numerical modelling of soil anchors

NUMERICAL MODELLING OF SOIL

ANCHORS

by

Richard Simon Merifield

B.E.(Hons 1)

A Thesis submitted for the Degree of

Doctor of Philosophy

at the University of Newcastle.

February 2002

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I hereby certify that the work embodied in this Thesis is the result of original researchand has not been submitted for a higher degree to any other University or Institute

(signed)

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ABSTRACT

The design of many engineering structures require foundation systems to resist vertical

uplift or horizontal pullout forces. These types of structures, which may include

transmission towers or earth retaining structures, are commonly supported directly by soil

anchors. As the range of applications for anchors increases, to include the support of more

elaborate and substantially larger structures, a greater understanding regarding their

behaviour is required.

During the last thirty years various researchers have proposed approximate techniques to

estimate the uplift capacity of soil anchors. The majority of past research has beenexperimentally based and, as a result, current design practices are largely based on

empiricism. In contrast, very few rigorous numerical analyses have been performed to

determine the ultimate pull-out load of anchors.

A rigorous study of geotechnical engineering problems such as plate anchors is made

difficult due to the inherent natural variation and complex behaviour of geomaterials. A

study of anchors is further complicated by the large number of secondary variables that

influence overall behaviour. These include anchor size, shape, embedment depth and

orientation, and must be considered in any analysis.

In this thesis, the results of a comprehensive numerical study into the behaviour of anchor

plates is presented. Consideration is given to the wide range of parameters that influence

anchor capacity. The aim of this research is to better understand anchor behaviour and to

develop rigorous stability solutions for earth anchors that can be used by design engineers.

The study is unique in that three distinctly different numerical methods have been used in

tandem to determine the ultimate capacity of anchors, namely the upper and lower bound

theorems of limit analysis and the displacement finite element method. A comparison of

the results from each technique provides an opportunity to validate the findings and gives

a rigorous evaluation of anchor capacity.

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ACKNOWLEDGMENTS

I gratefully acknowledge the financial assistance received through the receipt of an

Australia Postgraduate Award during my candidature. I am also thankful for the top up

scholarship provided by the Department of Civil, Surveying and Environmental

Engineering at the University of Newcastle.

I am indebted to Prof. Scott Sloan for his interest, guidance and provision of financial

assistance during this research. Prof. Sloans commitment and assistance was limitless and

greatly appreciated.

Finally, thank you to my wife Cindy and baby daughter Sarah, my parents Fai and John,

and my brother Andrew for their support and encouragement throughout the period of my

studies.

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ABSTRACT ii. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ACKNOWLEDGMENTS iii. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

PREFACE ix. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

NOTATION xi. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1. INTRODUCTION 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 INTRODUCTION 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 SOIL ANCHORS 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.2.1 In-direct application of anchor theory 5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 THESIS OUTLINE 6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2. HISTORICAL REVIEW 9. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 INTRODUCTION 10. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 PREVIOUS EXPERIMENTAL INVESTIGATIONS 11. . . . . . . . . . . . . . . . . . . . . . . . . .

2.2.1 Anchors in purely cohesive soil 12. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1.1 Other investigations 15. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.2.2 Anchors in cohesionless soil 17. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 PREVIOUS THEORETICAL ANALYSES 23. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.3.1 Anchors in purely cohesive soil 23. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Anchors in cohesionless soil 24. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2.1 Other investigations 28. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.4 SUMMARY 29. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3. NUMERICAL FORMULATIONS 31. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 INTRODUCTION 32. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 NUMERICAL METHODS IN GEOMECHANICS 32. . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 THEORY OF LIMIT ANALYSIS 36. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.3.1 The assumption of perfect plasticity 37. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 The stability postulate of Drucker 38. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3 Yield criterion 40. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.4 Flow rule 41. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.5 Small deformations and the equation of virtual work 42. . . . . . . . . . . . . . . . . . . 3.3.6 The limit theorems 43. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.4 LOWER BOUND LIMIT ANALYSIS FORMULATION 48. . . . . . . . . . . . . . . . . . . . . . 3.4.1 Constraints from equilibrium conditions 51. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Constraints from stress boundary conditions 53. . . . . . . . . . . . . . . . . . . . . . . . . 3.4.3 Constraints from yield conditions 55. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.4 Formation of the objective function 58. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.5 UPPER BOUND LIMIT ANALYSIS FORMULATION 59. . . . . . . . . . . . . . . . . . . . . . . 3.5.1 Constraints from plastic flow in continuum 60. . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.2 Constraints due to plastic shearing in discontinuities 63. . . . . . . . . . . . . . . . . . .

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3.5.3 Constraints due to velocity boundary conditions 67. . . . . . . . . . . . . . . . . . . . . . 3.5.4 Formation of objective function: Power dissipation in continuum 68. . . . . . . . . 3.5.5 Formation of objective function: Power dissipation in velocity discontinuities 693.5.6 Upper bound linear programming problem 70. . . . . . . . . . . . . . . . . . . . . . . . . . .

3.6 NONLINEAR FORMULATION OF LOWER BOUND THEOREM 71. . . . . . . . . . . . . 3.7 DISPLACEMENT FINITE ELEMENT STUDY OF ANCHORS 73. . . . . . . . . . . . . . . .

3.7.1 Smooth hyperbolic approximation of Mohr-Coulomb yield criterion 73. . . . . . 3.7.2 Integration of elasto-plastic constitutive laws 75. . . . . . . . . . . . . . . . . . . . . . . . . 3.7.3 Solution of elasto-plastic load-displacement relations 76. . . . . . . . . . . . . . . . . .

3.8 CONCLUSIONS 76. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4. ANCHOR BEHAVIOUR AND NUMERICAL MODELLING 79. . . . . . . . . . 4.1 INTRODUCTION 80. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 GENERAL ANCHOR BEHAVIOUR 80. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.2.1 Anchors in purely cohesive soil 80. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Anchors in purely frictional soil 83. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.3 PLANE STRAIN LIMIT ANALYSIS MODELLING 85. . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Mesh details 85. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Anchor interface conditions 89. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 Collapse load determination for plate anchors 95. . . . . . . . . . . . . . . . . . . . . . . .

4.4 3-D LIMIT ANALYSIS MODELLING 95. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 DISPLACEMENT FINITE ELEMENT MODELLING 100. . . . . . . . . . . . . . . . . . . . . . . .

4.5.1 Anchor interface conditions 100. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.2 Mesh optimisation and arrangement 102. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.3 Initial stress conditions 104. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5. HORIZONTAL ANCHORS IN PURELY COHESIVE SOIL 109. . . . . . . . . . . 5.1 INTRODUCTION 110. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 ANALYTICAL SOLUTIONS 112. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.2.1 Limit analysis solutions 113. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Cavity expansion approach 116. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.3 NUMERICAL LIMIT ANALYSIS RESULTS 118. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Horizontal anchors in homogeneous soil () 119. . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Comparison with existing theoretical and laboratory solutions 122. . . . . . . . . . . 5.3.3 Effect of overburden pressure 126. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.4 Effect of increasing strength with depth 130. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.5 Anchors at great depth 135. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.4 DISPLACEMENT FINITE ELEMENT RESULTS 135. . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Anchors in homogeneous soil 136. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.2 Load-displacement response 137. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.3 Computed failure modes and plastic zones 140. . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.4 Effect of overburden pressure 150. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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5.4.5 Effect of anchor interface details 154. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.6 Effect of mesh arrangement 158. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.5 EFFECT OF ANCHOR ROUGHNESS 159. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 EFFECT OF SOIL SUCTION - NO BREAKAWAY 160. . . . . . . . . . . . . . . . . . . . . . . . . .

