung, Finite-Elemente-Methode. Bautechnik 83 (1): 35–45. Hettler, A., Triantafiylidis, Th. and Weißenbach, A. (2018). Baugruben, 3. Auflage, Berlin: Ernst & Sohn. Jung, S. and Vrettos, C. (2016). Bettungsmodul für Trägerbohlwände in mitteldichtem Sand aus einem Großversuch validierten FEM-Berechnungen. Geotechnik 39 (4): 252-262. Jung, S. (1970). Nichtlinearer horizontaler Bettungsmodulansatz für Trägerbohlwände in mitteldichtem Sand. Veröffentlichungen Fachgebiet Bodenmechanik und Grundbau, TU-Kaiserslautern. Heft 12. Nendza, H. (1983). Sicherung tiefer Baugruben neben Bauwerken. Tiefbau, Ingenieurbau, Straßenbau 8: 698–702. Pelz, G. (2011). Die Berücksichtigung einer Vorbelastung bei der Mobilisierung des passiven Erddrucks für körnige Böden. TUM Zentrum Geotechnik, Heft 48. Vogt, N. (1984). Erdwiderstandsermittlung bei monotonen und wiederholten Wandbewegungen in Sand. Baugrundinstitut Stuttgart, Mitteilung 22.

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Weißenbach, A. (1962). Der Erdwiderstand vor schmalen Druckflächen. Bautechnik 39 (6): 204-211. Weißenbach, A. and Hettler, A. (2011). Baugruben – Berechnungsverfahren. 2. Auflage, Berlin: Ernst & Sohn. Weißenbach, A. (1985). Baugruben, Teil II, Berechnungsgrundlagen. Berlin: Ernst & Sohn, 1. Nachdruck.

285

16 Application rules 16.1 Earth pressure inclination and angle of wall friction A strict differentiation has to be made between the inclination angle of the earth pressure and the wall friction angle, which physically forms the upper limit of the earth pressure inclination angle. In older publications and also in DIN 4085:1987-02, the two terms have often been used synonymously. The inclination angle of the earth pressure was sometimes also reduced in order to compensate for errors in the failure model, i.e. differences between failure models using Coulomb’s wedge theory or curved slip surfaces. The wall friction angle depends essentially on the condition of the wall surface and the friction angle φ′ k of the soil. DIN 4085:2017-08 distinguishes between interlocked, rough, less rough and smooth wall surfaces. For simplified use, the wall friction angles shown in Table 16.1 may be used. The full wall friction may only be applied with curved slip surfaces. In the case of Coulomb’s wedge, the values corresponding to the right-hand column in Table 16.1 have to be reduced to compensate for model error. Coulomb’s wedge theory may be taken as the basis for active earth pressure, irrespective of the friction angle φ′ k , for passive earth pressure only for φ′ k ≤ 35∘ . Apart from the Rankine case for δ = β = α = 0, Coulomb’s wedge model is subject to model errors. The active earth pressure results in smaller values than with curved slip surfaces. In the case of passive earth pressure, larger values are obtained. In order to compensate for these errors, calculations for interlocked and rough walls as well as sheet pile walls can be performed with an earth pressure inclination angle of at most δ = 2/3 ⋅ φ′ k . According to EAB (2012), this applies to the active earth pressure regardless of the friction angle, and for earth resistance to a maximum friction angle of φ′ k = 35∘ . More precise studies on the size of the model error are not available, in particular for the case of active earth pressure. Based on an evaluation of the present literature, Weißenbach (1985) provides the following recommendation for the wall friction angle of untreated steel: (16.1a) δ = φ′ − 2.5∘ for φ′ ≤ 30∘ p

k

δp = 27.5∘

k

for φ′k ≥ 30∘ .

(16.1b)

Accordingly, the friction angle of the soil is decisive until φ′ k = 30∘ . After this, the shear band lies in the soil. At φ′ k > 30∘ , the surface condition is decisive, and Earth Pressure, First Edition. Achim Hettler and Karl-Eugen Kurrer. © 2020 Ernst & Sohn Verlag GmbH & Co. KG. Published 2020 by Ernst & Sohn Verlag GmbH & Co. KG.

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16 Application rules

Table 16.1 Angle of wall friction according to DIN 4085:2017-08. Wall friction angle

Condition of the

Curved slip surfaces

Coulomb’s wedge

Interlocked: The wall concrete is placed in such a way that interlocking is created with the adjacent soil. a)

φ′ k

2 3

⋅ φ′k

Rough: Untreated surfaces of steel, timber or brickwork

≤ 27.5∘ ≤ φ′ k – 2.5∘

2 3

⋅ φ′k

Less rough: Wall coverings of weathering resistant, plastically undeformable plastic sheets.

1 2

1 2

⋅ φ′k

Smooth: Strongly greasy backfill; confining layer which cannot transfer shear forces.

0

wall surface

⋅ φ′k

0

a) Applies approximately also for corrugated, driven, vibration-installed or jacked sheet pile walls since the development of the wall is longer than the length of the wall axis.

shearing takes place directly on the wall surface. The results obtained in equation (16.1) have also been adopted in the EAB (2012) as well as DIN 4085:2017-08. The frequently used value of δ = 2/3 ⋅ φ′ k , e.g. in DIN 4085:2011-05, is on the other hand on the safe side as long as φ′ k does not exceed 40∘ . In this context, steel sheet pile walls do not fall into the categories “rough” or “untreated steel”. Since a failure surface between the sheet pile wall and the soil • either lies predominantly in the soil in a continuous surface in the plane of the back of the sheet piles • or due to the corrugated shape, has a total surface of approximately 50% larger than the continuous surface, the wall friction angle may approximately be assumed at δ = φ′ k . The actual, physically effective inclination angle of the earth pressure, whose absolute upper limit is the wall friction angle, depends essentially on the relative movements between the soil and the structure. However, the wall friction angle must first be determined. In the case of a confining stratum which cannot transmit any shear forces, the earth pressure inclination angle is δ = 0, independent of the shear strength of the soil and the relative movement. If, on the other hand, a wall is interlocked, an inclination angle in the range of −φk ≤ δ ≤ +φk can be reached. It is not always easy to estimate the relative movement between the soil and the wall correctly. In addition, the equilibrium of vertical forces must be maintained. The following example is intended to illustrate the relationships. In the case of a rough wall supported twice by horizontal stiffeners, it is generally common to assume that δa = 2/3 ⋅ φ for active earth pressure (Table 16.1). If the wall has a sufficient embedment depth, the vertical component Eav of the resultant active earth pressure can be absorbed at the toe of the wall by appropriate soil reactions with the resultant Bv (Fig. 16.1a). If there is no embedment as in Fig. 16.1b, one

16.2 Magnitude of earth pressure depending on the wall displacement

Fig. 16.1 Inclination angle of active earth pressure. a) δa > 0, if vertical equilibrium is possible, b) δa = 0, c) δa < 0.

has to set δa = 0. If, in addition, an external surcharge Pv is added as in Fig. 16.1c, the inclination δa < 0 must be selected in such a way that the negative vertical component of the earth pressure can absorb Pv with the required safety margin or, if not possible, another solution for the absorption of Pv in the soil has to be found. If a negative inclination angle δp is specified for earth resistance, the vertical forces for the serviceability limit state must always be verified, see e.g. EAB (2012). If the inclination has been selected as too large, its amount has to be correspondingly reduced. As shown by more accurate measurements, e.g. by Bartl (2004), the earth pressure inclination angle can change with the degree of mobilisation and the local earth pressure inclination angle can be different on one wall. In practice, this effect is often neglected. In contrast, a non-constant inclination angle δp is suggested in DIN 4085:2017-08, Table D 1 for the passive case and rotation about the toe of the wall. For the approach of at-rest earth pressure, the inclination angle is assumed to be parallel to the ground surface. This is the only way to ensure the equilibrium of stresses in the soil.

16.2 Magnitude of earth pressure depending on the wall displacement To mobilise the entire passive earth pressure, large wall displacements are required, see chapter 15.3. Therefore, it should generally be checked whether the deformations associated with the mobilisation are compatible with the structure. For this purpose, the approaches summarised in chapter 15.4 can be used. However, displacement dependencies must be taken into account not only in case of passive earth pressure but also for the approach of active earth pressure or for an increased active earth pressure. As in the passive case, deformations can be estimated by using mobilisation functions.

287

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16 Application rules

Table 16.2 Approach for earth pressure depending on the type of retaining wall for permanent structures according to DIN 4085:2017-08.

Row

Elasticity of the support system

Structure (examples)

Approach for earth pressure

1

Yielding

Retaining walls which can and have to undergo slight deformation in the direction of the earth pressure loading during their entire service life, e.g. waterfront walls, retaining walls founded on loose rock.

active earth pressure

2

Slightly yielding

Retaining walls according to row 1 where deformation in the direction of the earth pressure loading is undesirable during their entire service life and have been constructed against undisturbed soil.

increased active earth pressure E′ah = 0.75 ⋅ Eah + 0.25 ⋅ E0h

3

Nearly inflexible

Retaining walls which at first slightly yield under earth pressure loading due to their construction, but which cannot or must not be deformed, e.g. basement walls and retaining walls which are incorporated into buildings and additionally supported by these, design of the vertical walls of Land T-cantilever retaining walls.

increased active earth pressure in the regular case: E′ah = 0.50 ⋅ Eah + 0.50 ⋅ E0h

Retaining walls which are largely inflexible due to their construction, e.g. retaining walls founded on rock under plane strain conditions and retaining walls founded on loose rock as spatial systems, such as abutments with rigidly connected parallel wing walls.

increased active earth pressure E′ah = 0.25 ⋅ Eah + 0.75 ⋅ E0h

4

Inflexible

in exceptional cases: E′ah = 0.25 ⋅ Eah + 0.75 ⋅ E0h

in exceptional cases to at-rest earth pressure

In practice, tables B1 and B2 in DIN 4085:2017-08 are of great importance. Without complex calculations, the earth pressure can be determined • for permanent structures depending on the yielding of the supporting structure (Table 16.2) and • for excavation walls or other short-term support structures depending on the yielding of the support (Table 16.3). For further information on the support of excavation walls and the approach of active earth pressure, an increased active earth pressure or at-rest earth pressure, refer to EAB (2012).

16.3 Earth pressure redistribution

Table 16.3 Approach for earth pressure depending on the elasticity of the support system to excavation walls or other temporary support systems according to DIN 4085:2017-08 based on EAB (2012).

Elasticity of the Row support system Structure (examples)

Prestressing of the support force related to the next Approach for excavation state earth pressure

1

Unsupported or yielding support

– Wall without upper support (struts, anchors) or with yielding support (e.g. anchor not or only slightly prestressed)

2

Slightly yielding support

Struts are at least tightly connected by frictional contact (e.g. by wedges) Prestressed anchors

3

Nearly inflexible support

Struts • with multiply braced 30 % sheet pile walls, braced concrete walls • with multiply braced 60 % soldier pile walls

Increased active earth pressure in simple cases E′ah = 0.75 ⋅ Eah + 0.25 ⋅ E0h

Prestressed anchors

100 %

in exceptional cases E′ah = 0.25 ⋅ Eah + 0.75 ⋅ E0h

Walls designed for a reduced or for the full at-rest earth pressure and whose supports are accordingly prestressed. When anchors are additionally anchored in an inflexible rock layer or are considerably longer than required by calculation. Struts 100 % Anchors 100 %

Increased active earth pressure E′ah = 0.25 ⋅ Eah + 0.75 ⋅ E0h

4

Inflexible support

80 % to 100 %

Non-redistributed active earth pressure

Redistributed active earth pressure

in regular case E′ah = 0.50 ⋅ Eah + 0.50 ⋅ E0h

in exceptional cases to at-rest earth pressure

16.3 Earth pressure redistribution The calculation of earth pressure from soil self-weight is started from a triangular, i.e. classic distribution. Strictly speaking, this only applies for rotation of the wall about the toe in the active case and in case of passive earth pressure only

289

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16 Application rules

for parallel movement. While for earth resistance the assumption of the classic earth pressure distribution is sufficient in many cases, consideration of the earth pressure distribution cannot be neglected with supported walls. Depending on the type of wall movement, there may be considerable earth pressure redistribution in the case of active earth pressure and increased active earth pressure, in particular for soldier pile walls and sheet pile walls, and to a lesser extent for in-situ concrete walls. Figure 16.2 shows the influence of support conditions on the earth pressure distribution with sheet pile walls. In the case of a non-supported wall fixed in the soil, rotation about a low point with the classical triangular distribution is established (Fig. 16.2a). In all other cases, there is a more or less pronounced earth pressure re-distribution in the direction of the support. The resulting combination of deflection and displacement always applies. If the wall has inflexible support at ground level and is fixed in the soil, then simultaneous rotation about the top and about a low point or pure deflection of the wall as shown in Fig. 16.2b occurs. If the wall has inflexible support at ground level and has a free earth support, then simultaneous deflection of the wall and foot displacement as shown in Fig. 16.2c occurs. In the case of multiply supported walls, parallel movement and an earth pressure distribution as shown in Fig. 16.2d can be expected. In general, the distribution of active earth pressure depends on a large number of influences, cf. e.g. Briske (1957), Weißenbach (1985), EAB (2012) or EAU (2012) and the literature cited therein. In particular, the following aspects exert the most influence: • yielding of the support, • the type and end fixing of the wall, • the deflection stiffness of the wall,

Fig. 16.2 Earth pressure distribution with sheet pile walls in simple cases according to Weißenbach (1985). a) Wall fixed in the soil and unsupported, b) Wall supported at ground level and fixed in the soil, c) Wall supported at ground level and free earth support, d) Wall supported several times and free earth support.

16.4 Earth pressure as a favourable action

• the number and arrangement of struts or anchors, • the size of the respective excavation section before the installation of the next layer of struts or anchors, • prestressing of the struts or anchors. Where the conditions are met, the proposals of the EAB (2012) for soldier pile walls, sheet pile walls and in-situ concrete walls with slightly yielding support can be applied. In case of doubt, limit investigations with various possible pressure diagrams should be carried out. Depending on the question, Finite Element calculations can be helpful.

16.4 Earth pressure as a favourable action If the earth pressure acts as a favourable action, such as the vertical component Eav in the uplift verification in Fig. 16.3, then a lower characteristic value must be assumed, cf. Eurocode 7 (2011), part 1, chapter 9.5.1, paragraph A (11). In the case of non-cohesive soils, the recommendation is to reduce the horizontal and therefore also the vertical earth pressure to half the value used as the basis for other structural design. The explanation lies in the fact that lower characteristic values of the friction angle are provided in soil surveys, which usually means smaller friction angles with respect to the probable mean value. This leads to a higher earth pressure and thus to loading and design on the safe side. In the case of favourable actions from earth pressure, the upper value of the characteristic friction angle would have to be applied, i.e. the friction angle should be raised above the mean value in order to obtain a smaller value for the earth pressure. To this extent, the reduction to half is a pragmatic solution in order to avoid complex instructions for the determination of characteristic values that are on the “safe” side. The same applies to cohesion, which in practice is often estimated to be relatively low for safety reasons. However, if it is present, the earth pressure can easily tend towards zero, or a gap can even open, for example in combination with temperature effects. For this reason, DIN 1054 or Eurocode 7, part 1 recommend setting of the earth pressure load Eah equal to zero in case of cohesive soils unless a more detailed investigation is available. Fig. 16.3 Earth pressure as a favourable action for the verification of heave illustrated by a narrow sheet pile wall excavation with thick concrete base for uplift control.

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References Bartl, U. (2004). Zur Mobilisierung des passiven Erddrucks in kohäsionslosem Boden. Institut für Geotechnik Technische Universität Dresden, Mitteilungen, Heft 12. Briske, R. (1957). Erddruckverlagerung bei Spundwandbauwerken. Berlin: Ernst & Sohn. EAU (2012). Recommendations of the Committee for Waterfront Structures Harbours and Waterways (EAU), 11th ed., Berlin: Ernst & Sohn. EAB (2012). Recommendations on Excavations (EAB), 5th ed., Berlin: Ernst & Sohn. Handbuch Eurocode 7 (2011). Geotechnische Bemessung, Band 1, Allgemeine Regeln, 1. Auflage, Berlin: Beuth Verlag. Weißenbach, A. (1985). Baugruben, Teil II, Berechnungsgrundlagen. Berlin: Ernst & Sohn, 1. Nachdruck.

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17 Commentary on DIN 4085:2017-08 17.1 Overview The explanations in chapters 2 to 16 contain the essential fundamentals of earth pressure determination and can therefore also be understood as a commentary. The following comments are intended as supplementary information. In this respect, the DIN standard is not dealt with chapter by chapter, but rather only through a few selected points that were partly subject to discussions in the DIN Committee or were also referred to the Committee as questions. As part of the preparation of the previous standard to the current issue from August 2017, a supplement with numerous examples was prepared. In order to make this knowledge available to the user in the future, the supplement has been revised. The new edition was published in December 2018, see chapter 17.6.

17.2 Active earth pressure General case according to Groß As explained in chapter 4.1, the tripartite approach for active earth pressure in eq. (4.1) or eq. (1) in DIN 4085 provides an approximation that e.g. assumes an earth pressure wedge according to Coulomb exactly for the case of a vertical wall with α = 0, a flat ground surface with β = 0 and an earth pressure inclination angle of δ = 0. In the general case, the equations are coupled, and the relevant slip surface inclination does not correspond to ϑa in eq. (4.11) or ϑag in eq. (8) of DIN 4085:2017-08. According to Groß (1981), the maximum earth pressure results at cos(φ − α) ⋅ sin(φ − β) cos(α + δ) + m ⋅ cos(φ − β − α − δ) ϑa = φ + arctan √ sin(φ − α) ⋅ sin(φ − β) ⋅ cos(α + δ) + m ⋅ sin(φ − β − α − δ) + b with b = [sin(φ − β) ⋅ cos(β − α) + m] ⋅ [sin(φ + δ) ⋅ cos(α + δ) + m] (17.1) Using: m:=

cos α ⋅ cos (β − α) ⋅ cos φ 2⋅c ⋅( ) γ⋅h 2 ⋅ p cos (β − α) + ⋅ cos α ⋅ cos β γ ⋅ h

(17.2)

Earth Pressure, First Edition. Achim Hettler and Karl-Eugen Kurrer. © 2020 Ernst & Sohn Verlag GmbH & Co. KG. Published 2020 by Ernst & Sohn Verlag GmbH & Co. KG.

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gives Ea =

] cos (β − α) 2 ⋅ p + ⋅ cos β cos α γ⋅h [ ] cos (ϑa − α) ⋅ sin (ϑa − φ) − m ⋅ . cos α ⋅ sin (ϑa − β) ⋅ cos (ϑa − φ − α − δ)

γ ⋅ h2 ⋅ 2

[

(17.3)

Obviously, the inclination of the slip surface and the earth pressure resultant depend on the cohesion and the surcharge; see also Gudehus (1981). In practice this difference is neglected, especially since deviations are usually low and the approach of an earth pressure wedge is also an approximation. Depending on type of wall movement, other failure mechanisms with different results occur. Earth pressure coefficients for cohesion according to Ohde In the context of internal comparison calculations carried out by the Federal Waterways Engineering and Research Institute (BAW), it was determined that using the earth pressure coefficient Kach =

2 ⋅ cos (α − β) ⋅ cos φ ⋅ cos (α + δa ) [1 + sin(φ + α + δa − β)] ⋅ cos α

(17.4)

see eq. (4.10c) or eq. (10) of the standard DIN 4085:2017-08 giving different results to the calculation with cos φ ⋅ cos δa (17.5) Eachϑ = −c ⋅ h ⋅ cos (ϑ − φ − δa ) ⋅ sin ϑ using ϑa from soil self-weight according to eq. (4.11) or eq. (8) in DIN 4085:2017-08. The answer can be found in Supplement 1 to DIN 4085 from February 1987, which states: Start of quote: For the calculation of the earth pressure coefficient due to cohesion, the formula of Ohde (1956) is provided. Here, Ohde has neglected the soil self-weight and has assumed that cohesion is equally distributed in the slip surface. The angle of the slip surface for cohesion according to Ohde is: φ δa,p α β + + (17.6) ϑa,p = 45∘ ± cal + 2 2 2 2 End of quote. That means, Ohde uses a different slip surface angle for the cohesion component than for the soil self-weight. Limits of application according to Franke DIN 4085:2011-05 provides application limits for the earth pressure coefficients based on Franke (1982), see equations (4.10a) to (4.10c) or equations (7), (9) and (10) of the standard DIN 4085:2017-08 δ ≥ 0∶ −20∘ ≤ α < −10∘ for 0 ≤ β ≤ φ (17.7a) a

−10∘ ≤ α ≤ αmax for − φ ≤ β ≤ φ

(17.7b)

17.4 Earth pressure due to compaction

2⋅φ (17.8) 3 with αmax as the angle between the opposite slip surface and the vertical, see Fig. 11.2. δa < 0∶ −20∘ ≤ α ≤ αmax for − φ ≤ β ≤

αmax = ϑag − φ

(17.9)

with ϑag for α = 0 and δa = β

(17.10)

In most cases, the conditions in equations (17.7) and (17.8) are fulfilled, so that the regulations in question were waived as part of the streamlining of the standard. In the case of extraordinary wall and terrain inclinations, it will be necessary anyway to question the models used and work out individual solutions.

17.3 Passive earth pressure The equations by Pregl, see chapter 6.2.2 and chapter 7.1, are used in the standard. However, DIN 4085 expressly allows other procedures as defined in chapter 5.1 of the standard. These include static stress field and kinematic mechanism methods, see chapters 3.2 and 3.3. Coulomb’s wedge theory may also be used if comparability is given. According to the EAB (2012), this is true as long as φ ≤ 35∘ , and the earth pressure inclination angle for negative slope is restricted to | δ | ≤ 2/3 ⋅ φ, see chapter 16.1 and the results in Figures 6.2 and 6.3. If methods are used for which only earth pressure coefficients from soil self-weight are available, then the approximations Kpph = Kpgh and Kpch ≈ 2 ⋅

(17.11)

√ Kpgh

(17.12)

may be used, see eq. (39) and eq. (40) in DIN 4085:2017-08. Although the investigations by Pregl contain no statements on kinematics, available comparative calculations with the Finite Element Method, see chapter 3.6.2, show that the earth pressure coefficients match parallel movement of the wall very well. The values by Pregl/Sokolowski may therefore be used for parallel movement. There are no publications for the exact derivation of the coefficients in the interpolation formulas, and a query by the DIN Committee to the successor of the deceased professor Pregl also provided no further information. However, all previous comparative studies indicate the good usability of the proposed formulas for practical application.

17.4 Earth pressure due to compaction Chapter 11 “Compaction earth pressure” in DIN 4085:2017-08 has been adopted virtually unchanged from the previous standard. However, table 5 of the standard

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has been supplemented by the figures for light compaction, which are based on Franke (2008). For the approach of earth pressure due to compaction on Land T-cantilever retaining walls, see chapter 17.6 and the discussion regarding example 9 of the supplement.

17.5 Spatial earth pressure The fundamentals of spatial active earth pressure are detailed in chapter 7.1. The equations listed in the standard follow previous investigations, see Fig 7.43, and are on the safe side. In individual cases, it may therefore be useful to look for more economical processes, e.g. to verify the stability of diaphragm walls. The equations for the calculation of earth pressure on retaining walls perpendicular to the slope and at an angle to the fall line of an embankment, see chapter 7.3, have not been included in the 2017-08 edition, in contrast to the previous standard. Various reasons were decisive for this. On the one hand, inquiries revealed that practical users do not use this relatively complex procedure as described in section 7.3, but make use of simple assumptions, such as that the earth pressure is continuously applied at the highest point. On the other hand, the derivation was unclear, and there was agreement to cut as much as possible. For spatial passive earth pressure, the formulas were basically adopted in comparison to the previous standard. However, in contrast to the previous standard, the equivalent lengths considered to be misunderstood were omitted from the previous standard, and the shape coefficients μ were converted analogously to the spatially active earth pressure as in the 1987 edition, cf. chapters 8.2 and 8.3. A detailed account of the development of spatial earth resistance over more than five decades can be found in Hettler (2013).

17.6 Advices on supplement DIN 4085:2018-12 The revised supplement contains a total of nine examples: Example 1: Earth pressure from soil self-weight and strip load on a yielding wall, Example 2: Earth pressure from soil self-weight and inclined ground surface on a yielding wall, Example 3: Active earth pressure and earth pressure from compaction at discontinuous wall and discontinuous ground surface, Example 4: Spatial active earth pressure on an excavation front wall, Example 5: Spatial earth resistance, Example 6: Earth pressure from compaction on an external basement wall, Example 7: Mobilisation of earth resistance, Example 8: Influence on the earth pressure slip surface from soil self-weight due to traffic loading, Example 9: Calculation of the earth pressure on a T-cantilever retaining wall.

References

Examples 3 to 8 were taken from the previous supplement with mostly unchanged content, adapted to the new standard DIN 4085:2017-08 and formally revised. Example 1 is new in principle and includes layered soil with a strip load and varying distance from the wall crest. Solutions are proposed e.g. how the relevant slip surface due to surcharges in soil stratification can be found, see also chapter 4.6 and the approximations listed there. Example 2 is new and has been included in the supplement following a question from a practical user. The case treats a diaphragm wall with inclined ground surface and a groundwater level directly below the upper edge of the wall. As will be shown, the procedure with the earth pressure coefficients reaches its limits and no longer delivers actions of earth pressure on the safe side. Therefore, alternative procedures such as the Culmann method can be used. Also new is example 9. For a T-cantilever retaining wall, the earth pressure is determined on both the equivalent wall and the vertical walls, and the differences in the approach of earth pressure from compaction are shown. As usual in practice and based on the specifications of DIN 4085 from 1987, earth pressure from compaction is only applied to the vertical wall. For the equivalent wall, it is assumed that the displacements of the wall, including the soil above the platform on the earth side, are sufficiently large that relaxation takes place and the earth pressure from compaction is thereby reduced. In contrast to example 6 “Earth pressure from compaction on an external basement wall”, where the relatively complex calculation according to chapter 11 of DIN 4085 is used, a simplified approach is proposed which takes also earth pressure from infinite uniformly distributed surcharges into account. An example of earth pressure on wing walls has been omitted for the reasons given in chapter 17.5.

References EAB (2012). Recommendations on Excavations (EAB). 5th ed., Berlin: Ernst & Sohn. Franke, D. (1982). Beiträge zur praktischen Erddruckberechnung. Von der Fakultät für Bau-, Wasser- und Forstwesen der Technischen Universität Darmstadt genehmigte Habilitationsschrift (Promotion B). Franke, D. (2008). Verdichtungserddruck bei leichter Verdichtung, Bautechnik 85 (3): 197-198. Groß, H. (1981). Korrekte Berechnung des aktiven und passiven Erddrucks. Geotechnik 2: 66–69. Gudehus, G. (1981). Bodenmechanik. Stuttgart: Enke. Hettler, A. (2013). Erddrucktheorie, der räumliche Erdwiderstand: Entwicklung über mehr als fünf Jahrzehnte, Bautechnik 90 (Sonderheft): 23-29. Ohde, J. (1956). Grundbaumechanik Hütte III, 26. Auflage, 912. Berlin: Verlag Wilhelm Ernst & Sohn.