5.6.1 Effect of inhomogeneous soils 168. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7 CONCLUSIONS 170. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6. VERTICAL & INCLINED ANCHORS IN PURELY COHESIVE SOIL 173. 6.1 INTRODUCTION 174. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 UPPER BOUND LIMIT ANALYSIS SOLUTIONS 175. . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 NUMERICAL LIMIT ANALYSIS RESULTS 177. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.3.1 Vertical anchors in homogeneous soil 178. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 Comparison with existing theoretical and laboratory results 180. . . . . . . . . . . . . 6.3.3 Effect of overburden pressure 183. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.4 Effect of increasing strength with depth 185. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.4 DISPLACEMENT FINITE ELEMENT RESULTS 188. . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 Anchors in homogeneous soil 188. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.2 Load-displacement response 189. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.3 Computed failure mechanisms and plastic zones 189. . . . . . . . . . . . . . . . . . . . . . 6.4.4 Effect of overburden pressure 198. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.5 Effect of anchor interface details 199. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.6 Effect of mesh arrangement 200. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.5 EFFECT OF ANCHOR ROUGHNESS 200. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 EFFECT OF SOIL SUCTION - NO BREAKAWAY 206. . . . . . . . . . . . . . . . . . . . . . . . . . 6.7 EFFECT OF ANCHOR INCLINATION 212. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.8 CONCLUSIONS 222. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7. HORIZONTAL ANCHORS IN COHESIONLESS SOIL 225. . . . . . . . . . . . . . 7.1 INTRODUCTION 226. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 ANALYTICAL SOLUTIONS 226. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7.2.1 Upper bound mechanisms for anchor plates in cohesionless soil 227. . . . . . . . . . 7.2.2 Cavity expansion approach 231. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7.3 NUMERICAL LIMIT ANALYSIS RESULTS 233. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 DISPLACEMENT FINITE ELEMENT RESULTS 241. . . . . . . . . . . . . . . . . . . . . . . . . . .

7.4.1 Break-out factors 242. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.2 Load-displacement response 242. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.3 Computed failure mechanisms and plastic zones 244. . . . . . . . . . . . . . . . . . . . . . 7.4.4 Effect of anchor interface details 255. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7.5 EFFECT OF ANCHOR ROUGHNESS 255. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6 EFFECT OF SOIL DILATION 256. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7.6.1 Half dilation 257. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6.2 No dilation 260. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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7.7 COMPARISON OF EXISTING THEORETICAL RESULTS 269. . . . . . . . . . . . . . . . . . . 7.8 COMPARISON OF EXISTING EXPERIMENTAL RESULTS 275. . . . . . . . . . . . . . . . . . 7.9 CONCLUSIONS 280. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8. VERTICAL & INCLINED ANCHORS IN COHESIONLESS SOIL 283. . . . 8.1 INTRODUCTION 284. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 UPPER BOUND MECHANISMS 284. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 NUMERICAL LIMIT ANALYSIS RESULTS 288. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 DISPLACEMENT FINITE ELEMENT RESULTS 297. . . . . . . . . . . . . . . . . . . . . . . . . . .

8.4.1 Break-out factors 297. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.2 Load-displacement response 299. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.3 Computed failure modes and plastic zones 299. . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.4 Effect of anchor interface details 308. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8.5 COMPARISON OF EXISTING THEORETICAL AND LABORATORY RESULTS 308. 8.6 EFFECT OF ANCHOR ROUGHNESS 318. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7 EFFECT OF ANCHOR INCLINATION 323. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.8 EFFECT OF SOIL DILATION 340. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.9 CONCLUSIONS 350. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9. ANCHORS IN COHESIVE-FRICTIONAL SOIL 353. . . . . . . . . . . . . . . . . . . . 9.1 INTRODUCTION 354. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 UPPER BOUND MECHANISMS FOR ANCHOR PLATES IN COHESIVE-FRICTIONAL

SOIL 355. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 NUMERICAL ANALYSIS RESULTS 361. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9.3.1 Horizontal anchors 361. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.2 Vertical anchors 369. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.2.1 Effect of anchor roughness 372. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9.4 SUPERPOSITION ERROR 372. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5 CONCLUSIONS 375. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10. SQUARE, RECTANGULAR AND CIRCULAR ANCHORS 379. . . . . . . . . . . 10.1 INTRODUCTION 380. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 SQUARE ANCHORS IN PURELY COHESIVE SOIL 380. . . . . . . . . . . . . . . . . . . . . . . .

10.2.1 Effect of anchor roughness 384. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.2 Effect of overburden pressure 385. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.3 Effect of increasing strength with depth 387. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10.3 CIRCULAR ANCHORS IN PURELY COHESIVE SOIL 390. . . . . . . . . . . . . . . . . . . . . . 10.3.1 Effect of overburden pressure 394. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.2 Effect of anchor roughness 396. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.3 Effect of increasing strength with depth 396. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10.4 RECTANGULAR ANCHORS IN PURELY COHESIVE SOIL 398. . . . . . . . . . . . . . . . . 10.4.1 Effect of overburden pressure 400. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10.5 SUGGESTED PROCEDURE FOR ESTIMATING 402. . . . . . . . . . . . . . . . . . . . . . . . . . .

viii

10.5.1 Example of application 403. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.6 CIRCULAR & SQUARE ANCHORS IN COHESIONLESS SOIL 403. . . . . . . . . . . . . . .

10.6.1 Square anchors 404. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.6.2 Circular anchors 406. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10.7 CONCLUSIONS 415. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.7.1 Anchors in purely cohesive soils 415. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.7.2 Anchors in cohesionless soils 416. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11. CONCLUDING REMARKS 417. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 SUMMARY 418. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 ULTIMATE CAPACITY OF STRIP ANCHORS 419. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 ULTIMATE CAPACITY OF INCLINED STRIP ANCHORS 420. . . . . . . . . . . . . . . . . . . 11.4 ULTIMATE CAPACITY OF CIRCULAR, SQUARE & RECTANGULAR ANCHORS 42011.5 FUTURE WORK 421. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

REFERENCES 423. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ix

PREFACE

The research work presented in this thesis was conducted in the Department of Civil,

Surveying and Environmental Engineering at the University of Newcastle from March

1997 to Nov 2001. This work was performed under the supervision of Prof. Scott Sloan.

During the term of the candidature, the following papers were published:

1. Merifield R. S. , Sloan S. W. and Yu H. S. (1999). Rigorous plasticity solutions forthe bearing capacity of two-layered clays. Geotechnique, 49(4), 471-490.Awarded 2000 Telford Medal, Institution of Civil Engineers, London.

2. Merifield R. S. , Sloan S. W. and Yu H. S. (2001). Stability of plate anchors inundrained clay. Geotechnique, 51(2), 141-153.

3. Merifield R. S. , Lyamin A. V. , Sloan, S. W. , and Yu, H. S. (2001).Three-dimensional stability analysis of plate anchors in clay. Journal ofGeotechnical and Geoenvironmental Engineering, ASCE. (Accepted).

4. Merifield R. S. , Pearce A. , Sloan S. W. and Yu H. S. (1999). Stability of anchorplates. Australian Geomechanics, Journal and News of the AustralianGeomechanics Society, 34(2), 55-64.

5. Merifield R. S. , Lyamin A. V. , Sloan S. W. and Yu H. S. (1999). Three dimensionalstability analysis of plate anchors in clay., Proceedings of the 7th InternationalSymposium on Numerical Models in Geomechanics (NUMOG), Pande,Pietruszczak & Schweiger (eds), Balkema, Graz, 481-487.

6. Merifield R. S. , Sloan S. W. and Yu H. S. (1998). The bearing capacity of footingson layered clay. Proceeding of the 3rd Australia-New Zealand YoungGeotechnical Professionals Conference, Melbourne, 95-103.

7. Merifield R. S. , Pearce A. , Yu H. S. and Sloan S. W. (1999). Stability of anchorplates. Proceedings 8th Australia New Zealand Conference on Geomechanics,Hobart (1999) , Vol 2, 553-559.

x

8. Merifield R. S. , Sloan S. W. and Yu H. S. (2000). Limit Analysis Solutions for thePullout Capacity of Anchors in Sand. GeoEng2000 , An International Conferenceon Geotechnical & Geological Engineering (CDROM), Melbourne, Australia,November 2000.

9. Merifield R. S. , Lyamin A. V., Pearce A., Yu H. S. and Sloan S. W. (2001). PulloutCapacity of Earth Anchors. Proceedings of XVth ICSMGE, Istanbul, Turkey2001, Vol 2, 1203-1206.

10. Merifield R. S. , Sloan S. W. , Abbo A.W. , and Yu H.S. (2001). The Ultimate PulloutCapacity of Anchors in Frictional Soils. IACMAG, Proceedings of the 10thInternational Conference on Computer Methods and Advances in Geomechanics,Tucson, Arizona January 2001, Vol 2, 1187-1192.

xi

NOTATION

All variables used in this Thesis are defined as they are introduced into the text. For

convenience, frequently used variables are described below. The general convention

adopted is that vector and matrix variables are shown in bold print while scalar variables

are shown in italic.

ai vector of constraint variables.

A total matrix of equality constraint gradients.

A area of anchor plate.

b right hand side for linear equalities.

B anchor width.

cT objective function.

c soil cohesion.

c drained soil cohesion.

cu undrained soil cohesion.

c vector of objective function coefficients.

D problem dimensionality, diameter of anchor plate.

E total number of elements in finite element mesh, Youngs Modulus.

Eu Undrained Youngs modulus.

E Drained Youngs modulus.

f Yield function/criterion.

F, Fk Mohr-Coulomb yield criterion.

Fi total body force components.

Fs anchor suction force.

g , gi vector/components of prescribed body force.

xii

H anchor embedment depth.

Ha depth from ground surface to anchor centre line.

Hcr anchor critical embedment depth.

i anchor inclination factor.

I index set of equality constraints.

J index set of inequality constraints.

J1 , J2, J3 stress invariants.

Ko at-rest earth pressure coefficient.

L length of anchor plate.

M material properties of domain.

Nc, Nco, N, Nco, Nco, Nc, Nc anchor break-out factors.

N anchor roughness factor.

N total number of nodes in finite element mesh.

Nl shape function for node l.

O index set of optimisable force components.

P index set of prescribed force components.

Qn net anchor pull-out capacity.

Qu ultimate anchor pull-out capacity.

qu ultimate bearing capacity/pressure of the anchor.

q , qi vector/components of optimisable surface traction.

Q objective function.

R dilation correction factor.

s vector of deviatoric stresses.

t , ti vector/components of prescribed surface traction.

xiii

Ti total surface traction components.

uj, u.

j displacement/velocity components.

V volume of body, upper bound velocity vector.

Wa weight of anchor plate.

xi, xi global/local Cartesian coordinates.

X global vector of unknown stresses.

x problem variables, vector of stress variables.

parameter of hyperbolic approximation of Mohr-Coulomb yield criterion.

unit weight of soil.

anchor displacement, soil anchor interface roughness/friction.

, j Lagrange multiplies for inequality constraints.

, i Lagrange multipliers for equality constraints.

, , local (parametric) coordinates.

change in soil cohesion with depth dcudz.

ij stress tensor.

i principal stresses.

xyi vector of stress variables.

, ij vector of stress variables.

m, , stress invariants.

friction angle of soil.

drained friction angle of soil.

u undrained friction angle of soil.

, dilation angle of soil.

Poissons ratio.

xiv

Drained Poissons ratio.

u Undrained Poissons ratio.

ij , .

ij strain/strain rate tensor.

1Chapter 1

CHAPTER 1

INTRODUCTION

2 Chapter 1

1.1 INTRODUCTION

The design of many engineering structures requires foundation systems to resist vertical

uplift or horizontal pullout forces. In such cases, an attractive and economic design

solution may be achieved through the use of tension members. These members, which are

referred to as soil anchors, are typically fixed to the structure and embedded in the ground

to sufficient depth so that they can resist pull-out forces with safety. Soil or ground

anchors are a lightweight foundation system designed and constructed specifically to resist

any uplifting force or overturning moment placed on a structure.

As the range of applications for soil anchors continues to increase to include support for

substantially larger and more elaborate structures, greater demands are placed on anchor

design and performance. One such recent application includes providing mooring support

to floating systems for offshore oil and gas facilities. Unfortunately, research into the

behaviour of plate anchors has not kept up with overall performance demand and is very

limited in comparison to other foundation systems. In particular, the numerical study of

soil anchor behaviour has attracted limited attention.

Current understanding regarding the behaviour of buried foundations, and anchor plates

in particular, is somewhat unsatisfactory. The complex nature of anchor behaviour, and the

shear number of variables that influence soil uplift capacity, has meant there are many

conflicting theories reported in the literature. Most currently proposed theories have

significant underlying assumptions based on experimental observations regarding the

likely failure mode of anchors. Unfortunately it would appear these assumptions are

responsible for the general lack of overall agreement on soil uplift theory. The advantage

of using rigorous numerical methods to study anchor behaviour is that a good indication

of the likely failure mechanism can be obtained without any assumptions being made in

advance.

The purpose of this Chapter is to provide an introduction to the topic of soil anchors and

general soil uplift capacity, and an overview of the thesis. In this context, the types of

anchors currently available and the range of direct applications for soil anchors will be

presented. Mention is also made of less obvious applications for soil uplift theory.

3Chapter 1

1.2 SOIL ANCHORS

Typically, soil anchors are used to transmit tensile forces from a structure to the soil. Their

strength is obtained through the shear strength and dead weight of the surrounding soil.

The types of soil anchors used in civil engineering practice vary considerably, however in

general, anchors can be divided into four basic categories. These are :

Deadman anchors/Plate anchors.

Screw anchors.

Grout injected anchors. Anchor piles.

The method of load transfer from the anchor to the surrounding soil provides the distinction

between these various forms of anchorage. Load can be transferred to the soil through

direct bearing (plate anchors, screw anchors), shaft friction (grout injected anchors), or acombination of both direct bearing and shaft friction (anchor piles). In this thesis, anchorsthat obtain some capacity through shaft friction are not considered and discussion will be

limited to those anchors that obtain their strength through direct bearing. These anchors

will be referred to as plate anchors.