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18 Forty selected brief biographies Baker, Sir Benjamin, born 31 Mar 1840, Keyford (now Frome), Somerset, UK, died 19 May 1907, Bowden Green, Pangbourne, UK

After leaving the grammar school in Cheltenham, the 16-year-old Benjamin Baker learned about the world of iron at the long-standing Neath Abbey Ironworks in South Wales. Following that, he furthered his technical knowledge of iron foundries, surveying and bricklaying in order to work under William Wilson (1822-1898) on the extension to Victoria Station in London. Finally, in 1862, Baker joined Metropolitan Railways and worked in the oFMohffice of John Fowler (1817-1898). It was during these years that he wrote his profound study of long-span iron railway bridges (Baker, 1867), presenting, among other things, the balanced cantilever principle; this article was later published in Germany, Austria, The Netherlands and the USA. By 1869 Fowler had already promoted him to the post of “Chief Assistant Engineer on the District Railway Section from Westminster to the City” (Hamilton, 1958, p. 105). Six years later he became Fowler’s Partner, a position he held until his death. This pair worked with great success. Baker was responsible for retaining walls up to 45 ft. high (total length: 9 miles) and timber-shored excavations as deep as 54 ft. (total length: 54 miles). It was this work that formed the background to his pioneering publication on the earth pressure on retaining walls (Baker, 1881) (see (Hamilton, 1958, p. 106ff.) and section 2.6.1), for which he was awarded the George Stephenson Prize by the Institution of Civil Engineers (ICE). He practically lived on the building site for the bridge over the Firth of Forth from 1883 to 1890 (Koerte, 1992, p. 207). Structurally, this bridge works according Earth Pressure, First Edition. Achim Hettler and Karl-Eugen Kurrer. © 2020 Ernst & Sohn Verlag GmbH & Co. KG. Published 2020 by Ernst & Sohn Verlag GmbH & Co. KG.

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to the balanced cantilever principle with separate sections spanning between the cantilevers. When the bridge was opened on 4 March 1890, the Prince of Wales granted him a peerage. Further honours followed, including two honorary doctorates and presidency of the ICE (1895-1896). Hamilton described Baker as follows: “Sir Benjamin Baker, like Telford, was essentially a practical man. Also like Telford, by private study (since no other means were available) he fully acquainted himself with the engineering theory of his day, but never allowed himself to be misled or intimidated by any doctrine that conflicted with his observations or his own carefully conducted field experiments. Mathematics should (so he held) be used to interpret experimental results, not to forecast them from uncertain assumptions” (Hamilton, 1958, p. 105). This appraisal applies to Baker’s paper on the earth pressure on retaining walls in particular. A memorial to this great British structural engineer was unveiled on 3 December 1909 in Westminster Abbey and can still be seen today. Main Works: Long-Span Railway Bridges (1867); The River Nile (1880a); Shipment and erection of Cleopatra’s Needle (contribution to discussion) (1880b); The Actual Lateral Pressure of Earthwork (1881); The Metropolitan and District railways (1885); Bridging the Firth of Forth (1887) Further biographical reading: (Hamilton, 1958); (Koerte, 1992, pp. 205-208) Photo courtesy of: (Hamilton. 1958, p. 104) Bélidor, Bernard Forest de, born 1697/1698, Catalonia, Spain, died 1761, Paris, France

Bélidor’s father, Spanish cavalry officer Jean-Baptiste Forest de Bélidor, and his mother Marie Hébert, both died when he was just five months old. Bernard Forest de Bélidor was therefore brought up by his godfather, de Fossiébourg, an artillery officer. Up until 1718, the young Bélidor assisted in the meridian measurements between Paris and the English Channel coast led by Jacques Cassini and Philippe de La Hire, the results of which were published in 1720. The Duke of Orléans became aware of Bélidor’s talents and set him up as professor of mathematics at the newly founded Artillery Academy in La Fère. During this

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period he published manuals entitled La science des ingénieurs and Architecture hydraulique – the very first civil engineering textbooks of the modern age; they were to remain influential until the early 19th century. Bélidor’s manuals influenced Lazare Carnot, Coulomb, Poncelet and Navier; the latter republished them with a comprehensive, critical commentary (Bélidor, 1813, 1819). His earth pressure model (see section 2.2.1) and the charts derived from it were still being used by practising engineers in the first years of the constitution phase of theory of structures (1825-1850). Main Works: La science des ingénieurs dans la conduite des travaux de fortification et d’architecture civile (1729); Architecture hydraulique (1737-1753); IngenieurWissenschaft bey aufzuführenden Vestungswerken und bürgerlichen Gebäuden, 1. Teil (1729/1757) Further historical reading: (Gillispie, 1970) Photo courtesy of: Collection École Nationale des Ponts et Chaussées Boussinesq, Joseph Valentin, born 13 Mar 1842, Saint-André-de-Sangonis, Hérault Département, France, died 19 Feb 1929, Paris, France

Boussinesq’s parents were very keen for their son to have a classical education. Therefore, his mother’s brother, Abbé Cavalier, instructed him in Latin, Greek and, of course, religion. His mother, the daughter of an industrialist, died early, in 1857. The father was counting on his son; he was to take over the farm later. Opposing the wish of his father, Boussinesq went to study mathematics at Montpellier University. Boussinesq was therefore on his own financially and had to fund his studies by teaching at a grammar school in Montpellier. After finishing his studies in 1861, Boussinesq taught at the grammar schools in the towns of Agde, Le Vigan and Gap in southern France (1862-1866). It was during this period that he became acquainted with Lamé’s books on elasticity and heat theory. Boussinesq’s dissertation on heat propagation in homogeneous media was approved in May 1867. From then on, Saint-Venant (1797-1886) supported this promising mathematician in the role of a scientific father and

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advised him to study physics. Boussinesq followed this advice in 1872, and success was not long in coming. In that same year, Boussinesq was awarded the Poncelet Prize of the Académie des Sciences. There was now nothing more standing in the way of his academic career: professor for differential and integral calculus at Lille University (1872-1886), successor to Eugene Rolland at the Académie des Sciences (1886), professor for experimental and physical mechanics at the Sorbonne (1886-1896) and Chair of Mathematical Physics and Probability Theory at the same university (1896-1918). His important writings on earth pressure theory (Boussinesq, 1876, 1882) and elastic theory (Boussinesq, 1885) were written completely during his time in Lille. Unfortunately, Boussinesq did not always formulate his work with the necessary clarity that would ensure adoption; adding detail to his line of thought was not his strong point either. This becomes clear in the presentation of the earth pressure theory (see section 3.4.3) at the conclusion of the paper by B. Baker (1881, see section 2.6.1). It would be left to Caqout to enter the maze of ideas in the mind of Boussinesq and use them productively for the further development of earth pressure theory (see section 2.5.3). Main Works: Essai théorique sur l’équilibre d’élasticité des massifs pulvérulents, compare à celui de massifs solides et sur la poussée des terres sans cohesion (1876); Sur la determination de l’épaisseur minimum que doit avoir un mur vertical, d’une hauteur et d’une densité données, pour contenir un massif terreux, sans cohésion, dont la surface supérieure est horizontale (1882); Application des potentiels à l’étude de l’équilibre et du mouvement des solides élastiques (1885); Notice sur la vie et les travaux de M. de Saint-Venant (1886) Further historical reading: (Mayer, A., 1954); (Bois, 2007) Photo courtesy of: (Mayer, A., 1954) Brennecke, Ludwig Nathaniel August, born 6 Mar 1843, Leitzkau, Prussia, died 10 Apr 1931, Buchschlag, German Empire

18 Forty selected brief biographies

Ludwig Brennecke was the son of a Lutheran pastor who was the maternal cousin of the author Paul Heyse (1830-1914). After leaving the grammar school in Stendal, he worked in a metalworking shop and studied mechanical engineering at the Berlin Trade & Industry Academy (1863–1866); during those years he also attended lectures at the Building Academy, Mining Academy, and university. Brennecke had in mind a career as a railway machinery master, and spent nine months on the footplate of a locomotive. Nevertheless, he had switched to civil engineering by 1867 and started working for the railway authorities, where he was involved with the bridges over the River Elbe at Hämerten, Dömitz and Lauenburg, also the bridge over the River Ilmenau at Lüneburg. It was during this period that Brennecke was promoted from building site manager to senior engineer responsible for design, construction and site operations. He accumulated knowledge and experience in engineering works, especially the construction of caissons and excavating using pneumatic caissons (Rollmann, p. 345). The latter came to the attention of the Russian businessman and engineer Amand J. von Struve (1835-1898), who appointed Brennecke to help with the building of the Alexander Bridge over the River Neva in St. Petersburg (now the Liteiny Bridge) in 1877 in order to prevent further difficulties with the excavation work under compressed air, which had already cost a number of workers their lives. Brennecke moved with his family to St. Petersburg and solved the foundation problems, enabling the bridge to be finally opened in 1879 and earning von Struve promotion to the rank of major-general. Brennecke continued to work successfully for von Struve, e.g. on the foundations and superstructures for large bridges over the Neva, Dnieper and Olza rivers; he also worked on plans for railways in Bulgaria. Brennecke and his family returned to Germany in 1881. He initially worked as a consulting engineer, joined the Ports Department of the German Imperial Naval Office in 1883 and finally passed his construction master examination in civil engineering. During the planning and construction of the Kaiser Wilhelm Canal (1887-1895, now the Kiel Canal), Brennecke quickly rose to become head of the engineering office of the Imperial Canal Commission in Kiel. It was during this period of creativity that he published (1887) the first systematic book on foundations in the German language, which was translated into several languages and was distinguished by the expert presentation of the mechanisation of this field. It is therefore no surprise that the then director of naval ports Georg Franzius (1842-1914) recommended to the German Imperial Naval Office that Brennecke should supply the designs for the new dry dock that would be required by the fleet-building programme of Alfred von Tirpitz (1849-1930). Brennecke therefore joined the German Imperial Naval Office once again in 1891, becoming director of naval ports in 1899, with responsibility for all the port and shipyard structures of the Imperial Navy in Kiel and Wilhelmshaven, before retiring to Buchschlag near Frankfurt a.M. for health reasons in 1904. His book on ship locks appeared in that same year and that was followed in 1906 by the third edition of his standard work on foundations, which later – in the fourth edition revised by Erich Lohmeyer (1886-1966) – was expanded to three volumes (Brennecke, 1927, 1930, 1942).

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The honorary doctorate awarded to him by Dresden TH in 1905 was just one of many honours. His practical and theoretical work on foundations, acknowledged nationally and internationally, made a crucial contribution to the initial phase of geotechnical engineering (1900-1925). Main Works: Der Grundbau (1887); Ergänzungen zum Grundbau (1895); Ueber Erddruck und Stützmauern (1896); Die Schiffsschleusen (1904); Der Grundbau (1906); Der Grundbau. 1, Baugrund, Baustoffe, Pfähle und Spundwände, Baugrube (1927); Der Grundbau. 2, Pfahlrostgründung (Bohlwerke, tiefer und hoher Pfahlrost) (1930); Der Grundbau. 3, Die einzelnen Gründungsarten mit Ausnahme der Pfahlrostgründung (1942) Further historical reading: (de Thierry, G., 1931); (Rollmann, 1931); (Eckardt, 1955) Photo courtesy of: (Rollmann, 1931, p. 345) Caquot, Albert, born 1 Jul 1881, Vouziers, Ardennes, France, died 28 Nov 1976, Paris, France

When he was 18, Caquot, the son of a farmer, left the grammar school in Reims and first went to study at the École Polytechnique and then the École Nationale des Ponts et Chaussées. He gained his first experience as an engineer in Troyes, where he improved the drainage system so well that the town was unaffected by the extreme floods of 1910. In 1912 he became a partner in Armand Considère consulting engineers, which would be renamed Pelnard-Considère & Caquot after Considère’s death. Following war service, Caquot worked for this consultancy from 1919 to 1928, 1934 to 1938 and from 1940 onwards. He was involved with more than 300 engineering works, many of them international projects. Caquot also made significant contributions to the founding of geotechnical engineering and to theory of structures. For example, in 1934 he developed the continuum mechanics model of earth pressure of Joseph Valentin Boussinesq and Jean Résal further (see section 3.4.3) – pioneering work that earned him membership of the Académie des Sciences in that same year. Together with his son-in-law Jean Kérisel, Caquot produced tables for geotechnical engineering (see section 2.5.3) which also became very popular among civil engineers even

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in non-French-speaking countries. His book on soil mechanics (1949, 1956, 1966), again produced in collaboration with Kérisel, was also widely used and was published in German (1967), Romanian (1968), Spanish (1969) and Japanese (1975). Without reservations, Caquot may be regarded as a protagonist of geotechnical engineering during its development and consolidation phases. And more besides: At an early age, he proved that elastic theory is not sufficient for describing reinforced concrete, and he contributed to the continuum mechanics of plastic bodies by generalising the Coulomb-Mohr yield condition (Caquot, 1930) – the list goes on (see (Kérisel, 2001, pp. 165-168)). So Caquot’s name also stands for the paradigm change from founding theory of structures on elastic theory to founding it on plastic theory during the invention phase of theory of structures (1925-1950). Caquot’s services to the advancement of France’s aviation industry are unique. During the First World War, he was in charge of an airship battalion and invented an airship with stabilisers at the rear which France also produced for the UK and the USA, was used by the Allies for aerial reconnaissance and also led to their superiority in the air. It is therefore no surprise to discover that, in 1918, Georges Clemenceau appointed Caquot to the post of technical director of the entire military aviation division. Numerous technical and organisational innovations in French aviation were the work of Caquot, a description of which would exceed the scope of this brief biography. In 1928 Caquot was appointed the first director of the newly founded Ministry of Aviation. Without Caquot, the upturn in the building of shells in France by Laffaille and Aimond would have been impossible. Owing to funding cuts, Caquot left the ministry and worked as a civil engineer, returned to the ministry in 1938, but left again in 1940 to continue his engineering activities. Caquot was the president of the Académie des Sciences from 1952 to 1961. The high status of civil engineering and aviation in France is down to Caquot. Caquot, together with Freyssinet, was one of the outstanding French civil engineering personalities of the 20th century. Caqout received countless honours. Since 1989, the Association Française de Génie Civil has presented the Prix Albert Caquot, which was awarded to Fritz Leonhardt in its first year.

Main Works: Idées actuelles sur la résistance des matériaux (1930); Equilibre des massifs à frottement interne. Stabilité des terres pulvérulentes et cohérentes (1934); Tables des poussée et butée et de force portante des fondations (1948); Traité de mécanique des sols (1949, 1956, 1966); Grundlagen der Bodenmechanik (1967) Further historical reading: (Picon, 1997, p. 109); (Coronio, 1997, pp. 173-175); (Kérisel, 2001); (Marrey, 1997, pp. 121-143) Photo courtesy of: (Coronio, 1997, p. 173)

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Considère, Armand, born 8 Jun 1841, Port-sur-Saône, Haute-Saône, France, died 3 Aug 1914, Paris, France

Following his studies at the École Polytechnique and École des Ponts et Chaussées, Considère joined the corps of engineers for road- and bridge-building in 1865. He contributed to the further development of the continuum mechanics model of earth pressure just five years later (see section 2.4). Considère left state service in 1875, but returned 10 years later and served as chief engineer of the Finistère Département from 1885 to 1901. It was during these years that he published his highly acclaimed first article on reinforced concrete theory (Considère, 1899) in which he considered the tensile strength of the concrete as well in the design of reinforced concrete beams. Considère chaired one of the three subcommittees of the Commission for Reinforced Concrete set up by the Paris Ministry of Public Works in December 1900. The first decade of the 20th century would make Considère known to an international public: It was in December 1901 that he applied for a patent for his encased reinforced concrete column (béton fretté). In a spectacular tour de force from invention to innovation to dissemination, Considère managed to establish the encased reinforced concrete column in just a few years. To do this, he carried out extensive series of tests together with Augustin Mesnager (1862-1933) at the École des Ponts et Chaussées and published continually on the theoretical and experimental findings regarding the strength behaviour of encased reinforced concrete columns. During the creation of the encasement model, Considère was able to make good use of his profound knowledge of earth pressure theory (see also (Seelhofer-Schilling, 2008, pp. 64-66)). In 1902 he was promoted to inspector-general for roads and bridges, a position he held until he founded his consultancy, Considère et Compagnie, in 1906. At the end of that year, the authorities in Paris signed the French reinforced concrete code. Considère influenced the scientific principles of reinforced concrete construction and encouraged international research into reinforced concrete up until the middle of the accumulation phase of theory of structures (1900-1925). His name stands for encased reinforced concrete in the history of construction. Considère died on 3 August 1914, the day that Germany declared war on France. That also marked the end of the belle époque of reinforced concrete construction. Main Works: Note sur la poussée des terres (1870); Influence des armatures métalliques sur les propriétés des mortiers et bétons (1899); Étude théorique de la résistance

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à la compression du béton fretté (1902a); Étude expérimentale de la résistance à la compression du béton fretté (1902b); Résistance à la compression du béton fretté (1902c); Influence des pressions latérales sur la résistance des solides à l’écrasement (1904); Experimental researches on reinforced concrete (1906); Essais de poteaux et prismes à la compression (1907a); Essais à la compression de mortier et béton frettés (1907b); Der umschnürte Beton (1910) Further historical reading: (Picon, 1997, p. 130); (Coronio, 1997, pp. 142-143); (Seelhofer-Schilling, 2008, pp. 63-84), (Trout, 2013, pp. 2-3) Photo courtesy of: (Coronio, 1997, p. 142) Coulomb, Charles Augustin, born 14 Jun 1736, Angoulême, France, died 23 Aug 1806, Paris, France

Charles Augustin Coulomb attended lectures at the Collège Mazarin and the Collège de France; in 1757 he became an associate member of the Société des Sciences de Montpellier, to which he contributed several articles on astronomy and mathematics. Afterwards, he studied at the École du Génie de Mézière, from where he graduated in 1761 with the rank of lieutenant en premier des Corps du Génie. It was while studying there that he established his lifelong friendship with his mathematics tutors Jean Charles Borda and Abbé Charles Bossut. His first activities as an officer in the engineering corps were in Brest and the French colony of Martinique (1764-1772), where he was in charge of building fortifications. Coulomb transformed his practical experiences into a book of theory that he presented to the Académie des Sciences in 1773; after favourable reviews by academy members Borda and Bossut, his theories were published in 1776 under the title Mémoires de mathématique et de physique presentés à l’académie royale des sciences par divers savants (Coulomb, 1773/1776). A German version produced by the Copenhagen-based professor of mathematics Joachim Michael Geuß, but without the last three chapters (masonry arches), appeared three years later in Andreas Böhm’s Magazin für Ingenieur und Artilleristen (Coulomb, 1779). The French original was published posthumously (Coulomb, 1821, pp. 318-363) and is also included in Heyman’s monograph on Coulomb (Heyman, 1972/1, pp. 1-40), which is followed by the English translation (Heyman, 1972/1, pp. 41-69). In his Mémoire, Coulomb solved

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Galileo’s beam problem and developed forward-looking earth pressure and masonry arch theories (see section 2.2.3). Coulomb’s 40-page Mémoire rounded off the preparatory period of theory of structures (1575-1825) quite conclusively and anticipated the theories of the coming discipline-formation period (1825-1900) to some extent. However, the ideas of his Mémoire were not fully adopted until 40 years after its publication. Coulomb also led the way in theories on electricity, magnetism and friction. In the meantime, Coulomb had been promoted to lieutenant-colonel, and in 1776 his proposal to restructure the Corps du Génie into a “Corps à talent” (Coulomb) was backed by the reform plans of the Turgot government. In 1781 he became a member of the Académie des Sciences and had a decisive influence on the profile of that institution until it was abolished in August 1793. He was critical of the French Revolution; prudently, he withdrew to his small estate near Blois in 1792. Only after the downfall of the Jacobins’ ‘Reign of Terror’ did he return to Paris and, from 1795 onwards, was responsible for experimental physics as an elected member of the newly founded Institut de France. In 1801 Coulomb became president of this highly respected scientific establishment. From 1802 until his death he was inspector-general of all public education and, in this capacity, contributed significantly to creating the French secondary school system. Main Works: Essai sur une application des règles des Maximis et Minimis à quelques Problèmes de statique relatifs à l’Architecture (1773/1776); Théorie des machines simples (1821, pp. 318-363); Versuch einer Anwendung der Methode des Größten und Kleinsten auf einige Aufgaben der Statik, die in die Baukunst einschlagen (1779) (German ed. of (Coulomb, 1773/1776, pp. 343-370)); On an application of the rules of maximum and minimum to some statical problems, relevant to architecture (see (Heyman, 1972, pp. 41-69)) Further historical reading: (Golder, 1948, 1953); (Gillmor, 1971a,b); (Heyman, 1972); (Kahlow, 1986); (Dietrich & Arslan, 1989); (Radelet-de Grave, 1994) Photo courtesy of: (Szabó, 1996, p. 386) Couplet, Pierre, born unknown, died 23 Dec 1743, Paris, France Pierre Couplet was the son of Claude-Antoine Couplet (1642-1722), the treasurer of the Académie Royale des Sciences in Paris. The young Couplet entered the Académie in 1696 as a pupil of his father. Afterwards, Couplet went to Lisbon, where he learned Portuguese, and then signed up for a two-and-a-half-year astronomy research expedition to Brazil. In 1699 he became a member of the reformed Académie and, in 1700, took part in the campaign to measure geographical longitude led by Cassini II. Later, Couplet was promoted to Professeur Royal de Mathématiques des Pages de la Grande Écurie and, in 1717, to treasurer of the Académie – posts that his father had occupied previously. He presented several essays on astronomy, earth pressure theory (see section 2.2.1),

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masonry arch theory, mansard roofs, pipe hydraulics and the mechanics of carriages and sledges to the Académie; most significant among these was his second essay on masonry arch theory (Couplet, 1730/1732), which began the tradition of analysing the collapse mechanisms of arches. In Heyman’s historical reconstruction of the history of arch theories from the viewpoint of the ultimate load method, Couplet’s arch theory therefore plays a very important role (Heyman, 1982). After Couplet’s death, the Académie refrained from the Éloge customary for its members. His work on arch theory was therefore gradually forgotten: “It is only unfortunate that the work was slowly forgotten, so that fifty years later Coulomb, in seeming ignorance of Couplet’s contribution, had to rediscover much of the theory” (Heyman, 1976, pp. 35-36). Main Works: De la poussée des terres contre leurs revestements, et de la force des revestemens qu’ou leur doit opposer (1726-1728); De la poussée des voûtes (1727/1729); Seconde partie de l’éxamen de la poussée des voûtes (1728/1730) Further historical reading: (Heyman, 1976) Culmann, Karl, born 10 Jul 1821, Bergzabern, Bavaria, died 9 Dec 1881, Zurich, Switzerland

After attending Wissembourg College (1835-1836), he moved to Metz, where his uncle, Friedrich Jakob Culmann (1787-1849), was a professor at the Artillery School; this awakened in him an interest in a career in engineering. From 1838 to 1841 he studied at Karlsruhe Polytechnic and was subsequently employed on public building works by Bavarian State Railways until 1855. With the help of his superior, Friedrich August von Pauli (1802-1883), Culmann spent the years 1849-1851 abroad in England, Ireland and the USA; his experiences were published in two travelogues that contained the theory of trussed frameworks. After leaving Bavarian State Railways, he became a full professor of engineering sciences at Zurich ETH, where, from 1860 onwards, he gave lectures on graphical statics which also included a graphical solution to the earth pressure problem (see section 2.3.2); he gained his doctorate there in 1880. Culmann placed graphical

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statics on a sound footing and made a great contribution to the establishment phase of theory of structures (1850-1875). Although his reasoning behind graphical statics was, in the end, rendered obsolete by projective geometry, together with Mohr he was the greatest structural engineer of the 19th century in the German-speaking world. Main Works: Der Bau der hölzernen Brücken in den Vereinigten Staaten von Nordamerika (1851); Der Bau der eisernen Brücken in England und Amerika (1852); Ueber die Gleichgewichtsbedingungen von Erdmassen (1856); Die graphische Statik (1864, 1866); Vorlesungen über Ingenieurkunde. I. Abtheilung: Erdbau (1872); Die Graphische Statik (1875) Further historical reading: (Tetmajer, 1882); (Ritter, W., 1903); (Stüssi, 1951, 1957, 1971); (Charlton, 1982); (Scholz, 1989); (Maurer, 1998); (Lehmann & Maurer, 2006) Photo courtesy of: Bibliothek der ETH Zürich De Josselin de Jong, Gerard, born 27 Mar 1915, Amsterdam, The Netherlands, died 2 Dec 2012, The Hague, The Netherlands

After leaving the grammar school in Haarlem, Gerhard De Josseling de Jong studied civil engineering at Delft TU, from where he graduated in 1941. The Netherlands were occupied by the German Armed Forces at that time, which put this young Jewish graduate in a life-threatening situation. In 1942 he decided, together with Pieter De Lint, to flee to England in a folding boat. They were 30 km out to sea when they were captured by the German Navy. De Jong was sentenced to death in November 1942, but the sentence was later commuted to 15 years in prison. After being freed from a north German prison by British troops, he returned to The Netherlands and spent several years painting and drawing. After he had gained his first experience of engineering practice in a design office in Paris (1947-1949), he joined the Soil Mechanics Laboratory of Delft TU as a researcher. His work resulted in pioneering publications on consolidation theory and the theory of granular materials (see (De Jong, 1959, pp. X-XI)). In 1959 he gained his doctorate at Delft TU with a dissertation on the statics and kinematics of granular materials (De Jong, 1959), which was supervised by von Prof. E. C. W. A. Geuze. After that he was engaged as a visiting professor

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by the University of California and succeeded Geuze at the renowned Chair of Soil Mechanics at Delft TU in 1960. De Jong’s scientific life’s work embraced the kinematics of granular media in the plastic zone, consolidation theory, the stability of vertical trenches in the soil and groundwater flows in porous media; he shone in all these fields. Fastidiously prepared illustrations and drawings were important to him in all this work; they had already characterised his dissertation. De Jong was granted emeritus status in 1980, but continued his scientific and artistic work. His contributions to soil mechanics characterise the consolidation period of geotechnical engineering (1950-1975), but with their thinking in terms of discontinuities, e.g. in the shape of the emerging discrete element method (DEM), they paved the way for the theoretical side of the integration period of geotechnical engineering (1975 to date). De Jong was succeeded by his student Arnold Verruijt, who, through his articles on foundation dynamics, computational geomechanics and other topics, strengthened the international reputation of this chair at Delft TU further and hence successfully completed the transition to the integration period of geotechnical engineering (1975 to date) in Delft. Main Works: Statics and kinematics in the failure zone of a granular material (1959); De mechanica toegepast op grond (1961); Consolidation models consisting of an assembly of viscous elements or a cavity channel network (1968); The double sliding, free rotating model for granular assemblies (1971); Improvement of the lowerbound solution for the vertical cut off in a cohesive, frictionless soil (1978); Diskontinuitäten in Grenzspannungsfeldern (1979); Application of thecalculus of variation to the vertical cut off cohesive frictionless soil (1980); Soil mechanics and transport in porous media: Selected Works of G. de Josselin de Jong (2006) Further biographical reading: (Schotting et al. 2006, pp. VII-VIII); (Verruijt, A., 2013) Photo courtesy of: (Verruijt, A., 2013, p. 891) Drucker, Daniel C., born 3 Jun 1918, New York, USA, died 1 Sept 2001, Gainesville, Florida, USA

Daniel C. Drucker studied at Columbia University, where he became interested in the conception, design and analysis of bridges. However, Raymond D. Mindlin

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suggested he write his dissertation on the subject of photoelasticity, which he completed in 1940. Afterwards, he lectured at Cornell University until 1943. Following military service, he worked for a short time at the Illinois Institute of Technology before transferring to Brown University, where he worked as a teacher and researcher from 1947 to 1967. It was at this university that William Prager founded the world-famous school of applied mathematics and mechanics in the 1940s and it was here that Drucker carried out his pioneering work on plastic theory. For example, he formulated the principles of classical plastic theory (Drucker, 1949) and, based on the energy criterion for the stability of the elastic equilibrium, he introduced the concept of material stability (Drucker, 1951), which today – in the form of Drucker’s stability postulate – enjoys an established place in the literature. Material stability, especially the stability of the infinitesimal, is crucial for dealing with the shake-down of loadbearing structures (Bleich-Melan shake-down principle) and the formulation of the stress-strain relationships in plastic theory. The work by Drucker, Prager and Greenberg on the upper/lower bounds theorem (see section 3.1.2) also played a key role in the further development of earth pressure theory. Drucker became dean of the Faculty of Engineering at the University of Urbana-Champaign in 1968. From 1984 until his retirement in 1994, he worked as a research professor at the University of Florida, and he was editor of the Journal of Applied Mechanics for 12 years. Drucker’s work has been honoured with numerous awards; for example, Lehigh, Brown, Northwestern and Urbana-Champaign universities plus the Haifa Technicon all awarded him honorary doctorates, and the American Society of Mechanical Engineers (ASME) inaugurated its Daniel C. Drucker Medal in 1997. Charles E. Taylor wrote the following memorable words about Daniel C. Drucker: “In all of the thousands of hours we spent together, I never heard him utter a single swear word. He had a great sense of humor, but he never told a joke and he never spread gossip. I have never met a more honest man or pure person. Dan Drucker was the kind of person that we all try to be” (Taylor, 2003, p. 159).