The range of applications for plate anchors has been widely reviewed in the literature (e.g.Das 1990) and includes the following:

Foundations for transmission towers, utility poles and marine moorings

(Figure 1.1(a)).

Tieback support for retaining structures (Figure 1.1(b)).

Break-out support for submerged pipelines and other structures subject to upliftpressures (Figure 1.1(c)).

As the range of applications for anchors expands to include the support of more elaborate

and substantially larger structures, a greater understanding of their behaviour is required.

Anchors are typically constructed from steel or concrete and may be circular (includinghelical), square or rectangular in shape. They may be placed horizontally, vertically, or atan inclined position depending on the load orientation or type of structure requiring

support. A general layout of the problem to be analysed is shown Figure 1.2.

4 Chapter 1

Transmission tower foundations

Figure 1.1 Applications for soil anchors.

or

Sheet Pile Wall

Plate/DeadmanAnchor

Tie rod or cable

backfill

Submerged Pipeline(b) (c)

(a)

Steel - Helical

Concrete

Plate anchors can be installed by excavating the ground to the required depth, placing the

anchor, and then backfilling with soil. For example, when used as a support for retaining

structures, anchors are installed in excavated trenches and connected to tie rods which may

be driven or placed through augered holes. This type of anchor is the subject of interestin this thesis.

5Chapter 1

D B

B

Circular anchor Square anchor (L=B) Strip anchor (L>>B)

(a)

(b)Horizontal anchor Vertical anchor

Figure 1.2 Anchor terminology (a) Anchor shapes, (b) Horizontal anchor,(c) Vertical anchor.

(c)

L

H embedment ratio = H/B

embedment ratio = H/B

H

B,D

B,D

In comparison, whilst still a direct bearing plate anchor, helical plate anchors are rotated

into the ground using truck mounted augering equipment. An axial load is applied to the

anchor whilst it is rotating. This type of plate anchor is becoming increasingly popular and

provides a versatile and cost effective support solution. To increase their capacity, helical

anchors often contain a number of helixes (see Figure 1.3). This type of anchor is notconsidered in the thesis.

1.2.1 In-direct application of anchor theory

The theory of soil uplift resistance may also be applied to geotechnical problems where

primary uplift resistance of a structure is not provided by the addition of soil anchors. In

6 Chapter 1

Multi-helix

Single-helix

Figure 1.3 Types of screw anchors.

fact, any buried foundation subjected to tensile loads may be considered to be an anchor.For example, a buried strip footing may be idealised as an anchor, provided the area of the

connecting member (in this case a column) is small compared to the overall size of thefooting. Other problems such as the uplift capacity of a buried pipeline can also be analysed

as a soil anchor. These types of problems are illustrated in Figure 1.4.

1.3 THESIS OUTLINE

The initial focus of the thesis is to provide a rigorous study into the behaviour of horizontal

and vertical plane strain strip anchors of infinite length (LB). Such a studyprovides an excellent insight into the general behaviour of plate anchors. Moreover, very

few numerical studies on plane strain anchors have fully addressed the large number of

variables that influence soil uplift capacity. This will become more apparent in Chapter 2.

As mentioned in the previous Section, anchors are typically square, circular or rectangular

in shape and the conditions of plane strain may not apply for all problems. Therefore, a

complete study of anchors should include a three dimensional analysis to determine the

effect of anchor shape. This will also be addressed and is one of the more significant

contributions made to anchor theory in the thesis.

7Chapter 1

failurezone

Strip FootingBuried Pipeline

Strip Anchor

Figure 1.4 Indirect application of anchor theory.

&

As an overview, the research presented in this Thesis can be divided into four principal

areas:

(1) The investigation of vertical and horizontal strip anchor capacity in purely cohesiveand cohesionless soil.

(2) The investigation of vertical and horizontal strip anchor capacity in cohesive-frictionalsoil.

(3) An investigation into the effect of anchor orientation on the anchor capacity in purelycohesive and cohesionless soil.

(4) The investigation and evaluation of the capacity of square, circular and rectangularanchors in purely cohesive and cohesionless soil.

The structure of the thesis reflects the four main topics listed above. Initially, Chapter 2

provides a background to subsequent Chapters by presenting a summary of research into

plate anchor behaviour.

Chapter 3 provides background to selected aspects of classical plasticity and discusses the

numerical formulations used in detail. In Chapter 4, more precise details are given as to

8 Chapter 1

how an anchor is studied using these numerical formulations. This includes a discussion

of the finite element mesh arrangements and anchor interface details.

Chapters 5 to 10 constitute the main portion of the thesis and present the results obtained

from the numerical studies for a wide range of anchor problems. A separate Chapter is

provided for each anchor analysis based on soil type, anchor orientation and anchor shape.

Where possible, a comparison is made between the results obtained in the current study and

existing theories and laboratory studies performed on soil anchors.

9Chapter 2

CHAPTER 2

HISTORICAL REVIEW

10 Chapter 2

2.1 INTRODUCTION

To provide a background to subsequent Chapters, a summary of research into plate anchor

behaviour is presented. A comprehensive overview on the topic of anchors is given by Das

(1990).

One of the earliest applications of soil anchors was in supporting transmission towers. This

application was responsible for the driving force behind a lot of the initial research into

anchor behaviour (Balla 1961). Initially these towers were supported by large deadweightconcrete blocks where the required uplift capacity was achieved solely due to the self

weight of the concrete. This simple design came at considerable cost and, as a result,

research was undertaken in order to find a more economical design solution. The result

was what is known as belled piers or mushroom foundations. As the range of applications

for anchors expanded to include the support of more elaborate and substantially larger

structures, a more concerted research effort has meant soil anchors today have evolved to

the point where they now provide an economical and competitive alternative to these mass

foundations.

Research into the behaviour of soil anchors can take one of two forms, namely experimental

or numerical/theoretical based studies. The brief summary of existing research herein has

been separated based on this distinction. No attempt is made to present a complete

bibliography of all research, but rather a more selective overall summary of research with

greatest relevance to the thesis is presented. For example, although portions of the theory

underlying the capacity of grouted anchors is also applicable to general soil uplift

behaviour, discussion is limited solely to the current thesis topic of plate anchors. In

addition, contributions made to the behaviour of multiple under-reamed or multi-helix

anchors has not been reviewed.

It will become clear that the majority of past research has been experimentally based and,as a result, current design practices are largely based on empiricism. In contrast, very few

thorough numerical analyses have been performed to determine the ultimate pullout loads

of anchors. Of the numerical studies that have been presented in the literature, few can be

considered as rigorous.

11Chapter 2

2.2 PREVIOUS EXPERIMENTAL INVESTIGATIONS

During the last thirty years various researchers have conducted laboratory studies to better

understand and predict the ultimate uplift capacity of anchors in a range of soil types. The

majority of this past experimental research has focused on predicting anchor behaviour andcapacity in cohesionless soil. The laboratory study of anchors embedded in undrained

purely cohesive soil has attracted only limited attention.