Main Works: Relation of experiments to mathematical theories of plasticity (1949); A more fundamental approach to stress-strain relations (1951); Extended limit design theorems for continuous media (1951); Soil mechanics and plastic analysis or limit design (1952); Coulombs friction, plasticity and limit loads (1953); On uniqueness in the theory of plasticity (1956); A definition of stable inelastic material (1959); On Structural Concrete and the Theorems of Limit Analysis (1961) Further historical reading: (Taylor, 2003, 2015) Photo courtesy of: (Taylor, 2003, p. 158)

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Feld, Jacob, born 3 Mar 1899, Austro-Hungarian Empire, died 17 Aug 1975, New York, USA

Jacob Feld was born in the Habsburg Monarchy. He was still quite young when his father, Israel Feld (1866-1938), and mother, Gussie Harzoff Feld (1869-1938), took their family to New York in 1906. After leaving New York’s City College in 1918, Jacob Feld attended Cincinnati University, where he obtained a PhD in 1922 with a dissertation on determining earth pressures experimentally (Feld, 1923), which was given a critical review by Terzaghi. After that, Feld worked for Turner Constructors and the Long Island Railroad. He met the renowned engineers David B. Steinman (1886-1960) and Henry C. Goldmark (1857-1941) in New York. Even as a young engineer, Feld had been interested in the failure of structures, e.g. retaining walls. He opened an engineering consultancy in New York in 1926 and became a partner of consulting engineers Kaminetzky & Cohen in 1966, where he worked successfully until his death. In his first period of creativity as a consulting engineer, his publications included a history of earth pressure theory in particular (Feld, 1928) and soil mechanics in general (Feld, 1948). But he achieved much more besides: From the mid-1950s onwards, Feld was regarded as an authority on the building of radio telescopes and was one of the leading structural engineers in New York. For example, he acted as a consultant for the construction of part of the city’s 6th Avenue Subway and was responsible for many loadbearing structures, including those for the New York Coliseum, Guggenheim Museum, Yonkers Raceway, Lincoln Center and Hudson River Water Pollution Control Plant. Feld’s experience as a consulting engineer, but especially his reports on structural failures, was assembled in monographs (Feld, 1964, 1968), which helped him become a pioneer of forensic engineering. His clients included not only building owners, architects and engineers, but also contractors, construction financers, insurance companies and municipalities; he also advised the US Air Force and the Highway Research Board. Feld was awarded the Order of Merit for Science and Technology of the French Republic in 1963; nine years later, New York’s City College awarded him an honorary doctorate in law. Feld was president of the New York Academy of Sciences and a fellow of the American Society of Civil Engineers (ASCE), which elected him ‘Metropolitan Engineer of the Year’ in 1969. He made his wealth of engineering knowledge available to Purdue University, Northwestern University and North Carolina State University as a visiting professor. Feld also served as governor of the Technion in Haifa and as vice-president of the Cejwin Educational Camp in Port Jervis (New York).

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Without doubt, Feld’s greatest services were in the field of forensic engineering. As recently as 1997, Kenneth L. Carper edited a much expanded edition of Feld’s Construction Failure (1968) (Feld & Carper, 1997) – a classic of forensic engineering. Main Works: Measured retaining-wall pressure from sand and surcharge (1922); Lateral earth pressure: the accurate experimental determination of the lateral earth pressure together with a resume of previous experiments (1923); History of the Development of Lateral Earth Pressure Theories (1928); Early History and Bibliography of Soil Mechanics (1948); A historical chapter: British royal engineers’ papers on soil mechanics and foundation engineering, 1837-1874 (1952/1953); Structural design study for a parabolic reflector six hundred feet in diameter (1957); Lessons from Failures of Concrete Structures (1964); Construction Failure (1968, 1997) Further biographical reading: (Feld, 1968, p. X) Photo courtesy of: (Feld & Carper, 1997) Hansen, Brinch Jørgen, born 29 Jul 1909, Aarhus, Denmark, died 27 May 1969, Copenhagen, Denmark

After graduating in civil engineering from Copenhagen TU, Hansen worked in Copenhagen in the engineering office of contractors Christiani & Nielsen, which had been founded in 1904. His work involved the construction of ports, harbours and quays at home and abroad. These complex engineering tasks forced Hansen again and again to become actively involved in the expanding system of knowledge of geotechnical engineering during its discipline-formation period (1925-1950). Earth pressure problems were of prime concern here. In 1953 Hansen, who had in the meantime risen to the post of senior engineer at Christiani & Nielsen, submitted his dissertation on earth pressure calculation (Hansen, 1953) to Copenhagen TU. This work can be seen as a milestone in the theoretical treatment of earth pressure in the early years of the consolidation period of geotechnical engineering (1950-1975). In his dissertation,

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Hansen develops his equilibrium method on the basis of Kötter’s mathematical earth pressure theory (see section 2.5.4). His slip plane is assumed to be a circular cylinder, which represented a good approximation and had already been employed by the Swedish school of earthworks as a slip circle method for investigating embankments, dykes and quayside structures (Kurrer, 2018, pp. 342-343). Hansen’s earth pressure theory examines the equilibrium at a plastic limit state using a specific boundary condition. He therefore contributed to the paradigm change in geotechnical engineering – in history of science terms comparable to the paradigm change from elastic to plastic theory in structural engineering (Kurrer, 2018, pp. 121-137), especially steel construction. Hansen recognised the advantages of the partial safety factor concept for geotechnical engineering at an early stage (Hansen, 1956) – something that did not become established in design practice until decades later. When Copenhagen TU split the Chair of Hydraulic & Foundation Engineering of Prof. Helge Lundgren (1914-2011) in 1955, they appointed Hansen as professor for soil mechanics and foundations, while Lundgren continued to be responsible for hydraulic engineering. At the same time, Hansen took over as director of the Geotechnical Institute of Denmark. This reorganisation points to the emancipation of geotechnical engineering from hydraulic engineering and is a characteristic of the consolidation period of geotechnical engineering (1950-1975). Of course, Hansen, together with his institute and university colleagues, took part in large Danish infrastructure projects, e.g. the crossings over the Great and Little Belts. Hansen was chair of the Danish Geotechnical Society, a member of the Danish Academy of Engineering Sciences and, in 1965, vice-president of the European section of the International Society of Soil Mechanics. Furthermore, Hansen contributed his knowledge to the German ‘Waterfront Structures’ Committee. His equilibrium method was incorporated in the new Danish geotechnical engineering standards in 1965 – the year in which Ghent University awarded him an honorary doctorate. Hansen published a Danish textbook on soil mechanics together with his hydraulic engineering colleague Lundgren (Hansen & Lundgren, 1958), a German version of which appeared two years later (Hansen & Lundgren, 1960). It quickly became the standard work on geotechnical engineering in the second half of that discipline’s consolidation period (1950-1975). Main Works: Earth Pressure Calculation (1953); Brudstadieberegning og partialsikkerheder i geoteknikken (1956); Geoteknik (1958); Hauptprobleme der Bodenmechanik (1960); Spundwandberechnung nach dem Traglastverfahren (1962) Further biographical reading: (Bjerrum, 1969) Photo courtesy of: (Bjerrum, 1969)

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Jáky, József, born 15 Jul 1893, Szeged, Austro-Hungarian Empire (now Hungary), died 13 Sept 1950, Hévíz, Hungary

Jáky studied civil engineering at Budapest TH from 1911 to 1915, after which the pioneer of reinforced concrete in Hungary, Prof. Szilárd Zielinski (1860-1924), appointed Jáky to be his assistant at his chair. The young and enthusiastic Jáky became acquainted with the wealth of literature on earth pressure theory in French and German. However, his learning was interrupted by the war. Earth pressure turned out to be an area in which Jáky would work with great success during the discipline-formation period of geotechnical engineering (1925-1950) (see section 2.7) and make him the founder of soil mechanics in Hungary. In the early 1920s, Jáky was active as a teacher at the Chair of Transport and Railways at Budapest TH, submitting his dissertation on railway transition curves in 1924 and also carrying out practical work. It was during the production of plans for bridges, railways, reinforced concrete structures and stream training works that Jáky also came into contact with earth pressure problems. He therefore published an article on embankments in 1925 which earned him the Hollán Prize of the Hungarian Engineers & Architects Association. Jáky was awarded a Jeremiah Smith Stipendium in 1927 which enabled him to spend a year studying soil mechanics at MIT in Cambridge (Mass.). “Later, Terzaghi … took him under his wing and allowed him to participate in his research” (Széchy & Kézdi, 1955, p. 7). After returning to Hungary, Jáky submitted his habilitation thesis on soil mechanics to Budapest TH, where he established Hungary’s first soil mechanics laboratory, fought for the recognition of soil mechanics in science and practice, published the first coherent Hungarian work on soil mechanics (Jáky, 1933a) and advanced earth pressure theory with his own special brand of enthusiasm. Jáky’s international breakthrough came in 1936 at the second IABSE congress in Berlin with the further development of earth pressure theory based on Kötter’s differential equation (see eq. (2-110) in section 2.5.4) (Jáky, 1937/1938). That same year, Budapest TH appointed him to the post of associate professor for transport and railways, and full professor three years later, which enabled Jáky to put roadbuilding on a scientific footing from the soil mechanics viewpoint (Jáky, 1937). He also contributed four papers to the international roadbuilding congress that took place in The Hague in 1938. In that same year he began giving lectures on soil mechanics to civil engineering students in their ninth semester who had chosen roads, railways and waterways as their specialisation; two years later,

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these lectures became compulsory for all students of civil engineering. Jáky took over the Chair of Railways and Earthworks at Budapest TH in 1941 and therefore was able to pursue his main scientific interests even more systematically. Of all his scientific work, it is that on the new earth pressure theory (Jáky, 1944a) that really stands out. In that work he criticises Terzaghi’s earth pressure theory and Hertwig’s observations (see section 2.7.1); Jáky proved that, in the end, both lead to Coulomb’s earth pressure theory. In 1944 Jáky and other professors opposed the relocation of Budapest TH to Germany, and he spent the rest of the time up until the end of the war in a friend’s apartment far away from Budapest. Following the liberation of his country, he helped to rebuild his Alma Mater and was active as a foundations expert for many public works. Jáky was awarded the Kossuth Prize for his contribution to the rebuilding of his country. He also continued his research into earth pressure theory, and in 1946 was awarded the Marcibányi Prize and Medal of the Hungarian Academy of Sciences for his work in this field. His work on the plasticity of bodies of soil (Jáky, 1947/1948) gained international recognition through professors Schultze, Terzaghi, Timoshenko, Frontard, Fröhlich and Nádái. Jáky submitted seven papers, including three on earth pressure theory (Jáky, 1948a,b,c), for the 2nd International Conference on Soil Mechanics and Foundation Engineering (Rotterdam, June 1948). The Hungarian Academy of Sciences elected him a full member in 1950. Unfortunately, Jáky, the father of Hungarian soil mechanics, died on 13 September 1950 and so did not live to see the print version of his debut presentation on the network of slip planes in a body of soil for determining the upper and lower bounds of earth pressure acting on a retaining wall (Jáky, 1953). Main Works: A talajmechanika alapfogalmai és technikai alkalmazásuk (The basic concepts of soil mechanics and their practical applications) (1933a); Mártony Károly, az els˝o magyar földnyomáskutató emlékezete (In memory of Károly Mártony, the first Hungarian to carry out research on earth pressure theory) (1933b); The Classical Earth Pressure Theory (1935); Stability of Earth Slopes (1936); Az útépítés talajmechanikája (Soil mechanics of roadbuilding) (1937); Die klassische Erddrucktheorie mit besonderer Rücksicht auf die Stützwandbewegung (1937/1938); Az új földnyomáselmélet (New earth pressure theory) (1944a); A módosított Coulomb-elmélet (Coulomb’s modified theory) (1944b); A nyugalmi nyomás tényez˝oje (Coefficient of earth pressure at rest) (1944c). Talajmechanika (soil mechanics) (1944d); Tévedések a földnyomáselméletben (Errors in earth pressure theory) (1946); Sur la stabilité des masses de terres complètement plastiques (1947/1948); Minimum Value of Earth Pressure (1948a); Novel Theory of Earth Pressure (1948b); Validity of Coulomb’s Law of Stability (1948c); Network of Slip Lines in Soil Stability (1953) Further biographical reading: (Széchy & Kézdi, 1955) Photo courtesy of: (Széchy, 1955)

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Kérisel, Jean Lehuérou, born 18 Nov 1908, Saint-Brieuc, France, died 22 Jan 2005, Paris, France

In 1928 Jean Kérisel, the son of a lawyer, completed his education at the École Polytechnique. He married Suzy Caquot in 1931 and graduated from the École Nationale des Ponts et Chaussées (ENPC) as a civil engineer in 1933. Influenced by his father-in-law, he became interested in soil mechanics while at the ENPC and gained his doctorate at the Sorbonne in July 1935 with a first thesis on a subject from soil mechanics (Kérisel, 1935a) and a second thesis on the hysteresis problem in mechanics (Kérisel, 1935b). Between 1933 and 1939, Kérisel earned a living for himself and his family working as a roads and bridges engineer in Orléans. During the Second World War, Kérisel served as an engineering officer and was awarded the Croix de Guerre for his bravery. In 1941 he was appointed director of the Commission for Reconstruction and, following liberation in 1944, served as director for planning and directing construction at the Ministry of Rebuilding Works until 1951. In recognition of his contribution to the rebuilding of his country, Kérisel was elected a Knight of the Legion of Honour in December 1945 and an Officer in May 1951. During his time as director, he published, together with his father-in-law, a book of tables of earth pressures and the load-carrying capacities of foundations (Caquot & Kérisel, 1948), second, third and fourth editions of which were published in 1973, 1990 and 2003 respectively; this important practical work on geotechnical engineering was also translated into English. A monograph on soil mechanics appeared one year later (Caquot & Kérisel, 1949). Kérisel was appointed an honorary professor for foundations and soil mechanics at the ENPC in January 1951, a position he held until 1969 – again following in the footsteps of his father-in-law. His Cours de Mécanique des Sols (1950/1951), which went through six editions up until 1963/1964, is witness to his very productive teaching activities at the ENPC. Kérisel founded the Simecsol engineering consultancy in 1952, which worked successfully on many major geotechnical projects (Bourru de Lamotte, 2005, p. 3) and today is part of the Dutch Arcadis Group. It was the synthesis of science, practice and new ideas that enabled Kérisel – similarly to Skempton – to become a historian of geotechnical engineering – as is shown by many presentations and publications. Kérisel was an obvious choice for managing organisational tasks in the engineering community. For example, he was president of the Société des Ingénieurs Civils de France (1968-1969), president of the Comité Français de Mécanique des Sols et Travaux de Fondations (1969-1973) and chair of the International Society of Soil Mechanics & Foundation Engineering (1973-1977). He was appointed

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ENCP honorary professor in 1970 and elected an honorary member of the Hungarian Academy of Sciences in 1973. He was awarded honorary doctorates by Liège University in 1975 and Naples University in 1996. After retiring, Kérisel published more on the history of construction and made valuable contributions to Egyptology. His final book, Of Stones and Man: From the Pharaohs to the Present Day (2005), appeared shortly after his death. Kérisel was survived by his three children, 12 grandchildren and 23 great-grandchildren; his wife had died earlier in 1998. Thierry Kérisel wrote a wonderful and accurate tribute to his father (Kérisel, T., 2006). Without doubt, Kérisel was one of the leading engineering personalities on the international scene during the consolidation period of geotechnical engineering (1950-1975) and belongs to the tradition of France’s great geotechnical engineers. Main Works: Contribution à l’étude du frottement des milieux pulvérulents et application à l’étude des fondations (1935a); L’hystérésis dans les phénomènes mécaniques (1935b); Cours de Mécanique des Sols (1950/1951); Historique de la méchanique des sols en France jusqu’au 20e Siècle (1956); La mécanique des sols: recherches et investigations récentes (1958); Bicentenaire de l’essai de 1773 de Charles Augustin Coulomb (1973); Old structures in relation to soil mechanics (1975); Persectiva historica de la ingeniera geotecnica (1982); The History of Geotechnical Engineering up until about 1700 (1985); Down to earth. Foundations past and present: The invisible art of the builder (1987); History of retaining walls design (1992); Albert Caquot 1881-1976 – Savant, soldat et bâtisseur (2001); Of Stones and Man: From the Pharaohs to the Present Day (2005) Further biographical reading: (Bourru de Lamotte, 2005); (Bourru de Lamotte & Gambin, 2005); (Kérisel, T., 2006) Photo courtesy of: (Kérisel, T., 2006, p. 18) Kézdi, Árpád, born 19 Nov 1919, Komárom, Hungary, died 20 Oct 1983, Budapest, Hungary

The town of Komárom was split in two by the new border between Hungary and the newly created state of Czechoslovakia in 1918. The Kézdi family therefore

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decided to move to Györ in Hungary, where Árpád Kézdi attended primary school and then the Benedictine grammar school for four years. His teachers soon recognised Kézdi’s talent for the natural sciences, music and literary studies. As his father, a waterways engineer, was transferred to Miskolc, it was in that town that Kézdi past his university entrance examination in 1937 with an overall grade of “excellent”. He studied civil engineering at Budapest TH, graduating in 1942. Prof. Jáky became aware of this talented student at an early stage. He appointed him an assistant at his chair and, in 1942, made him his personal assistant, entrusting him with his lectures on soil mechanics, waterways and railways as his illness progressed. One year later, Kézdi became an official assistant at Jáky’s chair. Kézdi had to serve in the army from 1944 until the end of the war and took over his teacher’s chair in late 1950 following Jáky’s early death. By 1960 Kézdi had written several books on geotechnical problems in Hungarian. His international breakthrough followed in 1962 with a book on earth pressure theories published by Springer. Kézdi was fluent in English, German and French and also had a good command of several Slavic languages. He shared his talent for languages with his wife Anna, who had been among Jáky’s circle of friends and whom Kézdi had met in 1946. Besides his publications in Hungarian, he also wrote works in English and German. His list of publications runs to 41 books, or contributions to books (including translations), and 158 articles for journals. He often undertook the translations himself, certainly with the help of his wife. For example, for his compendium on soil mechanics in the Soviet Union, Kézdi translated the articles from Russian into German (Kézdi, 1981, p. 11). However, his masterpiece is his multi-volume work on soil mechanics which appeared in Hungarian, German (Kézdi, 1968, 1970, 1973, 1976), English, Spanish and Russian and which can be regarded as the standard work on soil mechanics during the consolidation period of geotechnical engineering (1950-1975). As is to be expected, Kézdi occupied several leading roles in the community of geotechnical engineers, e.g. in the International Soil Mechanics & Foundations Association. Kézdi’s services to geotechnical engineering were honoured many times: Hungarian State Prize (1966), member of the Hungarian Academy of Sciences (1976), honorary doctorates from Dresden TU (1971) and the University of Natural Resources & Life Sciences, Vienna (1972), and many more. Kézdi’s 65th birthday was to be honoured at the 6th Budapest Conference on Soil Mechanics & Foundation Engineering, which was held in Budapest in October 1984, by handing him a commemorative document dedicated to him. Unfortunately, a severe illness killed him shortly before his 64th birthday, which meant that this document first appeared in early 1985 in the Acta Technica journal (Petrasovitis, 1985, p. 5). Main Works: Über die Tragfähigkeit und Setzung von Pfahlgründungen (1955); Erddrucktheorien (1962); Bodenmechanik. Band 1. Bodenphysik (1968); Bodenmechanik. Band 2. Bodenmechanik im Erd-, Grund- und Straßenbau (1970); Bodenmechanik. Band 3. Bodenmechanisches Versuchswesen (1973); Bodenmechanik. Band 4. Anwendung der Bodenmechanik in der Praxis (1976); Bodenmechanik in der Sowjetunion (1981)

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Further biographical reading: (Petrasovitis, 1985); (Steinfeld, 1984) Photo courtesy of: (Steinfeld, 1984, p. 156) Kötter, Fritz, born 8 Nov 1857, Berlin, Prussia, died 17 Aug 1912, Schopfheim, German Empire

Fritz Kötter was the son of a civil servant working for the district court accounts department who was also the brother of the mathematician Ernst Kötter (1859-1922). Kötter attended a secondary school in the Luisenstadt district of Berlin and studied mathematics at Berlin University from 1878 to 1882, and by February 1883 he had already gained his Dr. phil. at Halle-Wittenberg University with a dissertation on the equilibrium of flexible, non-extensible surfaces. Following teaching work at a secondary school in the Königstadt district of Berlin, he submitted his habilitation thesis at Charlottenburg TH (later Berlin TH) in the spring of 1887 and was given a permit to teach pure mathematics and mechanics. It was during this period that he and other colleagues supported the edition of the collected works of the mathematician Karl Weierstraß (1815-1897). After that, Kötter was allowed to give lectures on higher mathematics at Berlin Mining Academy as a trial for one year before being taken on by that establishment in July 1889. In the autumn of 1900 he was appointed to the newly created Chair of Mechanics in the Civil Engineering Faculty at Charlottenburg TH, a post that Kötter held until he died; he was succeeded by Hans Reissner (1913-1934), Friedrich Tölke (1937-1945) and Istvan Szabó (1948-1971). In the 1890s it is Kötter’s historico-critical contribution to earth pressure theory that really stands out (see section 2.5.4). This laid the foundation for a new direction in research and can be regarded as an excellent example of a “historical engineering science” (see (Kurrer, 2018, pp. 948-961)) within the field of geotechnical engineering. Kötter worked together with Heinrich Müller-Breslau at the Civil Engineering Faculty. As a member of the Royal Prussian Academy of Sciences, Müller-Breslau presented several works by Kötter after 1900, which were then published in the minutes of the proceedings (Kötter, 1903, 1908a,b, 1909, 1910). In a review of Mohr’s Abhandlungen aus dem Gebiete der Technischen Mechanik (treatises from the field of applied mechanics, 1906), Kötter took Müller-Breslau’s side, which had been published in his book Erddruck auf Stützmauern (earth pressure on

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retaining walls, 1906) (Kötter, 1907). Müller-Breslau encouraged Hans Reissner to write his dissertation on vibrations in trussed frameworks (Reissner, 1902), which Kötter supervised, assisted by Müller-Breslau. Kötter was awarded the Order of the Red Eagle, 4th class, and, in 1910, elected to the ‘Leopoldina’ (German National Academy of Sciences). Shortly before his death, a commemorative document published to mark the 60th birthday of Müller-Breslau included a fundamental contribution to the extension of plate theory (Kötter, 1912). Kötter’s fundamental research into applied mechanics, earth pressure theory and his close collaboration with Müller-Breslau showed him to be an early advocate of the Berlin school of theory of structures (Kurrer, 2018, p. 524ff.). Numerous researchers, e.g. H. Reissner (1910, 1924), J. Jáky (1937/1938), J. Ohde (1938) and J. B. Hansen and H. Lundgren (1960) would use his earth pressure theory (see section 3.4.1) as their starting point. Main Works: Über das Problem der Erddruckbestimmung (1888); Beitrag zur Lehre vom Fachwerk (1890); Über die Bewegung eines festen Körpers in einer Flüssigkeit (1891); Die Entwicklung der Lehre vom Erddruck (1893); Der Bodendruck von Sand in vertikalen zylindrischen Gefäßen (1899); Die von Steklow und Liapunow entdeckten integrabelen Fälle der Bewegung eines starren Körpers in einer Flüssigkeit (1900); Die Bestimmung des Drucks an gekrümmten Gleitflächen, eine Aufgabe aus der Lehre vom Erddruck (1903); Die Bestimmung des Drucks an gekrümmter Gleitfläche, ein Beitrag zur Lehre vom Erddruck (1908a); Über die Torsion des Winkeleisens (1908b); Über den Druck von Sand gegen Öffnungsverschlüsse im horizontalen Boden kastenförmiger Gefäße (1909); Über die Spannungen in einem ursprünglich geraden, durch Einzelkräfte in stark gekrümmter Gleichgewichtslage gehaltenen Stab (1910); Über das Gleichgewicht elastischer Platten und langer Streifen (1912) Further biographical reading: (anon., 1912a); (anon., 1912b) Photo courtesy of: Leopoldina archives, M 3/8a, M No. 3313 Krey, Hans Detlef, born 8 Oct 1866, Osterbünge, Holstein, Prussia, died 15 Jul 1928, Berlin, Weimar Republic

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Following elementary and private education, Hans Krey, the son of a farming family, started attending the grammar school in Altona in 1879, where he passed his university entrance examination in the autumn of 1886. This enabled him to study civil engineering – initially in Munich, later in Berlin; he completed his studies in 1891. Those were followed by the posts typical of a civil servant in the Prussian building authority: foreman (1891), supervisor (1896), hydraulic structures inspector (1904) and executive officer for building (1906). During his time in Berlin (1901-1906), Krey worked as an assistant to professors Müller-Breslau and Grantz. Working with Müller-Breslau, he was introduced to earth pressure tests at the testing institute for theory of structures – work he was able to exploit successfully later on. Krey became head of the Lünen Canal-Building Department in 1906 and in this capacity worked on the building of the Mittelland Canal. By the end of March 1910 he was called back to the Prussian Ministry of Public Works in Berlin for a second time, where he took over the Royal Testing Institute for Hydraulics and Shipbuilding (VWS), which had been founded in 1903 and which he raised to the level of an internationally renowned testing institute. Besides Krey’s contributions to hydraulic structures – together with Hubert Engels (1854-1945) he can be regarded as one of the founders of scientific experiments with models –, his research into earthworks in general and earth pressure theory in particular earned international recognition. Dresden TH awarded him an honorary doctorate in 1920. One year later, Krey was appointed senior executive officer for building. Krey’s publication on sheet pile walls alone shows his skill in handling the complex interaction of anchor forces, hydrostatic pressure and active and passive earth pressures (1912a) in simple, yet not too simple, structural analysis models. It was in that same year that he published the first edition of his book Erddruck, Erdwiderstand und Tragfähigkeit des Baugrundes (active and passive earth pressures and bearing capacity of subsoils, 1912/2), which enjoyed five editions and placed earth pressure theory firmly on an experimental basis: “The uncertainty regarding the stability of earthworks is not due to … the shortcomings in our methods of computation, but, in the first place, to a lack of knowledge and insufficient studies of the types of soil that are also affected and, further, incorrect knowledge about the most unfavourable stress state” (Krey, 1927, p. 485). In 1927 Krey founded and took charge of an earthworks laboratory at the VWS. This is where Johann Ohde (1905-1953) carried out earth pressure research starting in that year. The year 1927 also saw Krey appointed an honorary professor at Berlin TH for the testing of hydraulic structures. One year prior to that, the Building Academy had awarding him the Medal for Outstanding Services to Building, and he was elected a member of the academy shortly before his death. His book on earth pressure is probably the most important of his 50 or so technical-scientific publications – an early example of a new style of theoretical treatment in soil mechanics during its constitution phase (1925-1950). Main Works: Praktische Beispiele zur Bewertung von Erddruck, Erdwiderstand und Tragfähigkeit des Baugrundes in grösserer Tiefe (1912a); Erddruck, Erdwiderstand und Tragfähigkeit des Baugrundes. Gesichtspunkte für die Berechnung. Praktische

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Beispiele und Erddrucktabellen (1912b, 1918, 1926b, 1932, 1936); Betrachtungen über Größe und Richtung des Erddruckes (1923); Die Widerstandsfähigkeit des Untergrundes und der Einfluß der Kohäsion beim Erddruck und Erdwiderstande (1924); Gebrochene und gekrümmte Gleitflächen bei Aufgaben des Erddruckes (1926a); Rutschgefährdete und fließende Bodenarten (1927) Further historical reading: (Ludin, 1928); (Schultze, 1954); (Jaeger, 1982) Photo courtesy of: (Ludin, 1928, p. 595) Lévy, Maurice, born 28 Feb 1838, Ribeauvillé, France, died 30 Sept 1910, Paris, France

Maurice Lévy studied at the École Polytechnique and the École des Ponts et Chaussées from 1856 to 1861. Thereafter, he spent several years in the provinces before gaining his doctorate in Paris in 1867 and working at the École Polytechnique as a mechanics tutor from 1862 to 1883. It was in this period that Lévy wrote his chief works – his contribution to earth pressure theory, his generalisation of Tresca’s strength hypothesis (Tresca, 1864) for the three-dimensional case (1871), his contribution to the elastic theory foundation of theory of structures (1873) and the first edition of his graphical statics (1874), which followed on directly from Cremona’s work. Wilhelm Ritter’s Anwendungen der graphischen Statik (applications of graphical statics) became a standard reference book in the German-speaking countries and Lévy’s four-volume work on graphical statics (2nd ed., 1886-1888) played the same role in the French-speaking world. In 1875 he was appointed professor of applied mechanics at the École Centrale des Arts et Manufactures and, in 1885, professor of analytical mechanics and celestial mechanics at the Collège de France, and was also promoted to the post of inspector-general of bridges and highways. From 1883 onwards, he was a member of the Académie des Sciences. An elegant solution for rectangular elastic plates with Navier boundary conditions (Lévy, 1899) is named after Lévy. Main Works: Note sur un système particulier de ponts biais (1869); Essai sur une théorie rationelle de l’équilibre des terres fraîchement remuées et de ses applications au calcul de la stabilité des murs soutènement (1869/1870); Extrait du Mémoire sur

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les equations générales des mouvements intérieures des corps solides ductiles au de là des limites où l’élasticité pourrait les ramener à leur premièr ètat (1871); Application de la théorie mathématique de l’élasticité à l’étude de systèmes articulés (1873); La statique graphique et ses applications aux constructions (1886-1888); Notes sur les diverses manières d’appliquer la règle du trapèze au calcul de la stabilité des barrages en maçonnerie (1897); Sur l’équilibre élastique d’une plaque rectangulaire (1899) Further biographical reading: (Picard, 1910); (Lecornu, 1915); (Koppelmann, 1973); (Becchi, 1998); (Maurer, 1998) Photo courtesy of: Collection École Nationale des Ponts et Chaussées Mohr, Otto, born 8 Oct 1835, Wesselburen, Holstein, died 2 Oct 1918, Dresden, German Empire

During his father’s period of office as the local mayor, the young Otto met Friedrich Hebbel, who was later to become famous as an author but at the time was the 14-year-old scribe employed in his father’s office. At the age of 16, Mohr went to the Polytechnic School in Hannover. After completing his studies, he worked for Hannover State Railways and then Oldenburg State Railways. Around 1860 he is supposed to have developed the method of sections (attributed to August Ritter) for analysing statically determinate trussed frameworks when working on a design for the first iron bridge with a simple triangulated form at Lüneburg. A little later, the young Mohr gained attention among his profession by publishing a paper on the consideration of displacements at the supports during the calculation of internal forces in continuous beams. But his work didn’t stop there: He introduced influence lines at the same time as Winkler in 1868 and discovered the analogy since named after him, which gave graphical statics an almighty helping hand. He was appointed professor of structural mechanics, route planning and earthworks at Stuttgart Polytechnic in 1867. During his time in Stuttgart, Mohr also examined the earth pressure problem intensely (see section 2.4.2). Six years later, Mohr accepted a post at Dresden Polytechnic as the successor to Claus Köpcke (1831-1911) and taught graphical statics plus railway and hydraulic engineering there until 1893. After the departure of Gustav Zeuner in 1894, he took on the subjects of applied mechanics and strength

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of materials in conjunction with graphical statics. Mohr gave up teaching in 1900, but continued working on the development of applied mechanics and theory of structures. His work on the fundamentals of theory of structures based on the principle of virtual forces (1874/1875) meant that he – alongside Maxwell (Maxwell, 1864b) – made the greatest contribution to classical theory of structures. Through his work, Mohr, like no other, provided impetus to the classical phase of the discipline-formation period (1825-1900) and the first half of the consolidation period (1900-1950) of theory of structures. Mohr argued with Müller-Breslau over the fundamentals of theory of structures and, later, over priority issues regarding essential concepts, theorems and methods in theory of structures. For example, Mohr’s stress circle (Mohr, 1882) is imprinted on the memory of every engineer and still plays a pivotal role, e.g. in the form of Coulomb-Mohr’s failure condition in cohesive soils; the stress circle also forms the starting point for further research in the field of strength of materials (see (Mertens, 2011), for example). Numerous personalities from the world of science and engineering, e.g. Robert Land, Georg Christoph Mehrtens, Willy Gehler, Kurt Beyer and Gustav Bürgermeister, were influenced by the founder of the Dresden school of applied mechanics. Hannover TH awarded him a doctorate. After lengthy deliberations, Mohr accepted the post of Working Privy Councillor with the title “Excellency”, which he had been awarded by the Saxony government. Main Works: Beitrag zur Theorie der Holz- und Eisenkonstruktionen (1868); Beitrag zur Theorie des Erddrucks (1871); Zur Theorie des Erddrucks (1872); Beitrag zur Theorie der Bogenfachwerksträger (1874a); Beitrag zur Theorie des Fachwerks (1874b); Beiträge zur Theorie des Fachwerks (1875); Über die Zusammensetzung der Kräfte im Raume (1876); Über die Darstellung des Spannungszustandes und des Deformationszustandes eines Körperelements und über die Anwendung derselben in der Festigkeitslehre (1882); Ueber das sogenannte Princip der kleinsten Deformationsarbeit (1883); Beitrag zur Theorie des Fachwerkes (1885); Über die Elasticität der Deformationsarbeit (1886); Die Berechnung der Fachwerke mit starren Knotenverbindungen (1892/93); Welche Umstände bedingen die Elastizitätsgrenze und den Bruch eines Materials? (1900); Abhandlungen aus dem Gebiete der Technischen Mechanik (1906, 1914, 1928); Eine neue Theorie des Erddrucks (1907) Further historical reading: (Gehler, 1916, 1928); (Grübler, 1918); (Steiding, 1985); (Knittel, 1994b), (Hänseroth, 2003) Photo courtesy of: Hebbel Museum, Wesselburen Möller, Max Carl Emil, born 19 Feb 1854, Hamburg, died 19 Dec 1935, Braunschweig, German Empire

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Max Möller was the son of a Hamburg businessman. After he died in 1863, the mother took her family to Flensburg, where Möller attended a grammar school, passed his university entrance examination in 1873 and then worked as an apprentice on the building of the Altona–Kiel railway. He began studying at the Berlin Building Academy in October 1874, attended lectures at the Berlin Trade & Industry Academy and passed his site manager examination in Hannover in November 1878. He worked as a site manager on the extension of Kiel naval base and – after passing the government building examination – started working for the City of Hamburg. It was there that he was able to use his knowledge of ports, bridges and warehouses with great success. In 1886 Möller was awarded a prize of 3000 marks for his experiments regarding the behaviour of columns in warehouses. Two years later, he was appointed associate professor for hydraulic engineering at Karlsruhe TH. An appointment as full professor for hydraulic engineering followed at Braunschweig TH in 1890. He remained in this post until being granted emeritus status in April 1925; his successor was Ludwig Leichtweiß (1878-1950), who was instrumental in establishing the scientific reputation of the Brauschweig Hydraulic Engineering Faculty. According to Walter Kertz, the years between 1888 and 1902 were the most productive in Möller’s creative output (Kertz, 1987, p. 11): in the mid-1890s he developed the reinforced concrete system that is named after him and which was employed to build more than 500 bridges after 1920 (see (Droese, 1999) and (Krafczyk, 2016)). Möller produced the first earth pressure tables for the German-speaking countries in 1902, and these proved to be extremely popular in practical foundation work during the first three decades of the 20th century. His monograph on hydraulic engineering followed four years later (Möller, 1906). In 1916 Möller was appointed to the international commission set up to investigate the collapse of Stigberg Quay in Gothenburg (Kurrer, 2018, pp. 341-343). Other important areas of Möller’s work were meteorology, natural philosophy and ether theory (see (Kertz, 1987)). The Civil Engineering Faculty of Dresden TH awarded him an honorary doctorate in 1920 for his services to hydraulic engineering and reinforced concrete construction. On the occasion of his 80th birthday, Braunschweig TH elected Möller as an honorary senator. Main Works: Über die Widerstandsfähigkeit auf Druck beanspruchter eiserner Baukonstruktionstheile bei erhöhter Temperatur (1888); Gurtträger-Decken. System Möller

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(1897); Erddruck-Tabellen mit Erläuterungen über Erddruck und Verankerungen (1902); Grundriß des Wasserbaues: für Studierende und Ingenieure (1906); Wahl des Größenwertes der Elastizitäts-Verhältniszahl n (1913); Erddruck-Tabellen mit Erläuterungen über Erddruck und Verankerungen (1922a); Erddruck-Tabellen. Lieferung 2: Erweiterte Zusammenstellung von Erddruck-Grundwerten mit neueren Erddruck-Untersuchungen (1922b) Further biographical reading: (Thierry, 1934); (Kertz, 1987); (Droese, 1999); (Krafczyk, 2016) Photo courtesy of: (Kertz, 1987) Mörsch, Emil, born 30 Apr 1872, Reutlingen, German Empire, died 29 Dec 1950, Weil im Dorfe near Stuttgart, FRG

Emil Mörsch studied civil engineering at Stuttgart TH from 1890 to 1894. Upon graduating, he worked as a senior civil servant and superintendent in the Ministerial Department for Highways & Waterways and afterwards was employed in the Bridges Department of Württemberg State Railways. He joined the Wayss & Freytag company in Neustadt, Palatinate, in February 1901 and it was here, commissioned by the company, that he published the first edition of his book Der Betoneisenbau. Seine Anwendung und Theorie (reinforced concrete, its application and theory), which later underwent numerous reprints (substantially enlarged) under the title of Der Eisenbeton. Seine Theorie und Anwendung. This book set standards in reinforced concrete writing during the consolidation period of theory of structures (1900-1950). Its theory based on practical trials made it the standard work of reference in reinforced concrete construction for more than half a century. In 1904 Mörsch was appointed professor of theory of structures, bridge-building and reinforced concrete at Zurich ETH. However, four years later, he returned to the board of Wayss & Freytag AG. From 1916 onwards, Mörsch worked as professor of theory of structures, reinforced concrete and masonry arch bridges at Stuttgart TH. The technical revolution in building connected with reinforced concrete during the first two decades of the 20th century led to completely new design forms for foundations as well. One example is the cantilever retaining wall, for which Mörsch had model tests

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carried out to determine the earth pressure (see section 2.6.3) and which he later wrote about (Mörsch, 1925a,b). He adhered rigorously to elastic theory for designing reinforced concrete components right up until his death. Emil Mörsch can be regarded as the founder of the Stuttgart school of structural engineering (Bögle & Kurrer, 2014, p. 831), which enjoys worldwide recognition. Among his numerous awards were honorary membership of the Concrete Institute (now the Institution of Structural Engineers) in 1913 and honorary doctorates from Stuttgart TH (1912) and Zurich ETH (1929). Main Works: Der Betoneisenbau. Seine Anwendung und Theorie (1901); Berechnung von eingespannten Gewölben (1906); Der Eisenbetonbau. Seine Theorie und Anwendung (1920); Die Berechnung der Winkelstützmauern (1925a, 1925b); Der Spannbetonträger. Seine Herstellung, Berechnung und Anwendung (1943); Statik der Gewölbe und Rahmen. (1947); Die Entwicklung des Eisenbetons seit dem Jahre 1900 (1949) Further historical reading: (Graf, 1951); (Bay, 1990); (Knittel, 1994a); (Picon, 1997, p. 318); (Ricken, 2001) Photo courtesy of: Stuttgart University archives Müller-Breslau, Heinrich, born 13 May 1851, Breslau, Prussia (now Wrocław, Poland), died 23 Apr 1925, Berlin-Grunewald, Weimar Republic

Following service in the Franco-Prussian War of 1870-1871, the young Müller, who a few years later was to change his name to Müller-Breslau, left the place of his birth to study at the Berlin Building Academy. However, the birth of a son in December 1872, who was also christened Heinrich (1872-1962), forced him to start earning money. He tutored his fellow students at the Building Academy in theory of structures in readiness for the dreaded state examination set by Schwedler, although he himself did not sit the examination. Müller-Breslau, however, turned duty into a virtue by publishing his theory of structures notes as a book in 1875 and setting himself up as an independent consulting engineer. In October 1883 he was appointed assistant and lecturer at Hannover TH and, in April 1885, professor of civil engineering at the same establishment before succeeding Emil Winkler in the Chair of Theory of Structures and Bridges at

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Berlin TH in October 1888. Taking the theorems of Castigliano and Maxwell’s trussed framework theory as his starting point, Müller-Breslau worked out a consistent theory of statically indeterminate trusses between 1883 and 1889, which culminated in his 𝛿 notation for the force method (Müller-Breslau, 1889, p. 477ff.). He therefore not only completed classical theory of structures (1875-1900), but also concluded the discipline-formation period of theory of structures (1825-1900). During the 1880s the dispute between Mohr and Müller-Breslau over the fundamentals of theory of structures led to the formation of the Dresden school of applied mechanics and the Berlin school of theory of structures, which also gained international recognition. Classical theory of structures was given a valid expression in Müller-Breslau’s Neuere Methoden der Festigkeitslehre (newer methods of strength of materials, five editions) and Graphische Statik der Baukonstruktionen (graphical statics of structures, six editions); both books were translated into several languages. Müller-Breslau’s experimental investigations of the earth pressure on retaining walls (see section 2.6.2) were also immensely important. He published a book about this work (Müller-Breslau, 1906, 1947), which would prolong the dispute between Mohr and Müller-Breslau. It was the earth pressure problem in particular that revealed quite remarkably the different stylistic elements of the Berlin school of theory of structures on one side and the Dresden school of applied mechanics on the other (see Fig. 7-61 in (Kurrer, 2018, p. 526)). Müller-Breslau’s appointment as a full member of the Prussian Academy of Sciences in January 1901 demonstrates the high status accorded to theory of structures and iron bridge-building – indeed, engineering sciences on the whole – by Imperial Germany. Main Works: Elementares Handbuch der Festigkeitslehre mit besonderer Anwendung auf die statische Berechnung der Eisen-Constructionen des Hochbaues (1875); Elemente der graphischen Statik der Bauconstructionen für Architekten und Ingenieure (1881); Über die Anwendung des Princips der Arbeit in der Festigkeitslehre (1883a); Noch ein Wort über das Princip der kleinsten Deformationsarbeit (1883b); Der Satz von der Abgeleiteten der ideellen Formänderungs-Arbeit (1884); Beitrag zur Theorie des Fachwerks (1885); Die neueren Methoden der Festigkeitslehre und der Statik der Baukonstruktionen (1886, 1893, 1904, 1913, 1924); Beiträge zur Theorie der ebenen elastischen Träger (1889); Die Graphische Statik der Baukonstruktionen (1887, 1892, 1901, 1903, 1905, 1907, 1908a, 1912, 1922, 1925, 1927); Beiträge zur Theorie der ebenen elastischen Träger (1889); Erddruck auf Stützmauern (1906, 1947); Bemerkungen über die Berechnung des Erddrucks auf Stützmauern (1908b) Further historical reading: (Hertwig & Reissner, 1912); (Mann, 1921); (Hertwig, 1921, 1951); (Bernhard, 1925); (Müller, 1925); (Pohl, 1925); (anon., 1925); (Hannover TH, 1956); (Hees, 1991); (Picon, 1997, p. 319); (Knittel, 1997) Photo courtesy of: (Hertwig, 1951, p. 53)

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Ohde, Johann, born 20 Nov 1905, Dümmer (Parum), German Empire, died 25 Jul 1953, Dresden, GDR

Johann Ohde was born into an ordinary working-class family, attended an elementary school for eight years, learned carpentry (1920-1923) and then worked as a carpenter. He studied civil engineering at the Hamburg School of Building from 1924 to 1926, graduating with distinction. His first practical experience was gained at Harburg Waterways Department, where he was entrusted with the design and construction of a tidal lock in concrete. During that work he was confronted with complicated foundation tasks, which made him aware of troublesome gaps in earth pressure theory – an area that soon became his field of research. In October 1927 Krey recruited this talented civil engineer for his Earthworks Department at the Prussian Testing Institute for Hydraulics & Shipbuilding. For Ohde that was the start of 16 successful years of learning and researching which, in the end, would make him the leading earth pressure expert in Germany (see section 2.7.3). In the Earthworks Department he was concerned not only with the conception and further development of methods for calculating earth pressure, but also with the perfection of testing equipment and processes. Ohde’s scientific activities were interrupted in May 1943 when he was conscripted into the army. However, just one year later he was able to join the Earthworks Institute at Dresden TH, whose director (1939-1945) since the dissolution of Franz Kögler’s (1882-1939) Freiberg Institute was Walter Bernatzik (1899-1953). Following the devastating bombing of Dresden on the night of 13/14 February 1945, which destroyed the Earthworks Institute totally, Ohde and his family moved to Hamburg, where he founded an office and laboratory for subsoils and earthworks. The new director of the Earthworks Institute was Dresden TH professor Friedrich Wilhelm Neuffer (1882-1960), who joined Ohde’s company as a managing director in November 1946. Ohde had been teaching ‘modern subsoil and earth pressure theory’ at Dresden TH since 1945, which, in 1950, was broadened to cover the subjects of foundation mechanics and building foundations; in 1951 he was appointed professor with teaching assignments. The Berlin authorities chose him to manage the Earthworks Department at the former Prussian Testing Institute for Hydraulics & Shipbuilding in February 1948. While in this post, Ohde worked on the following projects: housebuilding programme for Stalinallee (now Karl-Marx-Allee), the Paretz–Niederneuendorfer

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Canal, the Rappbode dam in the Harz Mountains, the extension to the Port of Wismar, the extension to the River Elbe port at Riesa and the Ohra rockfill dam. After Neuffer was granted emeritus status, Dresden TH tried to recruit Ohde full-time. Although he was still able to influence the layout and equipment concepts for the new institute building, on which work began in June 1953, a severe illness prevented him from doing any further work. At the transition from the discipline-formation period (1925-1950) to the consolidation period (1950-1975) of geotechnical engineering, Ohde’s contributions to earth pressure in general and earth pressure theory in particular represent undisputed milestones. Main Works: Zur Theorie des Erddruckes unter besonderer Berücksichtigung der Erddruckverteilung (1938); Die Berechnung der Sohldruckverteilung unter Gründungskörpern (1942); Zur Erddruck-Lehre (1948-1952); “Boden”-Mechanik? Einige Vorschläge für einheitliche Fachausdrücke in der Baugrundlehre (1949); Grundbaumechanik (1951, 1956); Gleit- und Kippsicherheit von Stützmauern (1952/1953); Die Entwicklung der Forschungsanstalt für Schiffahrt, Gewässerund Bodenkunde seit dem Jahre 1945, ihre Aufgaben und Arbeiten (1953) Further historical reading: (Brendel, 1953); (Lorenz, 1953); (Muhs, 1954); (anon., 1992); (Franke, 1992) Photo courtesy of: (anon., 1992, p. 1) Peck, Ralph Brazelton, born 23 Jun 1912, Winnipeg, Manitoba, Canada, died 18 Feb 2008, Albuquerque, New Mexico, USA

Ralph B. Peck was born to Ethel Huyck Peck and Orwin K. Peck, a bridges engineer for a railway company. Their son’s path was therefore predestined: civil engineering exams in 1934 and PhD at the Rensselaer Polytechnic Institute in Troy, New York, in 1937; afterwards, a post as a detail designer at the American Bridge Company in Ambridge until 1938. In that same year Peck switched to soil mechanics at Havard University and served as a laboratory assistant to Arthur Casagrande. Between 1939 and 1942 he deputised for Terzaghi on the building of the underground railway in Chicago. There followed an interlude as senior engineer in Marion (Ohio). December 1942 saw Peck move to the University of

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Illinois in Urbana, where he advanced to become professor for foundations in 1948. Following his transfer to emeritus status in 1974, he continued to work with great success as a subsoils consultant. Peck supervised 34 doctoral theses and more than 5000 students between 1942 and 1974. His list of publications contains 260 items. Besides his extensive consultancy activities for his country, he also wrote reports for 33 other countries, working on a total of more than 1000 civil engineering projects (e.g. the building of the World Trade Center in New York) (Mesri, 2011, p. 257). Peck was president of the International Society of Soil Mechanics & Foundation Engineering from 1969 to 1973 – just one example of his organisational commitment. This godfather of geotechnical engineering had a crucial impact on its consolidation period (1950-1975) not only through teaching and research, but also in practice. It is therefore not surprising that Peck received many honours, including honorary doctorates from Rensselaer Polytechnic Institute (1974) and the Laval University in Quebec (1987). The American president Gerald Ford (1913-2006) awarded him the National Medal of Science in 1975 “for his development of the science and art of subsurface engineering, combining the contributions of the sciences of geology and soil mechanics with the practical art of foundation design” (cited after (Mesri, 2011, p. 258)). Main Works: Soil Mechanics in engineering practice (1948, 1996); Bodenmechanik in der Baupraxis (1951); Foundation Engineering (1953); Judgement in Geotechnical Engineering – The Professional Legacy of Ralph B. Peck (1984); The last sixty years (1985) Further historical reading: (Peck et al., 1984); (Dunnicliff & Young, 2006); (Mesri, 2011) Photo courtesy of: (Mesri, 2011, p. 254) Poncelet, Jean-Victor, born 1 Jul 1788, Metz, France, died 22 Dec 1867, Paris, France

Poor and thus excluded from good schooling, but far more talented than any of his fellow pupils, Poncelet was quickly top of his class. He was therefore able to attend the Impériale Secondary School in Metz as an external student and enrol at the École Polytechnique in 1807. His tutors there were Ampère, Fourier,

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Lacroix, Legendre, Poinsot and Poisson. Poncelet quickly moved on to the École d’Application de Metz in 1810, but was ordered to leave for fortification works on Walchem Island in February 1812. After that, he took part in Napoleon’s field campaign against Russia and was taken prisoner by the Russians in November 1812. During his years as a prisoner, he extended the Géométrie descriptive of Monge (Monge, 1794/1795) and turned the principles into his famous book on projective geometry (Poncelet, 1822). Following his return to France in September 1814, he worked as an engineering officer on various military engineering projects, including the fortifications in Metz. In 1824 Poncelet was finally appointed a professor at the École d’application de l’Artillerie et du Génie in Metz and taught – with elegance, simplicity and clarity – the fundamentals of an applied mechanics based on machines to the officers attending the famous course Mécanique appliquée aux machines from 1825 to 1834; his lectures were published as lithographed editions (Poncelet, 1826-1832). Furthermore, between 1827 and 1830, he presented the popular evening lectures on applications of geometry and mechanics in industry to workers and industrialists in Metz (Cours de mécanique industrielle), which were published with the title Introduction à la mécanique industrielle (Poncelet, 1829b,c, 1841, 1870). Both Poncelet’s Cours de mécanique appliquée aux machines and his Introduction à la mécanique industrielle can be regarded as the two most important founding documents of applied mechanics. Like Navier’s Résumé des Leçons (Navier, 1826) forms the principal work of the constitution phase of theory of structures (1825-1850), the two works by Poncelet made a vital contribution to the constitution phase of applied mechanics (1825-1850). Based on his work on projective geometry (Poncelet, 1822), he also solved problems in masonry arch theory (Poncelet, 1822) and earth pressure theory. For example, Poncelet’s Mémoire sur la stabilité des revêtements et de leurs fondation (1840), which was translated into German and extended by J. W. Lahmeyer (Poncelet, 1844), contains the graphical determination of the earth pressure acting on retaining walls (see section 2.3.2). Poncelet became a member of Metz City Council, Secrétair du conseil général du département de la Moselle (1830), a member of the Paris Académie des Sciences (1834) and, from 1838 to 1848, was engaged as a professor at the Faculté des Sciences in Paris. His military career is impressive, too: He reached the rank of brigadier-general in 1848 and in that same year was appointed commander of the École Polytechnique, and in this capacity was appointed commander-in-chief of the National Guard of the Seine Département. Poncelet retired at the end of October 1850. The French government sent Poncelet to serve on the juries at the World Expositions in London (1851) and Paris (1855), and he reported on these in detail in books. Rühlmann called Poncelet the “Euler of the 19th century” because, like Euler, Poncelet was a “creator of totally new theories, a promoter of abstracts and empirical sciences ... He was blessed with being able to take part in the most important period of the emergence and development of industry, building and machine mechanics ... Like Euler, though, Poncelet was also an excellent teacher who, with the simplest of presentations and with moderate thoroughness, knew how to captivate his students and make them enthusiastic for science” (Rühlmann, 1885, pp. 387-389).