Although there are no entirely adequate substitutes for full-scale field testing, tests at

laboratory scale have the advantage of allowing close control of at least some of the

variables encountered in practice. In this way, trends and behaviour patterns observed in

the laboratory can be of value in developing an understanding of performance at larger

scales. In addition, observations made in laboratory testing can be used in conjunction withmathematical analyses to develop semi-empirical theories. These theories can then be

applied to solve a wider range of problems.

Experimental investigations into plate anchor behaviour have generally adopted one of two

approaches; namely, conventional methods under normal gravity conditions or

centrifuge systems.

Centrifuge systems use physical scaling laws to match the model and prototype behaviour

and can be used to study plate anchors. These investigations are based on generating soil

stress fields which are in proportion to the size of the model anchors. In this way, a

particular anchor size buried at a constant depth can also be used to investigate a range of

burial depths simply by varying the stress field. The stress field itself is induced through

centrifugal force, as the name suggests. While at rest, the set-up is subject to a staticgravitational force equal to 1g. By rotating the model in a centrifuge motion, gravitational

forces greater than 1g can be obtained which generate the required stress field and, in turn,

simulate insitu stresses for various burial depths. Unfortunately, because of the significant

equipment and set-up costs, only a few research institutions have such a tool at their

disposal.

In comparison to centrifuge testing, the more conventional gravity method is generally a

cheaper testing alternative due to the ease of set-up and the need for only simple equipment.

Quite often, a conventional gravity method can be incorporated into a civil engineering

12 Chapter 2

research laboratory by making use of existing equipment. Unfortunately, full scale testing

of foundations is costly, time consuming and in most cases unfeasible. For these reasons,

testing is generally limited to small scale model tests as these provide a cost effective and

convenient alternative. One concern associated with small scale conventional (normalgravity) testing is the presence of scale effects. These are most pronounced for cohesionlessgranular soils such as sands, and also afflict centrifuge testing.

Of course, both methods have advantages and disadvantages associated with them, and

these must be borne in mind when interpreting the results from experimental studies of

anchor behaviour.

The following sections provide a brief summary of past experimental research into the

behaviour of plate anchors in purely cohesive and cohesionless soil.

2.2.1 Anchors in purely cohesive soil

Laboratory based research into the capacity of anchors in purely cohesive soil is limited

in comparison to studies on anchors in purely frictional soil. The key reason for this is the

inherent difficulty and additional time required when preparing large clay samples for

testing. Samples need to be hydraulically consolidated or manually compacted in testing

chambers of various sizes which can be tedious and time consuming. This type of sample

preparation is far more difficult than that required to prepare sand samples for testing,

where simple raining or vibration techniques are used to achieve a particular density.

Interestingly, despite the obvious advantages in terms of sample size requirements, the

Author is unaware of any published centrifuge experiments on plate anchors in cohesive

soils.

In an attempt to reduce sample preparation times, many researchers choose to adopt small

scale model testing. Model anchors can be as small as 15mm in width/diameter, but most

researchers adopt anchors between 38mm and 50mm in size. The size of testing chamber

generally ranges from 500mm in width/diameter up to 1000mm.

Most of the results from studies of anchors in purely cohesive soil either consist of simple

approximate solutions or are derived empirically from laboratory model tests. These

results can be found in the works of Adams and Hayes (1967), Meyerhof and Adams(1968), Kupferman (1971), Vesic (1971, 1972), Meyerhof (1973), Das (1978,1980),

13Chapter 2

Ranjan and Arora (1980), and Das et al. (1985a,1985b,1989). The uplift capacity ofanchors is typically expressed in terms of a break-out factor, which is a function of the

anchor shape, embedment depth, overburden pressure and soil properties.

Initial research into plate anchor behaviour in purely cohesive soils revealed two modes

of failure, namely shallow and deep. An anchor is classified as shallow or deep depending

on the observed failure mechanism. Shallow anchors are characterised by a failure

mechanism which propagates to the ground surface, while deep anchors are characterised

by a failure mechanism that is localised around the anchor. In either case, the shape of the

true failure mechanism for anchors in cohesive or cohesionless soils remains unclear.

Earlier model tests on circular anchors have been performed by Spence (1965), Langley(1967), Adams and Hayes (1967), Ali (1968) and Kupferman (1971). In theseinvestigations, anchors were positioned horizontally (and pulled vertically) in remouldedsoils ranging from soft to stiff in strength. It was found that the anchor capacity increases

with embedment depth before finally reaching a constant value. This transition was

defined as deep anchor behaviour and occurred at embedment ratios HB ranging from

1.5 to 5.

Meyerhof and Adams (1968) and Meyerhof (1973) estimated break-out factors forhorizontal anchors based on a limited number of laboratory model tests. Meyerhof and

Adams (1968) performed laboratory tests using circular anchors, and Meyerhof (1973)performed tests on both circular and strip anchors. Using their results, a general theory for

the uplift resistance for both circular and strip anchors was proposed. This theory is

discussed in Section 2.3. Their solutions are only approximate, however, as they make

several key assumptions regarding the anchor failure mechanism and the earth pressure

distribution along the failure surface.

Meyerhof and Adams (1968) attempted to address the difference between short term(undrained) versus long term (drained) soil uplift capacity. Long term tests were performedby adding small increments of load each day until pullout occurred. Their results indicated

that in stiff clay the long term capacity is a small fraction of the short term capacity, whereas

in soft clay the long term capacity is a much higher percentage of the short term capacity.

It was suggested that if negative pore pressures develop in a clay under uplift loads, the soil

may soften and the long term capacity will be less than the short term capacity. This

14 Chapter 2

conjecture remains unproven, however, and is likely to be an issue only for anchors thatare close to the ground surface.

Vesic (1971) performed a number of laboratory pullout tests on horizontal circular plateanchors in soft and stiff clays and compared the results with the analytical solutions he

derived. Unfortunately, few details are presented regarding the number of tests, anchor

sizes or loading rates. It is believed that the laboratory results presented by Vesic may have

been from the study of Ali (1968), who investigated the pullout resistance of anchor platesand anchor piles in soft bentonite clay. Vesic reported a significant variation between the

theoretical and laboratory estimates of anchor capacity.

Das (1978,1980) provided tentative procedures, based on model laboratory tests, for theestimation of the ultimate uplift capacity of square, circular, and rectangular anchors

embedded horizontally in purely cohesive soil. These tests were mostly performed in soft

clays with a limited number of tests performed in stiff clays. The model anchors used had

widths of 38-50mm and lengths of 38-190mm and were vented at the base to eliminate

suction effects by the insertion of a hollow tube. Similarly, Ranjan and Arora (1980)performed model laboratory tests on vertical square, rectangular and strip anchors

embedded in very soft or soft clay.

Rowe (1978) studied the uplift behaviour of horizontal rectangular anchors, with the resultsbeing summarised by Rowe and Davis (1982a). Thirty uplift tests were performed onanchors with widths ranging from 13-38mm and lengths ranging from 64-190mm. This

equated to aspects ratios (L/B) between 3 and 8. A technique of underlaying the anchorwith filter paper was adopted to prevent adhesion between the underside of the anchor plate.