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Main Works: Traité des propriétés projectives des figures (1822); Cours de mécanique appliquée aux machines (1826-1832); Mémoire sur les centres de moyennes harmoniques; pour faire suite au traité des propriétés projectives des figures et servir d’introduction à la Théorie générale des propriétés projectives des courbes et surfaces géometriques (1828); Mémoire sur la théorie générale des polaires réciproques; pour faire suite au Mémoire sur les centres des moyennes harmoniques (1829a); Analyse des transversales appliquée à la recherche des propriétés projectives des lignes et surfaces géométriques (1832); Introduction à la mécanique industrielle (1829b, 1829c, 1841 and 1870); Solution graphique des principales questions sur la stabilité des voûtes (1835); Mémoire sur la stabilité des revêtements et de leurs fondations (1840); Über die Stabilität der Erdbekleidungen und deren Fundamente (1844); Examen critique et historique des principales théories ou solutions concernant l’équilibre des voûtes (1852) Further historical reading: (Eude, 1902, pp. 241-243); (Chatzis, 1998, 2008) Photo courtesy of: (Rühlmann, 1885, p. 398) Prager, William, born 23 May 1903, Karlsruhe, German Empire, died 16 Mar 1980, Zurich, Switzerland

Prager studied at Darmstadt TH and gained his doctorate there in 1926 under Prof. Wilhelm Schlink. His habilitation thesis at the same university was a dissertation on the theory of structures on elastic supports (1927/1). He started work as a private lecturer at Göttingen University in 1929 and was Ludwig Prandtl’s deputy at the Institute of Applied Mechanics at the same university until 1932. He was not quite 30 years old when he was appointed professor of applied mechanics at Karlsruhe TH. However, this appointment was later revoked by the Baden government to comply with an instruction of 28 March 1933 from the party leaders of the NSDAP regarding the implementation of anti-Semitic measures. Prager initially found a job at the Fieseler Works in Kassel. His former colleague from Göttingen, Kurt Hohenemser, was working there. Together they published the book Dynamik der Stabwerke (dynamics of trusses), which appeared in October

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1933. In the summer of that year, Prager had been invited to take over the Chair of Applied Mathematics and Mechanics at the newly founded Istanbul University, and he took up the post in December 1933. Prager had already arranged two further book projects concerning mathematical instruments and photoelastic methods with Julius Springer jr. (1880-1968) during his time in Göttingen. The first of these remained a plan. But the book on photoelastic methods was to be written together with his former assistant, Gustav Mesmer, and only failed due to the Nazi regime’s discrimination against Jewish authors. Owing to that, Prager felt “so repelled and his pride was so wounded that he refused more and more collaborative projects” (cited after (Sarkowski, 1992, p. 337)). Spannungsoptik (photoelasticity) appeared in August 1939, with Mesmer as the sole author. In his preface, the author thanked “Prof. Dr.-Ing. W. Prager (Istanbul) for the original idea for producing this book” (cited after (Sarkowski, 1992, p. 338)). In Istanbul, Prager was only able to teach and research until 1941. As Hitler’s troops advanced to within 200 km of Istanbul, Prager and his family fled by road via the Middle East and Pakistan to India, from where they boarded a ship and finally reached New York. Prager managed, within a very short time, to establish his world-famous school of applied mathematics and mechanics at Brown University. By 1943 he had founded the journal Quarterly of Applied Mathematics. It was in this journal that Drucker, Prager and Greenberg published their theories that are also fundamental to soil mechanics (see section 3.1.2). He was involved in teaching and research work at Brown University until 1973, except for the years 1963-1968. Prager knew how to present difficult problems with an amazing degree of simplicity – whether in discussions, lectures, presentations, papers or books. Following his transfer to emeritus status, he and his wife settled in Savognin, Switzerland. Prager was the author of extraordinary contributions in the fields of fluid mechanics, elastic theory, plastic theory, dynamics, numerics, transport technology and structural optimisation. This latter field was researched particularly intensively by Prager in his final period of creativity. In terms of theory of structures, it was his work – in cooperation with the group of researchers around J. F. Baker from Cambridge University – on the principles of the ultimate load method that proved crucial, because it initiated the paradigm change from elastic to plastic methods of design worldwide. Prager therefore also had a lasting influence on the style of theoretical treatment in the innovation phase of theory of structures. He was awarded countless honours for his scientific work: honorary doctorates from the universities of Brussels, Hannover, Liège, Manchester, Milan, Stuttgart, etc., membership of scientific academies, prizes and medals too many to mention. Main Works: Beitrag zur Kinematik des Raumfachwerkes (1926); Zur Theorie elastisch gelagerter Konstruktionen (1927a); Die Formänderungen von Raumfachwerken (1927b); Dynamik der Stabwerke. Eine Schwingungslehre für Bauingenieure (1933); Theory of perfectly plastic solids (1951); Soil mechanics and plastic analysis or limit design (1952); Theorie ideal plastischer Körper (1954); Probleme

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der Plastizitätstheorie (1955); An Introduction to Plasticity (1959); Limit analysis: the development of a concept (1974) Further historical reading: (Hopkins, 1980); (Drucker, 1984); (Rozvany, 1989); (Sarkowski, 1992, pp. 336-338); (Rozvany, 2000); (Wauer, 2017, pp. 98-101) Photo courtesy of: (Drucker, 1984, p. 232) Rankine, William John Macquorn, born 5 Jul 1820, Edinburgh, UK, died 24 Dec 1872, Glasgow, UK

William Rankine attended Ayr Academy (1828-1829) and Glasgow High School (1830). Illness forced him to leave the latter and so, for six years, he was taught by his father, David Rankine, a respected railway engineer at the Edinburgh & Dalkeith Railway and later the Caledonian Railway Company. When he was 14, his father gave him a copy of Newton’s Principia. The young Rankine absorbed this work written in Latin and thus laid the foundation for his knowledge of higher mathematics, dynamics and physics. He began studying chemistry, natural history, botany and natural philosophy at Edinburgh University in 1836, but had to interrupt his studies after two years for personal reasons – to assist his father for 12 months in his work for the Edinburgh & Dalkeith Railway. After that, Rankine worked for several years under Sir John MacNeill on the railways and canals of Ireland. He returned to Edinburgh in 1842 and worked for railway companies and consulting engineers and, later, in the London office of Lewis Gordon, who at that time held the Chair of Civil Engineering and Mechanics set up in 1840 at Glasgow University. However, it was not until 1855 that Rankine would succeed him in this position. In his inaugural lecture, The harmony between theory and practice in engineering, “Rankine distinguished between theoretical science, which is concerned with what we are to think, and practical science, where the question is what we are to do, often in situations where scientific theory and existing data can be insufficient” (Sutherland, 1999, p. 183). Rankine was active in both fields, making valuable additions to thermodynamics, elastic theory and hydrodynamics from 1848 onwards. Maxwell even went so far as to say that Rankine was one of the three founding fathers of thermodynamics. On

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the other hand, Rankine pushed back the boundaries of engineering science disciplines decisively, primarily through his Manuals of applied mechanics (Rankine, 1858) and civil engineering (Rankine, 1862), both of which enjoyed numerous editions and became standard works of reference in the process of creating a scientific basis for engineering. In 1872, after a long battle, the authorities agreed to Rankine’s proposal of awarding the academic degree of bachelor of science for the study of engineering at British universities – the first step on the way to raising the status of the engineering sciences in universities. During the discipline-formation period of theory of structures, Rankine came to the fore through his contributions to earth pressure theory (Rankine, 1857) (see section 2.4.2), masonry arch theory (Rankine, 1865) and graphical statics (Rankine, 1858, 1864, 1870). For example, he formulated the equilibrium criterion of kinematically and statically determinate plane frames – Rankine’s theorem (Maxwell, 1864a) – and found, by implication, the reciprocity between trussed framework geometry and the polygon of forces. Rankine did not explore this duality further and published a supplement to his space frames theorem (Rankine, 1864), but without any proof. Rankine also stood out in shipbuilding theory (Rankine, 1866c): The method for calculating the longitudinal strength of a ship, divided into smooth sea and additional wave loads, including the graphic presentation still common today, is attributed to Rankine (Lehmann & Fricke, 2001, p. 290). Rankine had a great effect on the establishment phase of theory of structures (1850–1875) – both in the UK and continental Europe. And his achievements in the creation of scientifically based shipbuilding theory and mechanical engineering remain undisputed. He was therefore in favour of combining the individual engineering sciences. Rankine was elected to the Royal Society in 1853 and was co-founder and first president of the Scottish Institution of Engineers & Shipbuilders in 1857. Interestingly, he was only an associate member of the Institution of Civil Engineers. He composed songs, wrote fables and humorous poems such as The Mathematician in Love. His biographer, H. B. Sutherland (emeritus professor of every chair Rankine held!) paid tribute to him with the following words: “Rankine was a wonderful combination of the man of genius and of humour. How much more pleasant and effective is the contribution, scientific or otherwise, when you know behind it lies a man capable of having a twinkle in his eyes. As Clerk Maxwell wrote, Rankine’s death at the early age of 52 was ‘as great a loss to the diffusion of science as to its advancement’” (Sutherland, 1999, p. 187). Main Works: On the stability of loose earth (1857); A Manual of Applied Mechanics (1858); A Manual of Civil Engineering (1862); Principle of the equilibrium of polyhedral frames (1864); Graphical measurement of elliptical and trochtoidal arcs, and the construction of a circular arc nearly equal to a given straight line (1865); Useful Rules and Tables Relating to Mensuration, Engineering, Structures, and Machines (1866a); Einige graphische Constructionen (1866b); A Manual of

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Applied Mechanics (1872); Diagrams of forces in frameworks (1870); A Manual of Civil Engineering (1872) Further historical reading: (Cook, 1951); (Parkinson, 1981); (Charlton, 1982); (Scholz, 1989); (Sutherland, 1999) Photo courtesy of: (Sutherland, 1999, p. 186) Rebhann, Georg Ritter von Aspernbruck, born 7 Apr 1824, Vienna, Austria, died 29 Aug 1892, Alt-Aussee, Austro-Hungarian Empire

After leaving secondary school, Rebhann studied at the Polytechnic Institute in Vienna, where he was a student of Adam Burg, Johannes Philipp Neumann, Simon Stampfer and Joseph Stummer. His work for the Austrian State Building Authority between 1843 and 1868 was certainly successful and took him from the Imperial-Royal provincial building departement in Lviv to the Building Council at the Interior Ministry in Vienna. Rebhann became interested in theory of structures at an early stage. For example, he published a short article on the graphical determination of earth pressure in Ludwig Förster’s Allgemeine Bauzeitung in 1850; one year later he reviewed Navier’s Mechanik der Baukunst (mechanics of architecture) (Navier, 1833/1851) in the Zeitschrift des Österreichischen Ingenieur- und Architekten-Vereins (Rebhann, 1851). Gießen University awarded him a doctorate in September 1855 for parts of his monograph on the theory of timber and iron structures (1856) – and that was after he had already written a habilitation thesis for structural mechanics at the Polytechnic Institute in Vienna in 1852 and had been giving lectures there on theory of structures in particular since that year. Rebhann’s monograph helped the theoretical treatment of structural analysis to become independent of authorities such as Navier and Redtenbacher. In 1868 he was invited to take up a post of professor for bridges and bridge theory at the Polytechnic Institute in Vienna (Vienna TH after 1872); and 1868 was also the year in which Emil Winkler became professor for railways and structural parts of bridges. Just a few years later, Rebhann published his second monograph on theory of structures (Rebhann, 1870/1871) (see section 2.3.2). Rebhann’s two monographs, which now fell under the heading of ‘higher engineering sciences’, turned him into a leading engineering figure during the establishment phase of theory of structures (1850-1875) – also beyond the borders of the German Empire. This

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becomes clear when we consider that Rebhann introduced structural mechanics as a subject at his Alma Mater – and the way those lectures were adapted creatively by Emil Winkler for theory of structures lectures at Berlin TH after 1877 – and carried out strength tests on Portland cements and loadbearing structure models in the early 1860s. After Winkler’s departure, Rebhann took charge of the subject of bridge-building in 1877 and published a book on normal clearance profiles, load tables and iron sections for use in design tasks for building bridges (Lechner, 1984, p. 3) in 1880. Rebhann served as rector of Vienna TH in 1882/1883. He was awarded many honours, including a peerage (1879) and the title of Privy Counsellor (1888). Main Works: Graphische Bestimmung des Erddrucks an Futtermauern und deren Widerstandsfähigkeit (1850); Theorie der Holz- und Eisen-Construktionen, mit besonderer Rücksicht auf das Bauwesen (1856); Theorie des Erddruckes und der Futtermauern, mit besonderer Rücksicht auf das Bauwesen (1870/1871) Further historical reading: (Paul, 1892); (Lechner, 1984) Photo courtesy of: Vienna TU archives Rendulic, Leo, born 1904, Austro-Hungarian Empire, died 1940, Berlin, German Empire

After graduating successfully in civil engineering from Vienna TH, Renulic was granted a doctorate in 1932 by the same university with a dissertation on a stability problem in structural steelwork (Rendulic, 1932) – work that he continued one year later (Rendulic, 1933). His two years in the building industry steered him towards geotechnical engineering. He was appointed Terzaghi’s assistent at Vienna TH and continued his work at Degebo, the German Society for Soil Mechanics. It was during this period that Rendulic carried out research into pore water pressure (Rendulic, 1936), checked Terzaghi’s concept of the effective stresses using triaxial tests and contributed to consolidation theory (Rendulic, 1937). He submitted his habilitation thesis on soil mechanics to Berlin TH in 1937 and worked as a structural engineer and soils mechanics specialist for a large contractor – initially in Berlin, later Frankfurt a.M. In Berlin he was involved in a refurbishment project for the north-south regional express railway where an excavation had collapsed in August 1935 resulting in 19 fatalities.

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Rendulic’s original contribution to earth pressure theory (see section 2.7.2) dates from this period of creativity. His hope to be appointed as professor for soil mechanics at Berlin TH, which had been offered to Terzaghi first, remained unfulfilled because he died unexpectedly at the age of just 36, leaving his wife and their two-year-old son. A comprehensive paper on earth pressure theory appeared shortly after his death (Rendulic, 1940). In his work he introduced the logarithmic spiral as a slip plane and formulated a general extremal theory of earth pressure that included Coulomb’s theory as a special case. “The theory gives us the chance to determine the relationship between the magnitude of the earth pressure and the position of the point of application. Rendulic was the first person to solve this problem without any errors” (Kézdi, 1962, p. 209). Three years later, Hans Lorenz (1905-1996) would take Rendulic’s theory further (Lorenz, 1943). In his book on earth pressure theories, Kézdi dedicated section 8.6 to Rendulic’s method (Kézdi, 1962, pp. 208-215). Unfortunately, the highly talented Rendulic was a “Nazi through and through, and had joined the NSDAP in Austria very early on” (Passer, 1941, p. 90). Rendulic is another example of how political and scientific intelligence can be worlds apart. Main Works: Über die Stabilität von Stäben, welche aus einem mit Randwinkeln vertärkten Blech bestehen (1932); Stabilität zusammengesetzter Querschnitte bei reiner Druckbeanspruchung (1933); Ein Beitrag zur Bestimmung der Gleitsicherheit (1935); Porenziffer und Porenwasserdruck (1936); Ein Grundgesetz der Tonmechanik und sein experimenteller Beweis (1937); Der Erddruck im Straßenbau und Brückenbau. Eine zusammenfassende Darstellung mit Berücksichtigung der neuesten Forschungsergebnisse (1938); Gleitflächen, Prüfflächen und Erddruck (1940) Further historical reading: (Hertwig, 1940); (Passer, 1941) Photo courtesy of: (Passer, 1941, p. 90) Résal, Jean, born 22 Oct 1854, Besançon, France, died 14 Nov 1919, Paris, France

Owing to his father, Jean Résal was practically preordained for a career in engineering. His father later became the Inspecteur général des Mines and a professor of mechanics at the Ècole Polytechnique and École des Mines.

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The career of the brilliant student of the École des Ponts et Chaussées was always an upward ladder: service in the Roads and Bridges Department at the Loire-Atlantique Département and thereafter in the shipping authority in Paris. Résal succeeded the student of Saint-Venant, Alfred-Aimé Flamant (1839-1915), at the Chair of Strength of Materials at the École des Ponts et Chaussées in 1892. Although Résal had already published a two-volume work on arch bridges together with Ernest Degrand (1822-1892) (Degrand/Résal, 1887, 1888), he concentrated on the theory and practice of steel bridges from a very early time (Résal, 1885, 1889) and had a profound influence on steel bridges at the transition from the discipline-formation (1825-1900) to the consolidation period (1900-1950) of theory of structures. The bold steel arches of the Pont General-de-la-Motte Rouge (1885) in Nantes, Pont Mirabeau (1896) in Paris, Pont de l’Université Lyon (1899) and Pont Alexandre III (1900) and Pont Notre Dame (1914) in Paris set standards for steel bridges. All those bridges listed could only be built as a result of Résal’s research into elasticity and the strength of structural steels, work that he summarised in a monograph (Résal, 1892). Furthermore, Résal made a lasting contribution to earth pressure theory (Résal, 1903, 1910), which Caquot would use successfully as his starting point. Almost all of Résal’s books appeared in the Encyclopédie des Travaux Publics founded by Médéric-Clément Lechalas (1820-1904) in 1884. So Résal’s works helped the encyclopaedic compilation of engineering science knowledge. Main Works: Ponts métalliques, Tome 1 (1885); Ponts en maçonnerie. Tome 1: Stabilité des voûtes (1887); Ponts en maçonnerie, Tome 2: Construction (1888); Ponts métalliques. Tome 2 (1889); Constructions métalliques: élasticité et résistance des matériaux fonte, fer et acier (1892); Emplacements, débouchés, fondations, ponts en maçonnerie (1896); Stabilité des constructions (1901); Poussée des terres. Première partie: Stabilité des murs de soutènement (1903); Poussée des terres. Deuxième partie: Théorie des terres cohérentes. Application Tables numériques (1910) Further biographical reading: (Martin & Londe, 1990); (Coronio, 1997, pp. 157-159); (Marrey, 1997, pp. 55-80) Photo courtesy of: (Coronio, 1997, p. 157) Scheffler, August Christian Wilhelm Hermann, born 10 Oct 1820, Brunswick, Duchy of Brunswick, died 13 Aug 1903, Braunschweig, German Empire

18 Forty selected brief biographies

Hermann Scheffler was the son of an army lieutenant who later became the treasurer of the town of Blankenburg. He attended the local grammar school, leaving in 1837 to begin two years of building studies at the Collegium Carolinum Braunschweig (now Braunschweig TU). After that, Hermann Scheffler held various posts in the building department of the Duchy of Brunswick: assistant, trainee and, finally, inspector (1846). A few years before that, Scheffler achieved excellent results in building examinations, which, in 1851, earned him a post with Duchy of Brunswick Railways, which had been founded in 1837; here, Scheffler progressed to executive officer (1854) and then senior executive officer (1870) for building. His achievements as a railway engineer earned him a reputation beyond the borders of the Duchy of Brunswick, and he received many awards. Scheffler campaigned, above all, for maximum safety in railway operations, introduced centralised points mechanisms in 1867, achieved progress in the building of railways, especially for tunnels and bridges, and, as a sideline, worked as an editor (1858-1863) for the Organ für die Fortschritte des Eisenbahnwesens. He wrote about many problems in theory of structures during the 1850s and early 1860s and thus made a significant contribution to shaping the establishment phase of theory of structures (1850-1875). Most important among these was his book Theorie der Gewölbe, Futtermauern und eisernen Brücken (theory of masonry arches, retaining walls and iron bridges) (Scheffler, 1857a), which contains his extensive observations regarding earth pressure (see section 2.4.1). Scheffler adopted Navier’s bending theory, but especially buckling theory (Scheffler, 1858c), which brought criticism from Franz Grashof. Scheffler’s aggressive spirit also shaped his final period of creativity, in which he wrote many articles on mathematics, physics, insurance, tax, physiological optics, physiology, hydraulics, biology and philosophy. In an article published in 1898, Scheffler disagreed with Charles Darwin’s (1809-1882) theory of evolution and its popularisation by Ernst Haeckel (1809-1882). Scheffler’s encyclopaedic output comprises about 100 items totalling 8000-9000 printed pages (Förster, 1983, p. 76). His attempt to explain the world to his contemporaries by way of principles remained unsuccessful. He himself put forward two reasons for this: “The success [of my writings] is, however, very small compared with the hopes I had … Things that I have regarded as important principles for the knowledge of the world for many decades, still lie buried and unknown today ... The outward obstacles consist partly in the reluctance of scholarly circles to pay any heed to writings coming from outside this circle, and partly in the fact that most of my writings have a mathematical basis …” (cited after (Förster, 1983, p. 76)). Philosophically, Scheffler was following the thinking of Kant, who emphasised the key role of mathematics in the preface to his book Metaphysical Foundations of Natural Science: “I assert, however, that in any special doctrine of nature there can be only as much proper science as there is mathematics therein” (Kant, 1786, preface, trans. M. Friedman). Scheffler was also committed to this principle, and tried to establish the knowledge of the world on the basis of mathematics (Scheffler, 1895, 1896). His love of mathematics was behind this. His father had recognised his mathematical talents at a very early age and actively encouraged and supported this through lessons. Scheffler published remarkable works on

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algebra, geometry, number theory and actuarial mathematics that still remain to be discovered by a historical study. Main Works: Ueber das Princip des kleinsten Widerstandes (1850); Ueber den Druck im Innern einer Erdmasse (1851); Theorie der Gewölbe, Futtermauern und eisernen Brücken (1857a); Über die Vermehrung der Tragfähigkeit der Brückenträger durch angemessene Bestimmung der Höhe und Entfernung der Stützpunkte (1857b); Festigkeits- und Biegungsverhältnisse eines über mehrere Stützpunkte fortlaufenden Trägers (1858a); Continuirliche Brückenträger (1858a); Theorie der Festigkeit gegen das Zerknicken nebst Untersuchungen über verschiedenen inneren Spannungen gebogener Körper und über andere Probleme der Biegungstheorie mit praktischen Anwendungen (1858c); Ueber das Gauss’sche Grundgesetz der Mechanik oder das Princip des kleinsten Zwanges, sowie über ein anderes neues Grundgesetz der Mechanik mit einer Excursion über verschiedene, die mechanischen Principien betreffenden Gegenstände (1858d); Ueber die Tragfähigkeit der Balken mit eingemauertem Ende (1858e); Continuirliche Brückenträger (1860); Ueber Gitter- und Bogenträger und über die Festigkeit der Gefässwände, insbesondere über die Haltbarkeit der Dampfkessel und die Ursachen der Explosionen. Zwei Monographien zur Erweiterung der Biegungs- und Festigkeitstheorie (1862) Further historical reading: (anon., 1863); (anon., 1907), (Foerster, 1983) Photo courtesy of: Braunschweig City archives Skempton, Alec Westley, born 4 Jun 1914, Northampton, UK, died 9 Aug 2001, London, UK

Regarded as the father of soil mechanics in the UK, Skempton finished his studies at Imperial College London in 1935. Influenced by Alfred Pippard, Skempton turned to civil engineering in this period. He joined the Building Research Station as a scientist in 1936. Leonhard Cooling had founded the Soil Mechanics Department there three years before, and, beginning in 1937, Skempton worked in that department. Ten years later, he presented the first seminars on soil mechanics in the UK at Imperial College London. He was finally appointed a professor for soil mechanics at his Alma Mater in 1955 and succeeded Pippard as the head of the Institute of Civil Engineering at Imperial College in 1957 – a post he held

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with great success until 1976. He transferred to emeritus status in 1981. Skempton received numerous honours for his services to geotechnical engineering, one of those being his knighthood in 2000. His scientific work was not confined to soil mechanics; he also played a great role in developing a historical study of construction in the UK. For example, Skempton’s publications on the history of geotechnical engineering formed a foundation for a historical study of that discipline that has yet to be surpassed. Main Works: Alexandre Collin (1808-1890), pioneer in soil mechanics (1945-1947); Earth pressure and the stability of slopes (1946); Alexandre Collin. A note on his pioneer work in soil mechanics (1949); Engineers of the English river navigations, 1620-1760 (1953-1955); William Strutt’s cotton mills, 1793-1812 (1955-1957); The origin of iron beams (1956a); Alexandre Collin and his pioneer work in soil mechanics (1956b); The evolution of steel-frame building (1959); Terzaghi’s discovery of effective stress (1960); The first iron frames (1962); Portland Cements, 1843-1887 (1962-1963); Long-term stability of clay slopes (1964); Alfred John Sutton Pippard 1891-1969 (1970); Telford and the design for a new London Bridge (1979); A Biographical Catalogue of the (Skempton) Collection of Works on Soil Mechanics 1764-1950 (1981a); John Smeaton FRS (1981b); Landmarks in early soil mechanics (1981c); Engineering in the Port of London, 1808-1834 (1981-1982); Selected Papers on Soil Mechanics. (1984); A history of soil properties (1985); British Civil Engineering 1640-1840: a Bibliography of Contemporary Printed Reports, Plans and Books (1987); Historical development of British embankment dams to 1960 (1990); Embankments and cuttings on the early railways (1995); Civil Engineers and Engineering in Britain 1600-1830. (1996); A Biographical Dictionary of Civil Engineers in Great Britain and Ireland, Vol. 1: 1500-1830 (2002) Further historical reading: (Chandler et al., 2001); (Niechcial, 2002); (Burland, 2008); (Llorente, 2015, pp. 89-118) Photo courtesy of: (Niechcial, 2002, p. 159) Sokolovsky, Vadim Vassilyev, born 17 Oct 1912, Kharkiv, Russia (now Ukraine), died 8 Jan 1978, Moscow, USSR

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After completing his studies in civil engineering at the Moscow State University of Civil Engineering in 1933, Sokolovsky became interested in shell theory. However, he soon turned to soil mechanics and would have a great impact on its mathematical foundation. During the 1930s, N. M. Gerssevanov (born in 1879) – driven by the first five-year plans (1928-1937) – made vital contributions to forming and establishing the Soviet school of soil mechanics (Tsytovich, 1981, p. 13). As the father of geotechnical engineering in the USSR, Gerssevanov, on the occasion of his 70th birthday in 1949, summarised the tasks and goals of this field in his speech entitled “New results in the field of foundations and soil mechanics, and perspectives for the future” (see (Goldstein, 1981, p. 32)). This speech is not only Gerssevanov’s scientific testament, but also concluded the discipline-formation period of geotechnical engineering (1925-1950) in the USSR and initiated its consolidation period (1950-1975). The results of this latter period have been reviewed in exemplary fashion in the book Bodenmechanik in der Sowjetunion (soil mechanics in the Soviet Union) edited by Kézdi (Kézdi, 1981). That outlines the history of science sounding board for Sokolovsky’s research. Another proponent of the Soviet school of soil mechanics was N. A. Tsytovich, who, in 1934, wrote the first soil mechanics textbook for the USSR’s technical universities and carried out research into the laws of dispersed media – an object of research to which Sokolovsky dedicated himself with great success. Sokolovsky investigated the theory of the limiting equilibrium condition of bodies of soil as early as the 1940s. He developed a method for the exact solution of Kötter’s differential equation (see section 3.4.1) and introduced the method of characteristics for solving partial hyperbolic differential equations into soil mechanics (see section 2.8.1). Sokolovsky became a corresponding member of the USSR Academy of Sciences in 1946 and a foreign member of the Polish Academy of Sciences in 1959. He turned to investigating the dynamics of viscoplastic and viscoelastic bodies in 1948, and his groundbreaking contributions to this field shaped further scientific development in the theory of materials. Sokolovsky’s monographs on plastic theory and the theory of dispersed media are standard works and were translated into several languages.