Hollow anchor rods were used to prevent the development of suction. This method is

similar to that used by Adams and Hayes (1967). Rowe and Davis concluded that anchorbehaviour can be divided into two categories, namely shallow anchor behaviour and deep

anchor behaviour. Shallow anchor behaviour, characterised by clearly defined collapse

loads, was observed for anchors with embedment ratios (HB) of less than or equal to 4.5.Deep anchor behaviour was observed when the embedment ratio is greater than 4.5. Rowe

and Davis observed that their laboratory findings showed encouraging agreement with

their theoretical solutions.

15Chapter 2

Although vertical plate anchors are widely used to support retaining structures, their load

capacity in purely cohesive soil has not received much attention in the literature. To the

Authors knowledge, the published works of MacKenzie (1955), Rangan and Arora (1980),and Das et al (1985a, 1985b) appear to summarise all the laboratory based research in thisarea. These Authors conducted a number of laboratory pullout tests on vertical anchors

with width to length ratios (L/B) varying from one (square) to five (rectangular) in very softto firm soils. Unfortunately, in these tests, the conditions behind the anchor are not clearly

defined and no attempt appears to have been made to measure or avoid the suction forces

that develop behind the anchor. It was observed that the anchor capacity increases with

embedment ratio before reaching a maximum value and remaining constant thereafter. Das

et al (1985a) defined the embedment depth at which time the anchor capacity reaches aconstant value as the critical embedment depth, and later presented simple empirical

relationships for estimating this value.

The important effect of anchor inclination has received very little attention by researchers.

A limited number of results for the capacity of inclined square and strip anchors can be

found in the works of Meyerhof (1973). The study of Das and Puri (1989) appears to bethe most significant attempt to quantify the capacity of inclined anchors. In their tests, the

capacity of shallow square anchors embedded in compacted clay with an average undrained

shear strength of 42.1kPa was investigated. Pullout tests were conducted on anchors at

inclinations ranging between 0(horizontal) and 90(vertical) for embedment ratios(HB) of up to four. A simple empirical relationship was suggested for predicting thecapacity of square anchors at any orientation which compared reasonably well with the

laboratory observations. Das and Puri (1989) also concluded that anchors with aspectratios (L/B) of 5 or greater would, for all practical purposes, behave as a strip anchor.

2.2.1.1 Other investigations

A number of Authors have conducted laboratory research into other aspects of plate anchor

behaviour. Parameters studied include the influence of soil suction, layered soil, sloping

ground and long term loading.

Stewart (1985) investigated the effectiveness of placing a cohesionless surcharge on theground surface in an attempt to increase the uplift capacity of circular horizontal anchors.

A limited number of pullout tests were conducted on shallow anchors in soft soil where

16 Chapter 2

HB 2. The results of these tests indicated the cohesionless overlay can lead to an

increase in the uplift capacity. This increase is composed of two parts; the first one is due

to the additional overburden pressure provided by the overlay, while the second one is due

to the mobilisation of the frictional resistance of the cohesionless overlay. It was concluded

that a shallow anchor in purely cohesive material can be transformed into a deep anchor

by placing a sufficient depth of cohesionless material on the ground surface. When deep

anchor conditions are established, the addition of more overburden has no effect on the

anchor capacity. In respect to mobilising the frictional capacity of the overlay, this could

be achieved only if the anchor was displaced almost entirely through the purely cohesive

material and into the overlay. The anchor would then behave as if it were embedded solely

within a cohesionless material. For this to occur, however, the displacements would be

unacceptably large for most foundation design requirements.

Soil suction effects were investigated by Das et al (1994) and Baba et al (1989). BothAuthors conducted a series of pullout tests on horizontal circular anchors in very soft soil

with and without venting to remove soil suction effects. It was concluded that the

magnitude of the suction can be significant and may be equivalent in magnitude to the net

capacity without suction for anchors at shallow embedment depths (Das et al (1994)).

The investigation of Baba et al (1989) was limited to anchors with an embedment depth(HB) of 6 while varying the soil moisture content and pullout rate. They concluded thatthe suction force component decreases with increasing moisture content and that the rate

of pullout significantly affects the suction force capacity. The observed suction force for

a given moisture content increased by as much as 200% when the pullout velocity was

varied from 1mm/min to 25 mm/min.

In the soil suction study of Das et al (1994), pullout tests were conducted in two differentsoils with significantly different moisture contents over a range of embedment ratios

varying from 1 to 5. The effect of pullout velocity was not investigated. Das concluded

that for a given clay and moisture content, the suction force generally decreases with an

increase in the embedment ratio. The ratio of the suction force to the net capacity of an

anchor without suction varied from about 1 at HB 1, to about 0.2 at HB 5. This

was true regardless of the type and amount of clay minerals present in the soil.

17Chapter 2

Rao & Prasad (1992) conducted a number of model tests on circular horizontal anchors insloped ground. The ground surface was varied between 0 and 45in 15 increments. As

expected, their results indicated that the anchor capacity decreases as the slope of the

ground surface increases from horizontal (0) to 45. The reduction in capacity wasobserved to be most significant for shallow anchors where HD 3. When deep anchor

failure is observed, the effect of ground slope is insignificant. Simple expressions are

suggested to calculate the uplift capacity based on the laboratory observations.

2.2.2 Anchors in cohesionless soil

Numerous investigators have performed model tests in an attempt to develop

semi-empirical relationships that can be used to estimate the capacity of anchors in

cohesionless soil. This is evidenced by the large number of studies shown in Table 2.1 and

Table 2.2. However, for the sake of brevity, discussions will be limited to those

investigations of greatest relevance to the thesis and/or those seen to have made the most

significant contribution to anchor uplift theory.

The works prior to 1970 have not been presented in Table 2.1 and Table 2.2. This includes

the field and/or model testing of horizontal circular anchors or belled piles by Mors (1959),Giffels et al (1960), Balla (1961), Turner (1962), Ireland (1963), Sutherland (1965),Mariupolskii (1965), Kananyan (1966), Baker and Konder (1966), and Adams and Hayes(1967). A number of these studies were primarily concerned with testing foundations fortransmission towers (Mors (1959), Balla (1961), Turner (1962), Ireland (1963)).

In the majority of earlier studies, a failure mechanism was assumed and the uplift capacitywas then determined by considering the equilibrium of the soil mass above the anchor and

contained by the assumed failure surface. Based on the underlying assumptions, these

methods of analysis can be separated into two approaches :

(1) The Soil cone method (Mors (1959)) in which the failure surface consists of atruncated cone extending from the anchor edges up to and intersecting the soil surface

at an angle of (45 2). The anchor capacity is assumed to equal the weight of

the soil contained within the area of the assumed failure surface. Any shearing

resistance developed along the failure surface is neglected.

(2) The Friction cylinder method (Downs & Chieurzzi (1966)), in which failure isassumed to occur along the surface of a cylinder of soil above the anchor. The anchor

18 Chapter 2

capacity is assumed to equal the sum of the weight of the soil contained within the area

of the assumed failure surface, and the frictional resistance derived along the failure

surface.

Table 2.1 Laboratory model tests on horizontal anchors in cohesionless soil.