Main Works: Theorie der Plastizität (1955); Statics of soil media (1960); Statics of granular media (1965) Further historical reading: (anon., 1979); (Hager, 2009, p. 1444) Photo courtesy of: (Hager, 2009, p. 1444)

18 Forty selected brief biographies

Straub, Hans, born 30 Nov 1895, Berg (Thurgau), Switzerland, died 24 Dec 1962, Winterthur, Switzerland

Hans Straub was the son of a priest and studied civil engineering, later architecture, at Zurich ETH from 1914 to 1919. He toured Italy for the first time in 1920, an undertaking that would determine his professional and spiritual development. After a short time as an assistant, he worked as a civil engineer for a building contractor in Rome. He very soon took responsibility for numerous engineering works, especially in the field of hydraulic engineering. In addition, he became interested in the history of civil engineering. While working in Rome, he discovered the report on the structural analysis of the dome of St. Peter’s compiled by the tre mattematici, Boscovich, Jacquier and Le Seur, in 1743, the purpose of which had been to establish the causes of damage and propose remedial measures. Since that discovery, the year 1743 has marked the birth of modern structural engineering. Straub’s most important work was his book Die Geschichte der Bauingenieurkunst (1949; English: A History of Civil Engineering, 1952), the fourth, revised and expanded, German edition of which was published in 1992 by Peter Zimmermann; it contains numerous passages relevant to the history of earth pressure theory. Main Works: Die Geschichte der Bauingenieurkunst (1949, 1992); Zur Geschichte des Bauingenieurwesens (1960a); Drei bisher unveröffentlichte Karikaturen zur Frühgeschichte der Baustatik (1960b) Further historical reading: (Halász, 1963) Photo courtesy of: Prof. Dr. E. Straub Terzaghi, Karl von, born 2 Oct 1883, Prague, Austro-Hungarian Empire (now Czech Republic), died 25 Oct 1963, Wincester, Massachusetts, USA

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Karl von Terzaghi was, without doubt, one of the outstanding civil engineering personalities of the 20th century. Regarded as the founder of soil mechanics in the discipline-formation period of geotechnical engineering (1925-1950), he had a lasting influence on the development of soil mechanics throughout the consolidation period (1950-1975) of this scientific discipline. By 1960 his published works numbered 256 and included numerous pioneering works that shaped geotechnical engineering in the 20th century. After his father, a lieutenant-colonel, retired, Terzaghi grew up in Graz. From the age of 10, he attended various grammar schools with a military orientation in Hungary, Moravia, Vienna and Graz. Following his schooling, he studied mechanical engineering at Graz TH, then switched to civil engineering and graduated in 1904. Terzaghi’s teachers included Philipp Forchheimer (1852-1933), professor of hydraulic engineering, and Ferdinand Wittenbauer (1857-1922), professor of mechanics, with whom he maintained contact in later years. Following military service, Terzaghi joined the Vienna-based contractor Adolph von Pittel in 1905, a company that was spreading reinforced concrete construction across Austria and had successfully tapped the new market for hydroelectric power. Terzaghi designed reinforced concrete structures and investigated foundation problems while working for this company. That work led to his 1911 dissertation on vessel theory – supervised by Forchheimer and bridge building professor Joseph Cecerle (Cecerle and Forchheimer, 1912) – which earned him a doctorate at Graz TH; working with Theodor Pöschl (1882-1955), the dissertation was published two years later (Pöschl & Terzaghi, 1913). Terzaghi visited several dam-building projects in the USA in 1912. During the First World War he served as a lieutenant with Richard von Mises (1883-1953) and Theodore von Kármán (1881-1863) on the extension to Aspern military aerodrome near Vienna. Forchheimer, appointed to reorganise the Imperial Ottoman Technical University in Constantinople (now Istanbul Technical University, ITU), remembered his talented student and recruited Terzaghi in 1916 as professor for foundations and roadbuilding. Terzaghi set up an earthworks laboratory and, following the defeat of the Central Powers, was able to continue his soil mechanics research at Robert College (now Bo˘gaziçi Üniversitesi) in Istanbul until 1925. He summarised his research findings from his Istanbul years in a monograph Erdbaumechanik auf bodenphysikalischer Grundlage (mechanics of soil in construction) (Terzaghi, 1925) (see section 2.6.4), which earned him a professorship at MIT in Boston, USA, in that same year. While at MIT, Terzaghi continued to work intensely on the constitution of the new civil engineering discipline of soil mechanics. It was during this period of creativity that Terzaghi carried out his experimental work on earth pressure on retaining walls for different kinematics – work that would later be included in his earth pressure theory (see section 2.7.1). Terzaghi served as a professor at Vienna TH from 1929 to 1938, managing to establish his international school of soil mechanics at that establishment. That work led to him initiating the founding of the International Society for Soil Mechanics & Foundation Engineering in 1936 – the first milestone on the road to the scientific organisation of this discipline, with the first conference being held that same year in Cambridge (Mass.). The middle of the discipline-formation period of geotechnical engineering (1925-1936) had thus been reached.

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Nevertheless, there were also critics who fought against Terzaghi in particular and soil mechanics in general. For example, Paul Fillunger (1883-1937), professor for applied mechanics at Vienna TH, who described soil mechanics as “a remarkable example of how a revolutionary theory enters the engineering sciences even though their advocates themselves have to admit every step of the way that nothing is yet complete and only the future will reveal the extraordinary benefits of their new theory” (Fillunger, 1936, pp. 42-43). Fillunger’s criticism had been triggered by the scientific controversy surrounding the book Theorie der Setzung von Tonschichten (theory of settlement of clay strata) (Terzaghi & Fröhlich, 1936) regarding the consolidation of deformable porous soils. This dispute ended tragically when Fillunger and his wife took their own lives on the night of 6/7 March 1937 (see (Boer, 1990, 2005)). In 1938 Terzaghi handed over a certificate to US President Herbert Hoover (1874-1964) awarding him an honorary doctorate from Vienna TH (Casagrande, 1960, p. 10). The attempt by the National Socialists to get Terzaghi to remain in Hitler’s Germany failed. Not until 1940 was a successor found for Terzaghi’s Chair of Foundations and Soil Mechanics at Vienna TH in the shape of Otto Karl Fröhlich (1885-1964). Terzaghi accepted a teaching contract at Havard University in the autumn of 1938 and finally settled in Winchester (Mass.) with his family in June 1939. He lectured in engineering geology and applied soil mechanics at Harvard University and was granted American citizenship in 1943. Terzaghi was active worldwide as a consultant: the Chicago underground railway, port facilities at Newport News in Virginia, dam-building projects for hydroelectric power plants in British Columbia, the Asswan dam, etc. His 1948 monograph written together with Ralph B. Peck, Soil Mechanics in Engineering Practice, was also published in German (Terzaghi & Peck, 1951), Serbo-Croatian (1951), Spanish (1955), Japanese (1955) and French (1957) editions. The throng of Terzaghi’s students includes many prominent members: Walter Bernatzik (1899-1953), Mikael Juul Hvorslev (1895-1989), Wilhelm Steinbrenner, Hubert Borowicka (1910-1999), Richard Jelinek (1914-2010), Leo Rendulic (1904-1940), Leo Casagrande (1903-1990), Wilhelm Loos (1890-1952), Christian Veder (1907-1984), Peter Siedek, Gregory Tschebotarioff (1899-1985), Arthur Casagrande (1902-1981), Ralph B. Peck (1912-2008) and Leonardo Zeevaert (1914-2010). Terzaghi’s honours were numerous. For example, nine universities awarded him an honorary doctorate, including Istanbul TU (1949), Berlin TU (1958) and Graz TH (1963). The American Society of Civil Engineers (ASCE) awarded him the Norman Medal in 1930, 1943, 1946 and 1955. Main Works: Berechnung von Behältern nach neueren analytischen und graphischen Methoden (1913); Old Earth-Pressure Theories and New Test Results (1920); Erdbaumechanik auf bodenphysikalischer Grundlage (1925); Die Ursachen der Schiefstellung des Turmes von Pisa (1934); Fünfzehn Jahre Baugrundforschung (1935); Distribution of the lateral pressure of sand on the timbering of cuts (1936); Theorie der Setzung von Tonschichten (1936); Theoretical Soil Mechanics (1943); Soil Mechanics in engineering practice (1948, 1996); Bodenmechanik in der

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Baupraxis (1951); Theoretische Bodenmechanik (1954); From theory to practice in soil mechanics. Selections from the writings of Karl Terzaghi (1960) Further historical reading: (Sattler, 1958); (Casagrande, 1960); (Bjerrum, 1960); (Brandl, 1983); (Goodman, 1999, p. 173) Photo courtesy of: (Goodman, 1999, p. 173) Vauban, Sébastien Le Pestre de, born 1 or 4 May 1633, Saint-Léger-de-Foucheret (now Saint-Léger-Vauban), France, died 30 Mar 1707, Paris, France

Vauban may be regarded as the progenitor of the modern civil engineer shortly before military and civil engineering were divorced from each other in a process that lasted throughout the 18th century. Vauban came from a simple Burgundian landowning family. Even as a schoolboy, he showed his mathematical talents at the Carmelite school in Semur-en-Auxois. At the age of 18, Vauban joined the regiment of Louis II de Bourbon, Prince de Condé (1621-1686), cousin to Louis XIV (1638-1715), as a cadet, and it was there that he gained his first experience in the building of fortifications. Prince Condé belonged to the aristocratic opposition to the absolutist central power. Following his imprisonment, Vauban was convinced by Cardinal Jules Mazarin (1602-1661) to switch to the Royal Army. It was there that he climbed the career ladder at a breathtaking pace: Assistant to Marshall Louis Nicolas de Clerville (1610-1677) in 1654, appointment as ‘Ingénieur ordinaire du roi’ in 1655, company commander under Marshall De la Ferté (1599-1681) in 1656, company commander in the Régiment de Picardie in 1665, appointment as Maréchal de camp in 1676, ‘Commissaire général des fortifications’ in 1678, lieutenant-general in 1688 and, finally, Maréchal de France in 1703. This list tells us nothing about Vauban’s participation in more than 100 battles and more than 50 sieges, and the construction and reconstruction of numerous fortifications, ports and canals. One of the most prominent of those was the building of the fortifications and port at Dunkirk, which began in 1668. According to Straub, the lock that closes off the inner harbour is “a masterpiece of the art of engineering” (Straub, 1992, p. 163). Apart from planning these works, Vauban also supervised the construction work personally.

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Vauban’s technical creativity is a blueprint for engineering methodology and clarity, is permeated by a spirit of scientific rationality fused with Jean-Baptiste Colbert’s (1619-1683) mercantilism to create a productive entity. This is shown not only by Vauban’s design rules for retaining walls (see section 2.1), but also by his specifications that contain exact provisions regarding how the individual building measures are carried out, the origin and quality of the building materials, the loads, the responsibilities of the companies, the site records and the accounts (see (Straub, 1992, p. 188)). It is not exaggerating to speak of Vauban as an infrastructure engineer on a national scale who played a major role in creating the general working resources of France and thus affected a wider audience, i.e. reached other European countries as well. But it is not right to reduce Vauban’s personality to this. Owing to his many journeys (he called himself “le vagabond du roi”), he became aware of and understood the suffering of his people, and incorporated his observations in his actions. For example, he wrote the following in a memorandum to the Duke of Burgundy: “The people are all born as simple citizens. It is only their actions that ennoble them” (cited after (Göggel, 2011, p. 134)). Vauban lived for this principle. His actions allowed him to shine because he embodied the performance principle in social responsibility, cared about the state and the “happiness of the people”. In his Memorandum on the repatriation of the Huguenots (1689), the catholic Vauban therefore warned Minister of War Louvois (1641-1691) of the terrible consequences of the revocation of the Edict of Nantes (1685) by Louis XIV, which not only disenfranchised the Huguenots, but might also cause France enormous damage. “The blood of the martyrs of all religions,” wrote Vauban in his memorandum, “was always very fertile and an infallible means of making those who are pursued stronger and greater” (cited after (Göggel, 2011, p. 142)). Vauban would be proved right. In early 1707 he published, anonymously, his Projet d’une dixme royale (project for a royal tithe) and distributed this among various friends and acquaintances with political interests. He called for a royal tithe to relieve the people from the oppressive taxes and rob “200 000 thieves” of the opportunity to continue to make themselves rich. He fell out of favour with Louis XIV and his publication was added to the blacklist on 14 February 1707. The folly of the ruling classes! Vauban died a few weeks later. Main Works: De l’attaque et de la defense des places (1737); Der Angriff und die Vertheidigung der Festungen (1744/1745); Oisivetés de M. de Vauban (1842/1843); Projet d’une Dixme Royale (1933) Further biographical reading: (Virol, 2003); (Prost, 2007); (Göggel, 2011); (Jordan, 2011); (Petzsch, 2011) Photo courtesy of: (Vauban, 1744, frontispiece) Vitruvius, lived in 1st century BCE, mainly in Rome

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Everything we know about the life of Vitruvius stems from his work De architectura libri decem (Ten Books on Architecture) – called De architectura (On Architecture) for short. This is the most complete publication on architecture and engineering in ancient Rome. It is unique in terms of its scope, systematic form and detail. Nevertheless, it follows in the footsteps of the technical literature of Aristotle, Archimedes, Hero of Alexandria and Apollodorus of Athens, containing passages on architecture and urban planning, land surveying, mechanics (in the sense of describing simple machines), astrology, clockmaking, water organs and hydraulic apparatuses, catapults, siege machines and fortifications. These were things that, according to Vitruvius, the “inventive spirit of the master-builder” had to deal with. De architectura contains very little on construction innovations in ancient Rome. Instead, it exudes the spirit of acquiring Greek-Hellenistic learning through the prevailing classes during the reign of Augustus (27 BCE to 14 CE). De architectura was written in Latin and translated into numerous languages, and in the 15th to 19th centuries it was one of the most important textbooks on architecture. His book “was of fundamental significance for the creation of a European theory of architecture during the Renaissance” (Schmidt, 2015, p. 298). This does not apply to modern civil and structural engineering, as the scientific self-image of the modern civil or structural engineer is a product of the Enlightenment of the 18th century. Nevertheless, De architectura is indispensable for understanding the development processes in civil engineering in general and the foundations of the ancient world (see section 2.2) in particular. For example, Hans Straub (1895-1962) refers to De architectura at 12 places in his monograph on civil engineering from ancient times to the modern age (Straub, 1992, pp. 17f., 26f., 35, 47, 52, 54-56, 58-60, 62, 94, 137, 147f. & 169); but Straub can say only very little about the life of Vitruvius. He specifies the year of his birth as 88 BCE, the year of his death as 10 BCE, but both dates carry a question mark (Straub, 1992, p. 414). It is also not possible to state the place of his birth with any certainty. We do not find much about the life of Vitruvius in Curt Fensterbusch’s (1888-1978) annotated translation of De architectura, either (Fensterbusch, 1981, pp. 1-3). According to Fensterbusch, Vitruvius received a sound education as architect and engineer from his parents. He joined the army at an early age, served on Caesar’s staff and was in charge of building war machines. Following the assassination of Julius Caesar in 44 BCE, he served Emperor Augustus (27 BCE to 14 CE) in the same way. After that, Vitruvius worked under Marcus Vipsanius Agrippa on the construction of systems for supplying water to private buildings – that was in 33 BCE. Vitruvius must have retired around about this time. Octavia, the sister of Augustus, interceded so that Vitruvius would receive payments to ensure that he would have no financial worries. He was therefore able to devote himself to the study of architecture. He must have started his De architectura prior to 33 BCE and finished it at the earliest in 14 BCE. Vitruvius never attempted to secure any building contracts and was not particularly well known as an architect during his lifetime. It seems almost certain that he was responsible for one structure only – the basilica in Fano. “His main activity as an architect seemed to be the building of artillery and supervising the construction of Roman water supply systems” (Fensterbusch, 1981, pp. 2-3).

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Writing about himself, Vitruvius said that material wealth was of little importance to him – modest possessions and a good reputation were far more important. Vitruvius hoped that his book would remain for posterity. He certainly achieved that. A systematic review of De architectura from the point of view of the history of construction has yet to be written. Main Works: Zehn Bücher über Architektur (Ten Books on Architecure) (1981) Further biographical reading: (Schmidt, 2015); (Haselberger, 2015) Walz, Bernhard Nikolaus, born 27 Jul 1939, Berlin, German Empire, died 26 Aug 2009, Wuppertal, Germany

After attending a classical grammar school, Walz, who came from a family of doctors, studied civil engineering at Berlin TU from 1959 to 1965, specialising in structural engineering. That was followed by a period working as a scientific assistant and later as a senior engineer at the Institute of Foundations & Soil Mechanics at Berlin TU, where Hans Lorenz (1905-1997) was in charge from 1947 to 1972. Walz was appointed a professor at Berlin TU in 1971 and gained his doctorate there in 1975 with a dissertation on the three-dimensional earth pressure acting on a circular caisson. Solving three-dimensional earth pressure problems represented one priority in his scientific work. In 1978 Wuppertal University invited him to take over the newly created Chair of Subsurface Structures, Foundations & Soil Mechanics, which Walz helped to achieve great esteem. He therefore turned down a request, in 1983, by Berlin TU to succeed Hanno Müller-Kirchenbauer (1934-2004). Besides investigating three-dimensional earth pressure issues, the new laboratory for scientific investigations that Walz set up in Wuppertal examined the theory behind and use of slurries for support in special foundations and tunnelling as well as the loadbearing behaviour of single piles and groups of piles in tension. His outstanding commitment to university teaching was rewarded when he was elected to the post of prorector for seminars and studies at Wuppertal University in 1991. Walz held this position very successfully for four years, developing it and proving very

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productive in the shaping of courses and seminars for civil engineering as head of the North Rhine-Westphalia Commission. Walz acted as supervisor or co-supervisor for more than 50 engineering doctorate dissertations and his list of scientific publications on construction numbers almost 100. For example, in 1999, working with Johannes Gerlach and Matthias Pulsfort, he published the 18th edition of the second volume of Baugruben und Gründungen (excavations and foundations) of the standard work Grundbau (foundations). Less than a month after celebrating his 70th birthday, Walz died after a severe illness surrounded by his family. His death meant that geotechnical engineering in Germany had lost a distinguished representative of his science with a practical agenda. Main Works: Größe und Verteilung des Erddrucks auf einen runden Senkkasten (1975); Lösung von Erddruckproblemen nach der Elementscheibenmethode (1979); Berechnungsverfahren zum aktiven räumlichen Erddruck (1979); Experimentelle und theoretische Untersuchungen zum Erddruckproblem auf radialsymmetrische Senkkästen und Schächte (1980); Berechnungen des räumlichen aktiven Erddrucks mit der modifizierten Elementscheibentheorie (1987); Räumlicher Erddruck auf Senkkästen und Schächte – Darstellung eines einfachen Rechenansatzes (1988); Erddruckabminderung an einspringenden Baugrubenecken (1994); Grundbau. Band 2: Baugruben und Gründungen (1999); Ermittlung von Erddruckverteilungen aus Dehnungsmessungen mit der inversen FE-Methode (iFEM) (2006) Further historical reading: (Pulsfort, 1999); (Pulsfort, 2004); (Pulsfort & Savidis, 2009) Photo courtesy of: (Pulsfort & Savidis, 2009, p. 817) Winkler, Emil, born 18 Apr 1835, Falkenberg near Torgau, Saxony, died 27 Aug 1888, Friedenau near Berlin, German Empire

Following his apprenticeship as a bricklayer, Emil Winkler attended the Building Trades School in Holzminden and then studied at Dresden Polytechnic (Dresden TH) from 1854 to 1858. He then became an assistant engineer in the Saxony

References

Waterways Department. He gained his doctorate in 1861 at the Faculty of Philosophy at Leipzig University with a dissertation on the pressure inside bodies of soil. He was employed as an assistant at Dresden Polytechnic from 1861 to 1865, where, from 1863 onwards, he taught design of engineering works alongside Prof. Johann Andreas Schubert. He was appointed professor of civil engineering at Prague Polytechnic (Prague TH) in the autumn of 1865 and, three years later, full professor of railway engineering and the structural parts of bridge-building at the Polytechnic Institute in Vienna (Vienna TH). November 1877 saw him accept an appointment as professor of theory of structures and bridges at the Berlin Building Academy (Berlin-Charlottenburg TH after 1879). He remained active in this post until his death following a stroke. Shortly before died, he was awarded an honorary doctorate by Bologna University. Winkler contributed pioneering work on bridge-building, theory of structures (see (Kurrer, 2018, pp. 465-476)) and earth pressure theory (see section 2.4.2). A description of these at this point would exceed the remit of a brief biography. Main Works: Über den Druck im Inneren von Erdmassen (1861); Beiträge zur Theorie der continuirlichen Brückenträger (1862); Die Lehre von der Elasticität und Festigkeit (1867); Vortrag über die Berechnung der Bogenbrücken (1868/1869); Neue Theorie des Erddruckes (1871a); Versuche über den Erddruck (1871b); Bemerkungen zum ‘Beitrag zur Theorie des Erddrucks vom Baurath Mohr’ (1871c); Neue Theorie des Erddruckes nebst einer Geschichte der Theorie des Erddruckes und der hierüber angestellten Versuche (1872); Vorträge über Brückenbau (1872-1886); Die Lage der Stützlinie im Gewölbe (1879/1880). Vorträge über Statik der Baukonstruktionen, I. Heft: Festigkeit gerader Stäbe. Teil I (1883); Über die Belastungs-Gleichwerthe der Brückenträger (1884); Ueber Erddruck auf gebrochene und gekrümmte Wandflächen (1885); Vorträge über Brückenbau. 1. Heft. Theorie der Brücken. Äußere Kräfte der Balkenträger (1886) Further historical reading: (Melan, 1888); (anon., 1888a); (anon., 1888b); (anon., 1888c); (Kurrer, 1988); (Knothe & Tausendfreund, 2000); (Knothe, 2004) Photo courtesy of: (Stark, 1906)

References anon. (1863). Scheffler, August Christian Wilhelm Herrmann. In: J. C. Poggendorf. Biographisch-Literarisches Handwörterbuch der exakten Naturwissenschaften, Zweiter Band: M-Z, pp. 782. Leipzig: Verlag von Johann Ambrosius Barth. anon. (1888a). Professor Emil Winkler (obit.). Zentralblatt der Bauverwaltung 8: 387–388. anon. (1888b). Professor Emil Winkler (obit.). Deutsche Bauzeitung 22: 434–436. anon. (1888c). Professor Dr. Emil Winkler (obit.). Stahl und Eisen 10: 717–718.

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anon. (1907). Scheffler, Hermann. In: Meyers Großes Konversations-Lexikon, Vol. 17, 6th ed., 721. Leipzig/Vienna: Bibliographisches Institut. anon. (1912a). Professor Fritz Kötter (obit.). Zeitschrift des Vereines Deutscher Ingenieure, Vol. 56: 1422. anon. (1912b). Fritz Kötter (obit.). Die Bauwelt 3 (35): 23. anon. (1925). Heinrich-Müller Breslau (obit.). Zeitschrift für Angewandte Mathematik und Mechanik 5 (3): 277–278. anon. (U.S.S.R. National Committee on Theoretical and Applied Mechanics, Academy of Sciences of the U.S.S.R., Moscow) (1979). Professor Vadim Vasilyevich Sokolovsky. Journal of the Mechanics and Physics of Solids, Vol. 27, Issue 2, pp. 175-176. anon. (1992). Professor Ohde. In: Mitteilungen des Instituts für Geotechnik der TU Dresden, No. 1., pp. 1-5. Dresden: TU Dresden. Baker, B. (1867). Long-Span Railway Bridges. London: E. & F.N. Spon. Baker, B. (1880a). The River Nile. In: Minutes and Proceedings of the Institution of Civil Engineers, Vol. 60, 367. Baker, B. (1880b). Shipment and erection of Cleopatra’s Needle (Contribution to discussion). In: Minutes and Proceedings of the Institution of Civil Engineers, vol. 91, 233. Baker, B. (1881). The Actual Lateral Pressure of Earthwork. In: Minutes and Proceedings of the Institution of Civil Engineers, Vol. 65, pp. 140-186. Correspondence on Pressure of Earthwork, pp. 209-241. Baker, B. (1885). The Metropolitan and District railways. In: Minutes and Proceedings of the Institution of Civil Engineers, Vol. 81, 1–33. Baker, B. (1887). Bridging the Firth of Forth. Engineering, p. 116, p. 148, pp. 170-171, p. 210 and p. 238. Bay, H. (1990). Emil Mörsch. Erinnerungen an einen großen Lehrmeister des Stahlbetonbaus. In: Wegbereiter der Bautechnik, ed. by VDI-Gesellschaft Bautechnik, pp. 47-66. Düsseldorf: VDI-Verlag 1990. Becchi, A. (1998). M. Lévy versus J. de La Gournerie: a debate about skew bridges. In: Proceedings of the Second International Arch Bridge Conference (ed. A. Sinopoli), 65–72. Rotterdam: Balkema. Bélidor, B.F. de (1729). La science des ingénieurs dans la conduite des travaux de fortification et d’architecture civile. Paris: C.-A. Jombert. Bélidor, B.F. de (1737-1753). Architecture hydraulique, ou l’art de conduire, d’élever et de ménager les eaux pour les différens besoins de la vie. Paris: C.-A. Jombert. Bélidor, B. F. de, (1729/1757). Ingenieur-Wissenschaft bey aufzuführenden Vestungswerken und bürgerlichen Gebäuden, 1. Teil; trans. from the French. Nürnberg: Christoph Weigels. Bélidor, B.F. de (1813). La Science des ingénieurs dans la conduite des travaux de fortification et d’architecture civile, par Bélidor. Nouvelle édition avec des notes par M. Navier. Paris: F. Didot. Bélidor, B.F. de (1819). Architecture hydraulique, ou l’art de conduire, d’élever et de ménager les eaux pour les différens besoins de la vie. Nouvelle édition avec des notes par M. Navier. Paris: F. Didot. Bernhard, K. (1925). Müller-Breslau (obit.). Die Bautechnik 3 (20): 261–262.