Author Type ofTesting

Anchorshape

Anchorsize

Frictionangles

AnchorRoughness

H/B

Hanna & Carr (1971) Chamber CIRC 38mm 37 ? 4-1 12Hanna et al (1971) Chamber &

FieldCIRC 38mm &

150mm37 ? 4-1 12

Das & Seeley(1975a)

Chamber SQRECT

51mmL/B=1-5

31 ? 1-5

Rowe (1978) Chamber SQRRECT

51mm 32 16.7 1-8

Andreadis et al(1981)

Chamber CIRC 50mm -150mm

37, 42.5 ? 1-14

Ovesen (1981) Centrifuge& field

CIRCSQR

20mm 29.5- 37.7 ? 1-3.39

Murray & Geddes(1987)

Chamber CIRCRECT

L/B=1-10

50.8mm 44 Dense36 Med

11 smooth42 rough

1-10

Frydman & Shamam(1989)

FieldChamber

(Summary)

STRIPRECT

19mm200mm

30 Loose45 Dense

? 2.5-9.35

Dickin (1988) CentrifugeChamber

SQRRECT

L/B=1-8

25mm

50mm

38-41 *Loose

48-51 *Dense

? 1-8

Tagaya et al (1988) Centrifuge CIRCRECT

15mm 42 ? 3-7.02

Murray & Geddes(1989)

Chamber SQRRECT

L/B=1-10

50.8mm 43.6 Dense36 Med

dense

10.6 1-8

Sarac (1989) ? CIRCSQR

? 37.5, 48 ? 0.35-4

Bouazza & Finlay(1990)

Chamber CIRC 37.5mm 33.8, 39,43.7 Layered

? 2-5

Dickin (1994) Centrifuge STRIP &Pipes

25mmDia

38-41 Loose48-51 Dense

Yes 2-7

Sakai & Tanaka(1998)

Chamber CIRC 30mm -200mm

? ? 1-3

Pearce (2000) Chamber CIRC 50mm -125mm

Loose to veryDense

? 2-15

* Plane strain friction angle

Subsequent variations upon these early theories have been proposed including that of Balla

(1961) who determined the shape of slip surfaces for shallow horizontal anchors in densesand and proposed a rational method for estimating the capacity of anchors based on the

19Chapter 2

observed shapes of the slip surfaces. Baker and Kondner (1966) confirmed Ballas majorfindings regarding the behavioural difference of deep and shallow anchors in dense sand.

Sutherland (1965) presented results for the pull-out of 150mm horizontal anchors in looseand dense sand, as well as large diameter shafts in medium dense to dense sands. It was

concluded that the mode of failure varied with sand density and that Ballas analytical

approach may give reasonable results only in sands of intermediate density. Kananyan

(1966) presented results for horizontal circular plate anchors in loose to medium densesand. He also performed a series of tests on inclined anchors and observed the failure

surface, concluding that most of the soil particles above the anchor moved predominantly

in a vertical direction. In these tests, the ultimate capacity increased with the inclination

angle of the anchors.

Table 2.2 Laboratory model tests on vertical anchors in cohesionless soil.

Author Type ofTesting

Anchorshape

Anchorsize

Frictionangles

AnchorRoughness

H/B

Neely et al (1973) Chamber SQRRECT

50.8mm 38.5 21 1-5

Das (1975b) Chamber SQRCIRC

38-76mm 34 ? 1-5

Akinmusuru (1978) Chamber STRIPRECT

SQR CIRC

50mmL/B=2,10

24, 35 ? 1-10

Ovesen (1981) Centrifuge& field

SQR 20mm 29.5-37.7

? 1-3.39

Dickin & Leung (1983)(1985)

CentrifugeChamber

SQRRECTSTRIP

25mm,50mm

41* Polished29

1-81-13

Hoshiya & Mandal(1984)

SandChamber

SQRRECT

L/B=2,4,6

25.4mm 29.5 ? 1-6

Murray and Geddes(1989)

SandChamber

SQRRECT

L/B=1-10

50.8mm 43.6Dense

10.6 1-8

* Mobilised plane strain friction anglemp

Das and Seely (1975a) performed pull-out tests for horizontal rectangular anchors(LB 5) in dry sand with a friction angle of 31 at a density of 14.8kNm3. Foreach aspect ratio (LB), the anchor capacity was found to increase with embedment ratiobefore reaching a constant value at the critical embedment depth. A similar investigation

was conducted by Rowe (1978) in dry sand with friction angles 31 33, dilation

20 Chapter 2

angles 4 10, and dry unit weight of 14.9kNm3. Polished steel plates were

used for the anchors and the interface roughness was measures as 16.7. Most tests

were performed on anchors with an aspect ratio LB of 8.75. In each test collapse was

clearly defined, but in general there was little or no evidence of anchor failure at the soil

surface. Rowe concluded that decreasing the aspect ratio (LB) leads to increases inanchor capacity (relative to LB 8.75) of 10%, 25%, 35%, and 120% for LB ratiosof 5, 3, 2 and 1 respectively. That is, the effect of shape is significant for LB 2 and

is of little importance for LB 5. This suggests that anchors with aspect ratios of

LB 5 effectively behave as a continuous strip and can be compared with solutions that

assume plane strain conditions. In contrast to the observations of Das and Seeley (1975a),Rowe (1978) did not observe a critical embedment depth and the anchor capacity was foundto continually increase with embedment ratio over the range of HB 1 to 8.

Extensive chamber testing programs have been performed by Murray and Geddes (1987,1989), who performed pull-out tests on horizontal strip, circular, and rectangular anchorsin dense and medium dense sand with 43.6and 36 respectively. Anchors

were typically 50.8mm in width/diameter and were tested at aspect ratios (LB) of 1, 2,5 and 10. Based on their observations, Murray and Geddes made several conclusions: (1)the uplift capacity of rectangular anchors in very dense sand increases with embedment

ratio and with decreasing aspect ratio LB ; (2) there is a significant difference betweenthe capacity of horizontal anchors with rough surfaces compared to those with polished

smooth surfaces (as much as 15%); (3) experimental results suggest that an anchor with anaspect ratio of LB 10 behaves like a strip and does not differ much from an anchor with

LB 5, and; (4) the capacity of circular anchors in very dense sand is approximately1.26 times the capacity of square anchors. Several of these conclusions confirm the

findings of Rowe (1978). It is also of interest to note that for all the tests performed byMurray and Geddes, no critical embedment depth was observed.

More recently, Pearce (2000) performed a series of laboratory pullout tests on horizontalcircular plate anchors pulled vertically in dense sand. These tests were conducted in a large

calibration chamber, with dimensions one meter in height and one meter in diameter.

Various parameters such as anchor diameter, pullout rate and elasticity of loading system

have been investigated. The model anchors used for the pullout tests varied in diameter

from 50-125mm and were constructed from 8mm mild steel. Large diameter anchors were

21Chapter 2

chosen (compared with previous research) due to the recognised influence of scale effectson the break-out factor for anchors of diameters less than 50mm (Andreadis et al, 1981).

Although not as popular as chamber testing, centrifuge testing of anchors has been

undertaken by a number of Authors (see Table 2.1). Dickin (1988) performed 41 tests on25mm anchor plates with aspect ratios of LB 1, 2, 5 and 8 at embedment ratios HB

up to 8 in both loose and dense sand. A number of conventional gravity tests were also

performed and compared to the centrifuge results. This comparison revealed a significant

difference between the estimated anchor capacities, particularly for square anchors where

the conventional test results gave anchor capacities up to twice that given by the centrifuge.

Without explaining why, Dickin concluded that direct extrapolation of conventional

chamber test data to field scale would provide over-optimistic predictions of the ultimate

capacity for rectangular anchors in sand.