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Bjerrum, L. (1960). Some notes on Terzaghi’s method of working. In: From Theory to Practice in Soil Mechanics, ed. by L. Bjerrum, A. Casagrande, R. B. Peck, A. W. Skempton, pp. 22-25. New York: Wiley & Sons. Bjerrum, L. (1969). Jörgen Brinch Hansen. Géotechnique 19: 436–438. Bögle, A. and Kurrer, K.-E. (2014). Das strukturale Komponieren von Tragwerken bei Jörg Schlaich. Beton- und Stahlbetonbau 109 (11): 829–837. Boer, R.d. (1990). Wiener Beitrag zur Theorie poröser Medien und zur theoretischen Bodenmechanik. Österreichische Ingenieur- und Architekten-Zeitschrift (ÖIAZ) 135 (10): 546–554. Boer, R.d. (2005). The Engineer and the Scandal. A Piece of Science History. Berlin/Heidelberg: Springer-Verlag. Bois, P.-A. (2007). Joseph Boussinesq (1842-1929): a pioneer of mechanical modelling at the end of the 19th Century. Comptes Rendus Mécanique 335: 479–495. Bourru de Lamotte, M. (2005). Hommage à Jean Lehuerou Kérisel (1908-2005). Revue Française de Géotechnique, No. 110, 1er trimestre, pp. 3-7. Bourru de Lamotte, M. and Gambin, M. (2005). Jean Kérisel. Géotechnique 55 (10): 765–767. Boussinesq, J.V. (1876). Essai théorique sur l‘équilibre d’élasticité des massifs pulvérulents, compare à celui de massifs solides et sur la poussée des terres sans cohesion. Brüssel: Imprimerie F. Hayez. Boussinesq, J. V. (1882). Sur la determination de l’épaisseur minimum que doit avoir un mur vertical, d’une hauteur et d’une densité données, pour contenir un massif terreux, sans cohésion, dont la surface supérieure est horizontale. Annales des ponts et chaussées, 6e série, Tomé 3, No. 29, pp. 625-643. Boussinesq, J.V. (1885). Application des potentiels à l’étude de l’équilibre et du mouvement des solides élastiques. Paris: Ed. Gautier-Villars. Boussinesq, J.V. and Flamant, A. (1886). Notice sur la vie et les travaux de M. de Saint-Venant. Annales des ponts et chaussées 12: 557–595. Brandl, H. (1983). 100 Jahre Karl v. Terzaghi. In: Mitteilungen für Grundbau, Bodenmechanik und Felsbau, Technische Universität Wien, No. 2, pp. 11-41. Wien: Eigenverlag der TU. Brendel, G. (1953). Prof. Ing. Johann Ohde (obit.). Bauplanung und Bautechnik 7 (10): 482. Brennecke, L. (1887). Der Grundbau. Berlin: Kommissions-Verlag von E. Toeche. Brennecke, L. (1895). Ergänzungen zum Grundbau. Berlin: Kommissions-Verlag von E. Toeche. Brennecke, L. (1896). Ueber Erddruck und Stützmauern. Centralblatt der Bauverwaltung, Vol. 16, pp. 178 and pp. 354-355. Brennecke, L. (1904). Die Schiffsschleusen. In: Der Wasserbau. III. Teil des Handbuchs der Ingenieurwissenschaften, ed. by F. Bubendey. 4th enlarged ed. Leipzig: Verlag von Wilhelm Engelmann. Brennecke, L. (1906). Der Grundbau, 3rd enlarged ed. Berlin: Verlag Deutsche Bauzeitung. Brennecke, L. (1927). Der Grundbau. 1, Baugrund, Baustoffe, Pfähle und Spundwände, Baugrube, 4th rev. ed. by E. Lohmeyer. Berlin: Wilhelm Ernst & Sohn.

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Brennecke, L. (1930). Der Grundbau. 2, Pfahlrostgründung (Bohlwerke, tiefer und hoher Pfahlrost), 4th rev. ed. by E. Lohmeyer. Berlin: Wilhelm Ernst & Sohn. Brennecke, L. (1942). Der Grundbau. 3, Die einzelnen Gründungsarten mit Ausnahme der Pfahlrostgründung, 4th rev. ed. by E. Lohmeyer. Berlin: Wilhelm Ernst & Sohn. Burland, J.B. (2008). The founders of Géotechnique. Géotechnique 58 (5): 327–341. Caquot, A., 1930. Idées actuelles sur la résistance des matériaux. Génie Civil, Numéro Spécial publié al’recasion du Cinquantenaire de la Fondation du ‘Génie Civile’, Nov., pp. 189-192. Caquot, A. (1934). Equilibre des massifs à frottement interne. Stabilité des terres pulvérulentes et cohérentes. Paris: Gauthiers-Villard. Caquot, A. and Kérisel, J. (1948). Tables des poussée et butée et de force portante des fondations. Paris: Gauthiers-Villard. Caquot, A. and Kérisel, J. (1949). Traité de mécanique des sols. Paris: Gauthiers-Villard. Caquot, A. and Kérisel, J. (1956). Traité de mécanique des sols, 2nd ed. Paris: Gauthiers-Villard. Caquot, A. and Kérisel, J. (1966). Traité de mécanique des sols, 3rd ed. Paris: Gauthiers-Villard. Caquot, A. and Kérisel, J. (1967). Grundlagen der Bodenmechanik. Trans. from the French by G. Scheuch. Berlin: Springer-Verlag. Casagrande, A. (1960). Karl Terzaghi – his life and achievements. In: From Theory to Practice in Soil Mechanics, ed. by L. Bjerrum, A. Casagrande, R. B. Peck, A. W. Skempton, pp. 1-21. New York: Wiley & Sons. Cecerle, J. and Forchheimer, P. (1912). Gutachten zur Dissertation Karl von Terzaghi. Nachlass Karl von Terzaghi. Graz: Archiv of the TU Graz. Chandler, R.J., Chrimes, M.M., Burland, J.B. and Vaughan, P.R. (2001). Alec Westley Skempton 1914-2001. Géotechnique 51 (10): 829–834. Charlton, T.M. (1982). A history of theory of structures in the nineteenth century. Cambridge: Cambridge University Press. Chatzis, K. (1998). Jean-Victor Poncelet (1788-1867) ou le Newton de la mécanique appliquée. Bulletin de la Sabix (19): 69–97. Chatzis, K. (2008). Les cours de mécanique appliquée de Jean-Victor Poncelet à l’École de l’Artillerie et du Génie et à la Sorbonne, 1825-1848. Histoire de l’éducation 9 (120): 113–138. Considère, A. (1870). Note sur la poussée des terres. Annales des ponts et chaussées, 4e série, 10e année (256): 547–594. Considère, A. (1899). Influence des armatures métalliques sur les propriétés des mortiers et bétons. Le Génie Civil, 19e année, tome 34, No 14-17, pp. 213-216, 229-233, 244-247 and 260-263. Considère, A. (1902a). Ètude théorique de la résistance à la compression du béton fretté. Comptes rendus des séances de l’académie des sciences, tome 135, Paris, 25.8.1902, pp. 365-368. Considère, A. (1902b). Ètude expérimentale de la résistance à la compression du béton fretté. Comptes rendus des séances de l’académie des sciences, tome 135, Paris, 8.9.1902, pp. 415-419.

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Considère, A. (1902c). Résistance à la compression du béton fretté. Le Génie Civil, 23e année, tome 17, No 1, pp. 5-7, 20-34, 38-40, 58-60, 72-74 and 82-86. Considère, A. (1904). Influence des pressions latérales sur la résistance des solides à l’écrasement. Comptes rendus des séances de l’académie des sciences, tome 138, Paris, 18.4.1904, pp. 945-949. Considère, A. (1906). Experimental researches on reinforced concrete. Second Edition, translated and arranged by Leon S. Moisseiff. New York: McGraw Publishing Company. Considère, A. (1907a). Essais de poteaux et prismes à la compression. In: Expériences, rapports & propositions instructions ministérielles relatives à l’emploi du béton armé, 246–340. Paris. Considère, A. (1907b). Essais à la compression de mortier et béton frettés. In: Expériences, rapports & propositions instructions ministérielles relatives à l’emploi du béton armé, 246–340. Paris. Considère, A. (1910). Der umschnürte Beton. Beton und Eisen 9 (6): 154–157. Cook, G. (1951). Rankine and the Theory of Earth Pressure. Géotechnique 2 (4): 271–279. Coronio, G. (ed.) (1997). 250 ans de l’École des ponts en cent portraits. Paris: Presses de l’école nationale des ponts et chaussées. Coulomb, C. A. (1773/1776). Essai sur une application des règles des Maximis et Minimis à quelques Problèmes de statique relatifs à l’Architecture. In: Mémoires de mathématique & de physique, présentés à l’Académie Royale des Sciences par divers savans, Vol. 7, année 1773, pp. 343-382. Paris. Coulomb, C. A. (1779). Versuch einer Anwendung der Methode des Größten und Kleinsten auf einige Aufgaben der Statik, die in die Baukunst einschlagen (German ed. of (Coulomb, 1773/1776, S. 343-370), trans. from the French by J. M. Geuß). In: Magazin für Ingenieur und Artilleristen, Vol. V, ed. by Andreas Böhm, pp. 135-151. Gießen: Joh. Chr. Krieger (the Younger). Coulomb, C.A. (1821). Théorie des machines simples. Paris: Bachelier. Couplet, P. (1726/1728). De la poussée des terres contre leurs revestements, et de la force des revestemens qu’ou leur doit opposer, premier partie, In: Mémoires de l’Académie Royale des Sciences, année 1726, pp. 106-164. Paris. Couplet, P. (1727/1729). De la poussée des terres contre leurs revestements, et de la force des revestemens qu’ou leur doit opposer, seconde partie. In: Mémoires de l’Académie Royale des Sciences, année 1727, pp. 139-184. Paris. Couplet, P. (1728/1730). De la poussée des terres contre leurs revestements, et de la force des revestemens qu’ou leur doit opposer, troisème partie. In: Mémoires de l’Académie Royale des Sciences, année 1728, pp. 113-138. Paris. Couplet, P. (1729/1731). De la poussée des voûtes. In: Mémoires de l’Académie Royale des Sciences, année 1729, pp. 79-117. Paris. Couplet, P. (1730/1732). Seconde partie de l’éxamen de la poussée des voûtes. In: Mémoires de l’Académie Royale des Sciences, année 1730, pp. 117-141. Paris. Culmann, K. (1851). Der Bau der hölzernen Brücken in den Vereinigten Staaten von Nordamerika. Allgemeine Bauzeitung 16: 69–129. Culmann, K. (1852). Der Bau der eisernen Brücken in England und Amerika. Allgemeine Bauzeitung 17: 163–222.

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Culmann, K. (1856). Ueber die Gleichgewichtsbedingungen von Erdmassen. Zurich: Druck von Züricher und Furrer. Culmann, K. (1864/1866). Die Graphische Statik. Zürich: Verlag von Meyer & Zeller. Culmann, K. (1872). Vorlesungen über Ingenieurkunde. I. Abtheilung: Erdbau. Zurich: Printed ms. Culmann, K. (1875). Die Graphische Statik, 2nd rev.e. Zürich: Meyer & Zeller. De Josselin de Jong, G. (1959). Statics and kinematics in the failure zone of a granular material. Delft: Uitgeverij Waltman. De Josselin de Jong, G. (1961). De mechanica toegepast op grond. Delft: Uitgeverij Waltman. De Josselin de Jong, G. (1968). Consolidation models consisting of an assembly of viscous elements or a cavity channel network. Géotechnique 18 (2): 195–228. De Josselin de Jong, G. (1971). The double sliding, free rotating model for granular assemblies. Géotechnique 21 (2): 155–163. De Josselin de Jong, G. (1978). Improvement of the lowerbound solution for the vertical cut off in a cohesive, frictionless soil. Géotechnique 28 (2): 197–201. De Josselin de Jong, G. (1979). Diskontinuitäten in Grenzspannungsfeldern. Geotechnik 2 (3): 125–129. De Josselin de Jong, G. (1980). Application of thecalculus of variation to the vertical cut off cohesive frictionless soil. Géotechnique 30 (1): 1–16. Degrand, E. and Résal, J. (1887). Ponts en maçonnerie, Tome 1: Stabilité des voûtes. Paris: Baudry. Degrand, E. and Résal, J. (1888). Ponts en maçonnerie, Tome 2: Construction. Paris: Baudry. Dietrich and Arslan, U. (1989). Über die Coulombsche Erddrucktheorie und deren Rezeption in Mitteleuropa. In: Bauingenieur, Vol. 64, 513–524. Droese, S. (1999). Eine fast vergessene Brückenbauweise: Hängebrücken “System Möller”. Bautechnik 76 (8): 625–633. Drucker, D.C. (1949). Relation of experiments to mathematical theories of plasticity. Journal of Applied Mechanics – Transactions of the American Society of Mechanical Engineers 16: 349–357. Drucker, D.C. (1951). A more fundamental approach to stress-strain relations. In: Proceedings of the 1st U.S. National Congress of Applied Mechanics, 487–491. Drucker, D.C., Prager, W. and Greenberg, H.J. (1951). Extended limit design theorems for continuous media. Quarterly of Applied Mathematics 9: 381–389. Drucker, D.C. and Prager, W. (1952). Soil mechanics and plastic analysis or limit design. Quarterly of Applied Mathematics 10: 157–165. Drucker, D.C. (1953). Coulombs friction, plasticity and limit loads. Transactions of the American Society of Mechanical Engineers 21: 71–74. Drucker, D.C. (1956). On uniqueness in the theory of plasticity. Quarterly Applied Mathematics 14: 35–42. Drucker, D.C. (1959). A definition of stable inelastic material. Journal of Applied Mechanics 26: 101–106. Drucker, D.C. (1961). On Structural Concrete and the Theorems of Limit Analysis. In: Abhandlungen der Internationalen Vereinigung für Brückenbau und Hochbau (IVBH), Vol. 21, 49–59. Zürich: IVBH.

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Drucker, D.C. (1984). William Prager. In: Memorial Tributes. National Academy of Engineering, Vol. 2, 232–235. Washington, D.C: National Academy Press. Dunnicliff, J. and Young, N.P. (2006). Ralph B. Peck – Educator and Engineer: The Essence of the Man. Vancouver: BiTech. Publishers. Eckardt, A. (1955). Brennecke, Ludwig Nathaniel August. In: Neue Deutsche Biographie, Vol. 2, 585–586. Eude, É. (1902). Histoire Documentaire de la Mécanique Française. Paris: Dunod. Feld, J. (1922). Measured retaining-wall pressure from sand and surcharge. Engineering News-Record 88 (3): 106–108. Feld, J. (1923). Lateral earth pressure: theaccurate experimental determination of the lateral earth pressure together with a resume of previous experiments. Proceedings of the American Society of Civil Engineers, Vol. 43, No. 4, pp. 603-660 (Discussion: No. 5, pp. 1007-1025). Feld, J. (1928). History of the Development of Lateral Earth Pressure Theories. In: Brooklyn Engineers’ Club Proceedings, January, pp. 61-104 Feld, J. (1948). Early History and Bibliography of Soil Mechanics. In: Proc. of the Second Intern. Conf. on Soil Mechanics and Foundation Engineering, Vol. 1, 1–7. Rotterdam/Boston: A. A. Balkema. Feld, J. (1952/1953). A historical chapter: British royal engineers’s papers on soil mechanics and foundation engineering, 1837-1874. Géotechnique III (6): 242–247. Feld, J. (1957). Structural design study for a parabolic reflector six hundred feet in diameter. In: Annals of the New York Academy of Sciences, 155–276. New York: Academy of Sciences. Feld, J. (1964). Lessons from Failures of Concrete Structures. Detroit: American Concrete Insitute. Feld, J. (1968). Construction Failure. New York: John Wiley & Sons, Inc. Feld, J. and Carper, K.L. (1997). Construction Failure, 2nde. New York: John Wiley & Sons, Inc. Fillunger, P. (1936). Erdbaumechanik? Vienna: Publ. by the author. Foerster, J. (1983). Nur ein Naturphilosoph? Sudhoffs Archiv: Zeitschrift für Wissenschaftsgeschichte 67: 74–79. Franke, D. (1992). Das Institut für Geotechnik. In: Mitteilungen des Instituts für Geotechnik der TU Dresden, No. 1. pp. I-V. Dresden: TU Dresden. Gehler, W. (ed.) (1916). Otto Mohr zum achtzigsten Geburtstage. Berlin: Wilhelm Ernst & Sohn. Gehler, W. (1928). Mohr, Christian Otto. In: Deutsches Biographisches Jahrbuch, ed. by Verbande der Deutschen Akademien. Überleitungsband II: 1917-1920, pp. 282-285. Berlin/Leipzig: Deutsche Verlags-Anstalt. Gerlach, J., Pulsfort, M. and Walz, B. (1999). Grundbau. Band 2: Baugruben und Gründungen. 18th rev. ed. Stuttgart/Leipzig: B. G. Teubner. Gillispie, C.C. (1970). Bélidor, Bernard Forest de. In: Dictionary of Scientific Biography, Vol (ed. I.C.C. Gillispie), 581–582. New York: Charles Scribner’s Sons. Gillmor, C.S. (1971a). Coulomb and the evolution of physics and engineering in eighteenth-century France. Princeton: Princeton University Press.

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383

Appendix A Terms, symbols, indices A1: Terms The following terms and symbols are based on DIN 4085 and are relevant in chapters 3 to 17. For historical reasons, different symbols may be used in chapter 2. In recent versions of the standards, a strict distinction is made between earth pressure as a contact potential between the soil and the structure and the resulting earth pressure, which is also referred to as earth pressure force. Previously, the term of earth pressure was used to describe both the contact potential and the resultant. Another new feature is the distinction between the earth pressure inclination angle which describes the actual inclination of the earth pressure or its resultant, and the angle of wall friction which is the physically largest inclination. In undisturbed, grown soil prevails at-rest earth pressure. The earth pressure is called active when it reaches a lower limit value while the structure moves away from the soil. When the structure moves towards the soil, the upper limit value is called passive earth pressure. The resultant of the passive earth pressure is often referred to as earth resistance. Earth pressure e Resulting effective stress from the effective normal stress and the shear stress on a wall Earth pressure coefficient K (with index) Ratio of the earth pressure to the vertical stress in the soil at a point on the wall Earth pressure force E Resultant of earth pressure active earth pressure ea Smallest possible earth pressure which occurs when a relaxation in the soil leads to a full utilization of the shear strength by means of wall and soil movements passive earth pressure ep Largest possible earth pressure which occurs when a compression in the soil leads to a full utilization of the shear strength by means of wall and soil movements Earth resistance Ep Resultant of passive earth pressure Earth Pressure, First Edition. Achim Hettler and Karl-Eugen Kurrer. © 2020 Ernst & Sohn Verlag GmbH & Co. KG. Published 2020 by Ernst & Sohn Verlag GmbH & Co. KG.

384

A Terms, symbols, indices

At-rest earth pressure e0 Earth pressure on an inflexible wall relative to unmoved soil in grown, undisturbed soil Earth pressure from compaction ev Earth pressure which is reached when the backfill is compacted Minimum earth pressure Earth pressure which is to be applied at least in case of cohesive soil for the dimensioning of a supporting structure in order to prevent a progressive decomposition of the soil by means of relaxation and under-dimensioning Increased active earth pressure e’ a Earth pressure resulting from soil self-weight, surcharges and other actions on a wall when the relaxation in the soil is not sufficient to generate active earth pressure Reduced passive earth pressure e’ p Earth pressure resulting from soil self-weight, surcharges and other actions on a wall when the movements between soil and wall are not sufficient to generate passive earth pressure Silo pressure es Earth pressure which is generated when the soil body behind a wall is geometrically limited in a way that the earth pressure on the wall is smaller as in case the soil body would not be limited Earth pressure inclination angle 𝛅 Angle between the earth pressure direction and the wall normal Angle of wall friction Angle of friction between wall and soil, the largest absolute value of earth pressure inclination angle

A2: Symbols Table A.1 Symbols Symbol

Denomination

Unit

b

width

m

c

cohesion

kN/m2

C

force due to cohesion

kN/m

d

anchoring depth of a wall

m

D

density



e

earth pressure

kN/m2

E

earth pressure force, resultant of earth pressure

kN or kN/m

G

Self weight of a soil body

kN or kN/m

h

height

m

H

horizontal line load

kN/m

i

hydraulic gradient



IC

consistency index

– (Continued)

A3: Indices

Table A.1 (Continued) Symbol

Denomination

Unit

K

earth pressure coefficient



l

length, length of a wall in top view

m

pv

Infinite uniformly distributed vertical load

kN/m2

pv‘

vertical strip load

kN/m2

Q

resultant from a normal force and friction force in a slip surface

kN or kN/m

s

wall displacement

m

V

vertical line load or resultant from a vertical strip load

kN/m

W

resultant of water pressure

kN/m

z

depth below the intersection line of the ground level with the wall

m

𝛂

wall inclination angle

𝛃

slope inclination angle

𝛄

Specific weight of the soil

𝛅

angle of wall friction and inclination angle of earth pressure

kN/m3 ∘

𝛗

friction angle of soil



𝛍

shape coefficient of spatial earth pressure



𝛝

inclination angle of slip surface



A3: Indices

Table A.2 Indices Index

Denomination

a

active status

c

due to cohesion

dyn

due to dynamic actions

g

due to soil self-weight

h

horizontal component

H

due to a horizontal line load

0

status of at-rest earth pressure

p

passive status

r

spatial

u

undrained status

v

vertical component

V

due to a vertical load

k

characteristic

d

design value

385

387

Appendix B Earth pressure tables

References Pregl, O. (2002). Bemessung von Stützbauwerken. Handbuch der Geotechnik, Bd. 16. Wien: Eigenverlag des Instituts für Geotechnik, Universität für Bodenkultur. Weißenbach, A. (1985). Baugruben, Teil II, Berechnungsgrundlagen. Berlin: Ernst & Sohn, 1. Nachdruck.

Earth Pressure, First Edition. Achim Hettler and Karl-Eugen Kurrer. © 2020 Ernst & Sohn Verlag GmbH & Co. KG. Published 2020 by Ernst & Sohn Verlag GmbH & Co. KG.

388

B Earth pressure tables

Table B.1 Earth pressure coefficients Kagh for horizontal ground surface and vertical wall according to Weißenbach (1985). Prerequisite: plane slip surface

Kagh = Kaph = Kah = [

√ 1+

δa

cos2 φ sin(φ + δa ) • sin φ cos δa φ=

10°

12.5° 15°

17.5° 20°

22.5° 25°

27.5° 30°

+ 23 • φ 0.647 0.583 0.525 0.473 0.426 0.384 0.346 0.31 +45° +42.5° +40° +37.5° +35° +32.5° +30° +27.5° +25° +22.5° +20° +17.5° +15° +12.5°

]2

32.5° 35°

37.5° 40°

42.5° 45°

0.279 0.251 0.224 0.200 0.179 0.159 0.140

0.125 0.142 0.128 0.161 0.145 0.130 0.182 0.164 0.148 0.133 0.205 0.186 0.168 0.151 0.135 0.230 0.209 0.189 0.171 0.154 0.138 0.257 0.234 0.213 0.193 0.174 0.157 0.140 0.288 0.263 0.239 0.217 0.197 0.178 0.160 0.143 0.322 0.294 0.268 0.244 0.222 0.200 0.181 0.162 0.145 0.359 0.329 0.300 0.274 0.249 0.226 0.204 0.184 0.165 0.148 0.401 0.367 0.336 0.307 0.279 0.254 0.230 0.208 0.187 0.168 0.150 0.448 0.410 0.375 0.343 0.313 0.285 0.259 0.235 0.212 0.191 0.171 0.152 0.500 0.458 0.420 0.384 0.351 0.320 0.291 0.264 0.239 0.216 0.194 0.174 0.155 0.559 0.512 0.469 0.429 0.393 0.359 0.327 0.297 0.270 0.244 0.220 0.198 0.177 0.158

+10°

0.625 0.573 0.525 0.481 0.440 0.402 0.367 0.334 0.304 0.275 0.249 0.224 0.201 0.180 0.160

+7.5°

0.642 0.588 0.539 0.493 0.451 0.412 0.376 0.342 0.311 0.281 0.254 0.229 0.205 0.183 0.163

+5°

0.660 0.605 0.554 0.507 0.463 0.423 0.385 0.350 0.318 0.287 0.259 0.233 0.209 0.187 0.166

+2.5°

0.680 0.623 0.570 0.521 0.476 0.434 0.395 0.359 0.325 0.294 0.265 0.238 0.213 0.190 0.169



0.704 0.644 0.589 0.538 0.490 0.446 0.406 0.368 0.333 0.301 0.271 0.243 0.217 0.194 0.172

–2.5°

0.733 0.669 0.610 0.556 0.506 0.460 0.418 0.378 0.342 0.308 0.277 0.249 0.222 0.197 0.175

–5°

0.769 0.698 0.635 0.577 0.524 0.475 0.431 0.390 0.352 0.317 0.284 0.254 0.227 0.201 0.178

–7.5°

0.820 0.736 0.665 0.602 0.545 0.493 0.445 0.402 0.362 0.325 0.292 0.261 0.232 0.206 0.182

–10°

0.970 0.791 0.704 0.632 0.569 0.513 0.462 0.416 0.374 0.335 0.300 0.267 0.238 0.210 0.186

–12.5° –15° –17.5° –20° –22.5° –25° –27.5° –30° –32.5° –35° –37.5° –40° –42.5° –45° − 23 • φ

0.953 0.761 0.672 0.599 0.537 0.482 0.432 0.387 0.346 0.309 0.275 0.244 0.215 0.190 0.933 0.730 0.639 0.567 0.505 0.450 0.402 0.358 0.319 0.283 0.251 0.221 0.194 0.910 0.698 0.606 0.534 0.473 0.420 0.373 0.330 0.292 0.258 0.227 0.199 0.883 0.665 0.572 0.501 0.441 0.389 0.344 0.303 0.267 0.234 0.204 0.854 0.631 0.539 0.468 0.410 0.359 0.315 0.276 0.242 0.211 0.821 0.596 0.504 0.435 0.378 0.330 0.288 0.250 0.217 0.787 0.560 0.470 0.403 0.348 0.301 0.261 0.225 0.750 0.525 0.436 0.371 0.318 0.273 0.235 0.711 0.488 0.402 0.339 0.288 0.240 0.671 0.452 0.369 0.308 0.260 0.629 0.416 0.336 0.278 0.587 0.380 0.303 0.544 0.345 0.500 0.800 0.752 0.704 0.657 0.611 0.567 0.523 0.481 0.441 0.402 0.365 0.330 0.296 0.265 0.235

References

Table B.2 Inclination angle of slip surface ϑa for horizontal ground surface and vertical wall according to Weißenbach (1985). Prerequisite: plane slip surface √ sin φ + tan ϑa = δa

tan φ tan φ + tan δa cos φ φ=

10°

+ 23 • φ 43.9

12.5°

15°

17.5°

20°

22.5°

25°

27.5°

30°

32.5°

35°

37.5°

40°

42.5°

45°

45.5

47.0

48.5

50.0

51.5

53.0

54.5

56.0

57.5

58.9

60.4

61.9

63.4

64.8

+45°

63.4

+42.5°

61.9

63.6

60.4 62.2

63.9

58.9

60.7 62.5

64.1

57.4

59.2

61.0 62.7

64.4

55.9

57.7

59.5

61.3 62.9

64.6

54.3 56.2

58.1

59.8

61.5 63.2

64.8

52.8

54.7 56.6

58.4

60.1

61.8 63.4

65.0

51.3 53.2

55.1 57.0

58.7

60.4

62.1 63.7

65.2

49.7

51.8 53.7

55.6 57.4

59.0

60.7

62.3 63.9

65.4

48.1

50.3

52.3 54.2

56.0 57.7

59.4

61.0

62.6 64.2

65.7

46.6

48.8

50.8

52.8 54.7

56.4 58.1

59.7

61.3

62.9 64.4

65.9

45.1

47.4

49.5

51.5

53.4 55.1

56.9 58.5

60.1

61.6

63.2 64.6

66.1

43.4

45.9

48.2

50.3

52.1

54.0 55.7

57.3 58.9

60.5

61.9

63.4 64.9

66.3

+40° +37.5° +35° +32.5° +30° +27.5° +25° +22.5° +20° +17.5° +15° +12.5° +10°

41.8

44.6

47.0

49.1

51.1

52.9

54.6 56.2

57.8 59.3

60.8

62.3

63.7 65.1

66.5

+7.5°

43.4

45.9

48.1

50.1

51.9

53.6

55.2 56.7

58.3 59.8

61.2

62.6

64.0 65.4

66.8

+5°

54.2

47.4

49.4

51.2

52.8

54.4

55.9 57.4

58.8 60.2

61.6

63.0

64.4 65.7

67.0

+2.5°

47.3

49.2

50.8

52.4

53.8

55.3

56.7 58.0

59.4 60.7

62.0

63.4

64.6 65.9

67.2



50.0

51.3

52.5

53.8

55.0

56.3

57.5 58.8

60.0 61.3

62.5

63.8

65.0 66.3

67.5

–2.5°

53.4

53.8

54.4

55.3

56.3

57.3

58.4 59.5

60.6 61.8

63.0

64.2

65.3 66.6

67.8

–5°

58.1

57.0

56.8

57.1

57.7

58.5

59.4 60.4

61.4 62.4

63.5

64.6

65.7 66.9

68.0

–7.5°

65.5

61.3

59.8

59.4

59.5

59.9

60.5 61.3

62.2 63.1

64.1

65.1

66.1 67.2

68.3

–10°

90.0

68.1

63.9

62.2

61.5

61.5

61.8 62.3

63.0 63.8

64.6

65.6

66.6 67.6

68.6

90.0

70.1

65.9

64.2

63.4

63.3 63.5

64.0 64.6

65.3

66.2

67.1 68.0

69.0

90.0

71.7

67.7

65.9

65.1 64.9

65.1 65.5

66.1

66.8

67.6 68.4

69.4

73.0

69.2

67.4 66.6

66.4 66.5

66.9

67.4

68.1 68.9

69.7

90.0

74.2

70.5 67.8

68.0 67.8

67.9

68.2

68.8 69.4

70.2 70.7

–12.5° –15° –17.5°

90.0

–20° –22.5°

90.0

–25° –27.5°

75.1 71.6

70.0 69.3

69.0

69.1

69.5 70.0

90.0 76.0

72.7 71.1

70.4

70.2

70.3 70.7

71.2

90.0

76.8 73.7

72.2

71.5

71.3 71.4

71.8

90.0 77.5

74.5

73.1

72.5 72.4

72.5

90.0

78.1

75.3

74.0 73.5

73.4

90.0

78.8

76.1 74.9

74.4

90.0

79.3 76.8

75.7

–30° –32.5° –35° –37.5° –40° –42.5°

90.0 79.9

77.5

90.0

80.3

–45° − 23 • φ

90.0 62.6

63.2

63.9

65.5

65.2

65.9

66.6 67.3

68.0 68.7

69.5

70.2

70.9 71.7

72.5

389

390

B Earth pressure tables

Table B.3 Earth pressure coefficients Kaph for line loads and strip loads at vertical wall according to Weißenbach (1985). Prerequisite: plane slip surface with ϑ = ϑa Kaph =

sin(ϑa − φ) • cos δa cos(ϑa − δa − φ)

δa

φ= 10°

12.5° 15°

17.5° 20°

22.5° 25°

27.5° 30°

32.5° 35°

37.5° 40°

42.5° 45°

+ 23 • φ 0.624 0.592 0.563 0.535 0.508 0.483 0.459 0.436 0.414 0.393 0.373 0.353 0.334 0.316 0.298 +45°

0.250

+42.5°

0.267 0.258

+40°

0.284 0.275 0.266

+37.5°

0.301 0.293 0.284 0.274

+35°

0.320 0.311 0.302 0.293 0.282

+32.5°

0.339 0.331 0.322 0.312 0.301 0.290

+30°

0.359 0.351 0.342 0.332 0.321 0.310 0.298

+27.5°

0.379 0.372 0.363 0.353 0.342 0.331 0.319 0.306

+25°

0.401 0.394 0.385 0.375 0.365 0.353 0.341 0.328 0.315

+22.5°

0.424 0.417 0.409 0.399 0.388 0.377 0.364 0.351 0.337 0.323

+20°

0.448 0.442 0.434 0.425 0.414 0.402 0.389 0.376 0.362 0.347 0.332

+17.5°

0.473 0.469 0.462 0.452 0.442 0.429 0.416 0.402 0.388 0.372 0.357 0.340

+15°

0.500 0.498 0.492 0.483 0.472 0.460 0.446 0.431 0.416 0.400 0.384 0.367 0.350

+12.5°

0.529 0.529 0.525 0.517 0.506 0.493 0.479 0.464 0.447 0.431 0.413 0.395 0.377 0.359

+10°

0.559 0.565 0.563 0.555 0.544 0.531 0.516 0.500 0.482 0.464 0.446 0.427 0.408 0.389 0.369

+7.5°

0.606 0.607 0.601 0.590 0.576 0.559 0.541 0.522 0.503 0.483 0.462 0.442 0.421 0.400 0.379

+5°

0.664 0.658 0.646 0.630 0.611 0.591 0.569 0.547 0.525 0.503 0.480 0.458 0.435 0.413 0.390

+2.5°

0.738 0.722 0.700 0.677 0.652 0.627 0.601 0.575 0.550 0.525 0.499 0.475 0.450 0.426 0.402



0.839 0.803 0.767 0.733 0.700 0.668 0.637 0.607 0.577 0.549 0.521 0.493 0.466 0.440 0.414

–2.5°

0.987 0.912 0.954 0.803 0.758 0.717 0.679 0.643 0.608 0.575 0.544 0.513 0.484 0.455 0.427

–5°

1.235 1.074 0.971 0.894 0.831 0.776 0.728 0.684 0.644 0.606 0.570 0.536 0.503 0.472 0.442

–7.5°

1.799 1.346 1.143 1.016 0.924 0.850 0.788 0.733 0.685 0.640 0.599 0.561 0.525 0.490 0.457

–10°



–12.5° –15° –17.5° –20° –22.5° –25° –27.5° –30° –32.5° –35° –37.5° –40° –42.5° –45° − 23 • φ

1.966 1.435 1.197 1.051 0.945 0.862 0.793 0.733 0.681 0.633 0.589 0.549 0.510 0.475 ∞

2.099 1.504 1.238 1.075 0.958 0.867 0.792 0.728 0.672 0.622 0.576 0.533 0.494 ∞

2.204 1.556 1.267 1.089 0.963 0.866 0.786 0.719 0.660 0.607 0.559 0.515 ∞

2.284 1.593 1.284 1.095 0.962 0.859 0.775 0.704 0.643 0.588 0.539 ∞

2.341 1.615 1.290 1.092 0.953 0.846 0.759 0.686 0.623 0.567 ∞

2.376 1.624 1.287 1.082 0.938 0.828 0.739 0.664 0.599 ∞

2.391 1.620 1.274 1.064 0.917 0.805 0.714 0.638 ∞

2.387 1.604 1.253 1.040 0.891 0.777 0.686 ∞

2.365 1.577 1.224 1.009 0.859 0.745 ∞

2.327 1.539 1.187 0.973 0.823 ∞

2.273 1.493 1.143 0.931 ∞

2.205 1.437 1.093 ∞

2.123 1.373 ∞

2.029 ∞

1.541 1.488 1.435 1.380 1.324 1.267 1.209 1.151 1.092 1.034 0.975 0.917 0.859 0.802 0.745

References

Table B.4 Earth resistance coefficients Kpgh for horizontal ground surface and vertical wall according to Weißenbach (1985). Prerequisite: plane slip surface

Kpgh = Kpph = Kph =

cos2 φ √

[ 1−

δp 2 3



sin φ

]2

cos δp φ=

10°



sin (φ − δp )

• φ 1.61

12.5°

15°

17.5°

20°

22.5°

25°

27.5°

30°

32.5°

35°

37.5°

40°

42.5°

45°

1.83

2.10

2.42

2.81

3.30

3.91

4.70

5.74

7.15

9.15

12.1

16.7

24.6

39.9

–45°



–42.5°

273

–40°

296

32.5

48.0 76.6

139

18.8 25.2

35.1 51.6

82.2

12.3

15.6 20.2

27.0 37.5

55.1

8.74 10.7

13.2 16.7

21.6 28.8

39.9

6.53

7.78 9.36

11.4 14.1

17.8 22.9

30.5

5.07 5.93

6.98 8.29

9.95 12.1

14.9 18.8

24.2

4.05

4.67 5.41

6.31 7.41

8.78 10.5

12.8 15.7

19.8

3.31 3.78

4.32 4.96

5.74 6.67

7.82 9.25

11.1 13.4

16.5

2.75

3.11 3.52

4.01 4.57

5.24 6.04

7.02 8.21

9.69 11.6

14.0

2.32 2.60

2.93 3.30

3.72 4.22

4.81 5.50

6.33 7.34

8.57 10.1

12.0

1.98

2.21 2.46

2.75 3.09

3.47 3.91

4.42 5.03

5.74 6.60

7.63 8.90

10.5

–37.5° –35° –32.5° –30° –27.5° –25° –22.5° –20° –17.5° –15° –12.5°

1118

70.9 128

–10°

1.70

1.89

2.10 2.33

2.60 2.89

3.24 3.63

4.08 4.61

5.23 5.96

6.84 7.90

9.20

–7.5°

1.63

1.80

1.99 2.21

2.45 2.72

3.02 3.37

3.77 4.24

4.78 5.41

6.16 7.06

8.14

–5°

1.56

1.72

1.89 2.09

2.30 2.55

2.82 3.14

3.49 3.90

4.37 4.93

5.57 6.34

7.25

–2.5°

1.49

1.64

1.80 1.97

2.17 2.39

2.64 2.92

3.24 3.60

4.01 4.50

5.06 5.71

6.49



1.42

1.55

1.70 1.86

2.04 2.24

2.46 2.72

3.00 3.32

3.69 4.11

4.60 5.17

5.83

+2.5°

1.34

1.47

1.60 1.75

1.91 2.10

2.30 2.53

2.78 3.07

3.39 3.77

4.19 4.68

5.25

+5°

1.26

1.38

1.50 1.64

1.79 1.96

2.14 2.35

2.58 2.83

3.12 3.45

3.83 4.25

4.75

+7.5°

1.16

1.28

1.40 1.53

1.67 1.82

1.99 2.18

2.39 2.62

2.87 3.17

3.49 3.87

4.30

+10°

0.97

1.17

1.30 1.42

1.55 1.69

1.85 2.02

2.20 2.41

2.64 2.90

3.19 3.52

3.90

0.95

1.17 1.30

1.43 1.56

1.71 1.86

2.03 2.22

2.43 2.66

2.92 3.21

3.53

0.93 1.17

1.30 1.43

1.57 1.71

1.87 2.04

2.22 2.43

2.66 2.92

3.20

+12.5° +15° +17.5°

0.91

+20° +22.5°

1.15 1.29

1.42 1.56

1.71 1.86

2.03 2.22

2.42 2.65

2.90

0.88 1.14

1.28 1.41

1.55 1.69

1.85 2.02

2.20 2.40

2.63

0.85

1.11 1.26

1.39 1.53

1.67 1.83

1.99 2.18

2.38

0.82 1.09

1.23 1.37

1.50 1.64

1.80 1.96

2.14

0.79

1.05 1.20

1.33 1.47

1.61 1.76

1.92

0.75 1.02

1.16 1.30

1.43 1.57

1.71

0.71

0.98 1.12

1.25 1.38

1.52

0.67 0.94

1.08 1.20

1.33

+25° +27.5° +30° +32.5° +35° +37.5°

0.63

+40° +42.5°

0.89 1.03

1.15

0.59 0.84

0.97

0.54

0.79

+45° + 23 • φ

0.50 1.20

1.25

1.32 1.34

1.39 1.43

1.47 1.52

1.55 1.58

1.62 1.65

1.67 1.69

1.71

391

392

B Earth pressure tables

Table B.5a Earth resistance coefficients Kpgh for horizontal ground surface and vertical wall at curved slip surfaces according to Pregl/Sokolowski (Pregl 2002), share from soil self-weight 1 + sin φ ⋅ (1 − 0.53 ⋅ δp )0.26+5.96⋅φ δp ≤ 0 1 − sin φ 1 + sin φ = cos δp ⋅ δp > 0 ⋅ (1 + 0.41 ⋅ δp )−7.13 1 − sin φ

Kpgh = cos δp ⋅ Kpgh δp

φ= 10°

12.5° 15°

17.5° 20°

22.5° 25°

27.5° 30°

32.5° 35°

37.5°

40°

42.5°

–45°

45°

23.01

–42.5°

17.98

22.11

14.17

17.27

21.15

11.26 13.61

16.52

20.14

9.03 10.82 13.02

15.73

19.09

8.68 10.35 12.40

14.919 18.01

–40° –37.5° –35° –32.5°

7.30

–30°

5.95 7.02

8.31

9.87 11.76

14.08

16.93

4.89

5.73 6.73

7.92

9.37 11.11

13.24

15.84

4.05 4.71

5.49 6.42

7.53

8.86 10.46

12.40

14.77

3.38

3.91 4.53

5.25 6.11

7.13

8.34

9.80

11.56

13.71

2.85 3.27

3.76 4.33

5.00 5.79

6.72

7.83

9.15

10.74

12.67

2.42

2.76 3.16

3.61 4.14

4.75 5.47

6.32

7.32

8.51

9.94

11.67

2.08 2.35

2.67 3.03

3.45 3.93

4.50 5.15

5.92

6.82

7.89

9.17

10.70

1.80

2.02 2.28

2.57 2.91

3.29 3.73

4.24 4.83

5.52

6.33

7.28

8.42

9.77

–27.5° –25° –22.5° –20° –17.5° –15° –12.5° –10°

1.57

1.76

1.96 2.20

2.47 2.78

3.13 3.52

3.98 4.52

5.13

5.85

6.70

7.70

8.89

–7.5°

1.54

1.71

1.90 2.12

2.37 2.64

2.96 3.32

3.73 4.21

4.75

5.39

6.13

7.01

8.05

–5°

1.50

1.66

1.84 2.04

2.26 2.51

2.79 3.12

3.48 3.90

4.39

4.94

5.60

6.36

7.26

–2.5°

1.46

1.61

1.77 1.95

2.15 2.37

2.63 2.91

3.24 3.61

4.03

4.52

5.08

5.74

6.52



1.42

1.55

1.70 1.86

2.04 2.24

2.46 2.72

3.00 3.32

3.69

4.11

4.60

5.17

5.83

+2.5°

1.25

1.37

1.50 1.64

1.80 1.97

2.17 2.39

2.64 2.93

3.25

3.62

4.05

4.55

5.13

+5°

1.10

1.20

1.32 1.44

1.58 1.74

1.91 2.11

2.33 2.58

2.86

3.19

3.57

4.01

4.52

+7.5°

0.97

1.06

1.16 1.27

1.39 1.53

1.68 1.85

2.05 2.27

2.52

2.81

3.14

3.53

3.98

+10°

0.85

0.93

1.02 1.12

1.23 1.35

1.48 1.63

1.80 2.00

2.22

2.47

2.77

3.11

3.51

0.82

0.90 0.99

1.08 1.19

1.31 1.44

1.59 1.76

1.96

2.18

2.44

2.74

3.09

0.79 0.87

0.95 1.05

1.15 1.27

1.40 1.55

1.72

1.92

2.15

2.41

2.72

0.76

0.84 0.92

1.01 1.12

1.23 1.37

1.52

1.69

1.89

2.12

2.40

0.74 0.81

0.89 0.98

1.09 1.20

1.34

1.49

1.67

1.87

2.11 1.86

+12.5° +15° +17.5° +20° +22.5° +25° +27.5° +30° +32.5° +35° +37.5° +40° +42.5° +45°

0.71

0.79 0.87

0.96 1.06

1.18

1.31

1.47

1.65

0.69 0.76

0.84 0.93

1.03

1.15

1.29

1.45

1.63

0.67

0.74 0.82

0.91

1.01

1.13

1.27

1.44

0.65 0.72

0.80

0.89

1.00

1.12

1.26

0.63

0.70

0.78

0.87

0.98

1.11

0.61

0.68

0.77

0.86

0.97

0.60

0.67

0.75

0.85

0.590

0.66

0.74

0.57

0.65 0.56

References

Table B.5b Earth resistance coefficients Kpph for horizontal ground surface and vertical wall at curved slip surfaces according to Pregl/Sokolowski (Pregl 2002), share from infinite uniformly distributed load 1 + sin φ ⋅ (1 − 1.33 ⋅ δp )0.08+2.37⋅φ 1 − sin φ 1 + sin φ = cos δp ⋅ ⋅ (1 − 0.72 ⋅ δp )2.81 1 − sin φ

Kpph = cos δp ⋅

δp ≤ 0

Kpph

δp > 0

δp

φ= 10°

12.5° 15°

17.5° 20°

22.5° 25°

27.5° 30°

32.5° 35°

37.5° 40°

42.5°

–45°

45°

16.52

–42.5°

13.45 16.29

–40°

11.01 13.23 15.98

–37.5°

9.06

10.81 12.95 15.60

7.50 8.89

10.57 12.62 15.15

6.24

7.35 8.68

10.28 12.24 14.64

5.22 6.11

7.17 8.43

9.96 11.82 14.08

4.39

5.11 5.95

6.96 8.16

9.61 11.35 13.48

3.72 4.30

4.98 5.78

6.73 7.86

9.22 10.86 12.84

3.16

3.64 4.19

4.83 5.58

6.48 7.54

8.81 10.33 12.17

2.71 3.09

3.54 4.06

4.67 5.37

6.21 7.20

8.37

9.78 11.48

2.33

2.65 3.01

3.44 3.92

4.49 5.15

5.92 6.84

7.92

9.22 10.77

2.02 2.28

2.58 2.93

3.32 3.77

4.30 4.91

5.63 6.47

7.46

8.64 10.06

1.76

1.98 2.23

2.51 2.83

3.20 3.62

4.10 4.66

5.32 6.08

6.98

–35° –32.5° –30° –27.5° –25° –22.5° –20° –17.5° –15° –12.5°

8.05

9.33

–10°

1.55

1.73

1.94 2.17

2.43 2.72

3.06 3.45

3.89 4.40

5.00 5.69

6.51

7.47

8.61

–7.5°

1.52

1.69

1.88 2.10

2.34 2.61

2.92 3.27

3.68 4.14

4.68 5.30

6.02

6.88

7.89

–5°

1.49

1.65

1.83 2.02

2.24 2.49

2.77 3.09

3.46 3.87

4.35 4.90

5.54

6.30

7.19

–2.5°

1.46

1.60

1.77 1.94

2.14 2.37

2.62 2.91

3.23 3.60

4.02 4.50

5.07

5.72

6.50



1.42

1.55

1.70 1.86

2.04 2.24

2.46 2.72

3.00 3.32

3.69 4.11

4.60

5.17

5.83

+2.5°

1.30

1.42

1.55 1.70

1.86 2.05

2.25 2.48

2.74 3.03

3.37 3.76

4.20

4.72

5.32

+5°

1.18

1.29

1.41 1.54

1.69 1.86

2.05 2.25

2.49 2.76

3.06 3.41

3.82

4.29

4.84

+7.5°

1.07

1.17

1.27 1.40

1.53 1.68

1.85 2.04

2.25 2.49

2.77 3.09

3.45

3.88

4.38

+10°

0.96

1.05

1.15 1.26

1.38 1.51

1.66 1.83

2.03 2.24

2.49 2.78

3.11

3.49

3.94

0.94

1.03 1.12

1.23 1.35

1.49 1.64

1.81 2.01

2.23 2.48

2.78

3.12

3.52

0.91 1.00

1.10 1.20

1.32 1.46

1.61 1.78

1.98 2.21

2.47

2.77

3.13

+12.5° +15° +17.5° +20° +22.5° +25° +27.5° +30° +32.5° +35° +37.5° +40° +42.5° +45°

0.88

0.97 1.06

1.17 1.29

1.42 1.58

1.75 1.95

2.18

2.45

2.77

0.85 0.93

1.03 1.13

1.25 1.38

1.54 1.71

1.92

2.15

2.43

0.81

0.89 0.99

1.09 1.21

1.34 1.49

1.67

1.88

2.12

0.77 0.85

0.94 1.04

1.16 1.29

1.44

1.62

1.83

0.73

0.81 0.90

0.99 1.11

1.24

1.39

1.57

0.69 0.76

0.85 0.94

1.05

1.18

1.34

0.64

0.71 0.79

0.89

1.00

1.12

0.59 0.66

0.74

0.83

0.94

0.54

0.61

0.68

0.77

0.49

0.56

0.63

0.45

0.50 0.40

393

394

B Earth pressure tables

Table B.5c Earth resistance coefficients Kpch for horizontal ground surface and vertical wall at curved slip surfaces according to Pregl/Sokolowski (Pregl 2002), share from cohesion (

) 1 + sin φ δp ≤ 0 − 1 ⋅ cot φ ⋅ (1 − 1.33 ⋅ δp )0.08+2.37⋅φ 1 − sin φ ( ) 1 + sin φ = cos δp ⋅ − 1 ⋅ cot φ ⋅ (1 + 4.46 ⋅ δp ⋅ tan φ)−1.14+0.57⋅φ δp > 0 1 − sin φ

Kpch = cos δp ⋅ Kpch δp

φ= 10°

12.5° 15°

17.5° 20°

22.5° 25°

27.5° 30°

32.5° 35°

37.5° 40°

42.5°

–45°

45°

13.69

–42.5°

11.83 13.49

–40°

10.26 11.64 13.24

–37.5°

8.94

10.08 11.40 12.92

7.81 8.76

9.86 11.11 12.55

6.85

7.65 8.56

9.59 10.77 12.13

6.03 6.70

7.46 8.32

9.29 10.40 11.67

5.33

5.90 6.53

7.25 8.05

8.96

9.99 11.17

4.74 5.21

5.75 6.34

7.01 7.75

8.60

9.55 10.64

4.23

4.63 5.08

5.58 6.13

6.74 7.44

8.21

9.09 10.09

3.79 4.14

4.51 4.93

5.39 5.90

6.46 7.10

7.81

8.61

9.51

3.42

3.71 4.03

4.38 4.76

5.18 5.65

6.17 6.74

7.39

8.11

8.93

3.10 3.35

3.62 3.91

4.23 4.58

4.97 5.39

5.86 6.38

6.96

7.60

8.33

2.83

3.04 3.27

3.51 3.78

4.07 4.39

4.74 5.12

5.54 6.00

6.51

7.09

7.73

–35° –32.5° –30° –27.5° –25° –22.5° –20° –17.5° –15° –12.5° –10°

2.60

2.78

2.97 3.18

3.40 3.64

3.90 4.19

4.49 4.83

5.20 5.61

6.07

6.57

7.13

–7.5°

2.56

2.72

2.89 3.08

3.28 3.49

3.72 3.97

4.25 4.54

4.87 5.22

5.62

6.05

6.54

–5°

2.51

2.65

2.80 2.97

3.14 3.33

3.53 3.75

3.99 4.25

4.53 4.83

5.17

5.54

5.95

–2.5°

2.45

2.57

2.71 2.85

3.00 3.17

3.34 3.53

3.73 3.95

4.18 4.44

4.73

5.04

5.38



2.38

2.49

2.61 2.73

2.86 2.99

3.14 3.30

3.46 3.65

3.84 4.06

4.29

4.55

4.83

+2.5°

2.30

2.39

2.48 2.57

2.68 2.79

2.90 3.03

3.16 3.31

3.47 3.64

3.83

4.04

4.27

+5°

2.22

2.28

2.35 2.43

2.51 2.60

2.70 2.80

2.91 3.03

3.16 3.31

3.46

3.64

3.83

+7.5°

2.13

2.18

2.24 2.30

2.36 2.43

2.51 2.60

2.69 2.79

2.90 3.03

3.16

3.31

3.48

+10°

2.05

2.09

2.13 2.17

2.22 2.28

2.35 2.42

2.50 2.58

2.68 2.79

2.91

3.04

3.19

2.00

2.02 2.06

2.10 2.14

2.20 2.26

2.32 2.40

2.49 2.58

2.69

2.81

2.94

1.92 1.95

1.98 2.01

2.06 2.11

2.17 2.24

2.31 2.40

2.50

2.61

2.73

1.84

1.86 1.90

1.93 1.98

2.03 2.09

2.16 2.23

2.32

2.42

2.54

1.76 1.78

1.81 1.85

1.90 1.95

2.01 2.09

2.17

2.26

2.37 2.21

+12.5° +15° +17.5° +20° +22.5° +25° +27.5° +30° +32.5° +35° +37.5° +40° +42.5° +45°

1.68

1.70 1.74

1.78 1.83

1.88 1.95

2.03

2.11

1.60 1.63

1.67 1.71

1.76 1.82

1.90

1.98

2.07

1.53

1.56 1.60

1.65 1.71

1.77

1.85

1.94

1.46 1.50

1.54 1.60

1.66

1.73

1.82

1.40

1.45 1.50

1.55

1.62

1.70

1.35 1.40

1.45

1.52

1.59

1.31

1.36

1.42

1.49

1.27

1.32

1.39

1.23

1.30 1.20