Tagaya et al (1988) also performed centrifuge testing on rectangular and circular anchors,although the study was limited in comparison to that of Dickin (1988) discussed above.

In contrast to the case of horizontal anchors, experimental investigations into the behaviour

of vertical anchors is limited, as evidenced by Table 2.2.

Neely et al (1973) reported results for small scale testing on vertical square and rectangularanchors in sand with a friction angle of 38.5. Anchors at aspect ratios 1 (square), 2 and5 were used and embedded up to HB 5. Very large displacements were observed for

square anchors when the embedment ratio was greater than 2. In fact it appears that the

load-displacement curves were still increasing when the test was terminated, and an

alternative criteria was used to define the ultimate load.

The capacity of deeper vertical anchors was reported by Akinmusuru (1978). Square,circular and rectangular anchors (LB 1, 10) were tested at embedment ratios rangingfrom 1 to 10. In a novel attempt to better observe the failure mechanism for anchors at

LB 10, the soil was simulated by steel pins (76mm length) placed to give a frictionangle of 24. The movement of the pins during each test was photographed with the aid

of long exposure film. It was observed that at HB 6.5 the failure mechanism does not

reach the soil surface and is a near circular shape immediately above the anchor. Although

this clearly defines the critical embedment depth, the anchor capacity continued to increase

22 Chapter 2

above HB 6.5 and no peak load was observed. For the remaining anchor shapes,

Akinmusuru placed the anchors in medium dense sand.

Dickin and Leung (1983,1985) conducted a very thorough investigation into the behaviourof vertical square and rectangular anchors in dense sand. Both centrifuge and conventional

chamber testing was performed and their results compared. It was found that a significant

difference existed between the conventional and centrifuge test results, with the former

being lower than the latter. In particular, the behaviour of small model anchors tested

conventionally at low stress levels is lower than that for anchors tested in a centrifuge at

stress levels similar to that expected in the field. In the worst case, the difference in the

estimated capacity for a square anchor varied by as much as 80%. The Authors concluded

that these differences stem from the inherent difficulty in obtaining reliable data from

conventional model tests, and the relevant shear strength tests at low stress levels.

Consequently, it was suggested that the results obtained from the centrifuge tests would

provide a more reliable basis for full scale anchor design.

Hoshiya & Mandal (1984) also investigated the capacity of square and rectangular anchors,but focused on their behaviour in loose sand. It is worth mentioning the small size of the

sand box used for testing, which measured only 300mm wide and 400mm long, is likely

to introduce edge effects into the results. The Authors concluded that the anchor break-out

factor increases with depth up to a certain embedment ratio before reaching a constant value

thereafter.

In addition to testing horizontal anchors, the extensive investigations of Murray and

Geddes (1989) also included pullout tests on vertical and inclined anchors in dense sand.In contrast to the findings of Hoshiya & Mandal (1984), no critical embedment depth wasobserved and the break-out factors were continually rising at HB 7. For inclined

anchors, they observed that the ultimate anchor capacity increases with embedment depth

and also the angle of loading. The greatest changes in capacity were observed between

loading angles of 45 and 90(vertical).

23Chapter 2

2.3 PREVIOUS THEORETICAL ANALYSES

In contrast to the variety of experimental results already discussed, very few rigorous

numerical analyses have been performed to determine the pullout capacity of anchors in

soil. This is particularly the case for anchors in purely cohesive soils.

Whilst it is essential to verify theoretical solutions with experimental studies wherever

possible, results obtained from laboratory testing alone are typically problem specific. This

is particularly the case in geomechanics where we are dealing with a highly nonlinear

material which often displays pronounced scale effects. As a result, it is often difficult to

extend the findings from laboratory research to full scale problems with different material

or geometric parameters. Since the cost of performing laboratory tests on each and every

field problem combination is prohibitive, it is necessary to be able to model soil uplift

resistance numerically for the purposes of design.

Existing numerical analyses generally assume a condition of plane strain for the case of a

continuous strip anchor or axi-symmetry for the case of circular anchors. The Author is

unaware of any three dimensional numerical analyses to ascertain the effect of anchor shape

on the uplift capacity.

2.3.1 Anchors in purely cohesive soil

The most rigorous numerical study of anchors in purely cohesive soils is due to Rowe and

Davis (1982a). In their paper, results were presented for both horizontal and vertical stripanchors embedded in homogeneous saturated purely cohesive soil. These were obtained

using an elasto-plastic finite element analysis which incorporated soil-structure interaction

theory at the soil/anchor boundary. The effect of anchor roughness, thickness and shape

were also considered.

Other displacement finite element studies on the behaviour of circular anchors in purely

cohesive soil have been made by Ashbee (1969), Davie and Sutherland (1977) andDewaikar (1988), although very limited results were reported.

In an early study, Vesic (1971) proposed an analytical approach for the pullout capacity ofhorizontal anchors, based on the solutions of Vesic et al. (1965) for the problem of anexpanding cavity close to the surface of a semi-infinite rigid plastic solid. These solutions

24 Chapter 2

give the ultimate radial pressure needed to break out a cylindrical or spherical cavity

embedded at a depth below the surface of a solid. The pullout capacities for strip and

circular anchors were then derived by assuming the pullout load was equivalent to the

ultimate cylinder and spherical cavity pressure, plus the weight of soil acting directly above

the anchor.

More recently, Yu (2000) derived an expression for the break-out factor based on moreaccurate analytical solutions for cavity expansion in cohesive-frictional soil. In this

solution it is assumed that break-out occurs if the boundary of the plastic zone (due to theanchor pullout action) predicted by cavity expansion theory is sufficiently close to or onthe ground surface. In other words, plate anchors break out when the plastic flow is not

confined by the outer elastic zone.

Although the limit theorems provide a simple and useful way of analysing the stability of

geotechnical structures, they have not been widely applied to the problem of anchors in soil.

Rowe (1978) and Gunn (1980) used the bound theorems to produce solutions for the caseof a horizontal strip anchor and trapdoor respectively. Due to the difficulty in manually

constructing statically admissible (lower bound) stress fields and kinematically admissible(upper bound) velocity fields, it is often difficult to bracket the pullout capacity to sufficientaccuracy. Rowes solutions, however, will later prove to be of good accuracy for the case

of deep anchor failure.

More recently, Kumar (1999) proposed a kinematic approach for the uplift of stripfoundations in clay. The method is based on the upper bound theorem of limit analysis and

satisfies the kinematic admissibility of the chosen collapse mechanism. The actual

break-out factors were given as a function of partial strength parameters which were

adopted as a means of accounting for the partial mobilisation of the material strength

throughout the soil mass.

2.3.2 Anchors in cohesionless soil

A summary of previous studies for horizontal and vertical anchors is provided in Table 2.3

and Table 2.4 respectively.

25Chapter 2

Table 2.3 Theoretical studies on horizontal anchors in cohesionless soil.

Author Analysis Method Anchorshape

AnchorRoughness

FrictionAngles

H/B

Meyerhof & Adams(1968)

Limit Equilibrium -Semi-analytical

STRIPSQR/CIRC

? - -

Vesic (1971) Cavity Expansion STRIP/CIRC

? 0 - 50 0-5

Rowe & Davis (1982b) Elastoplastic FiniteElement

STRIP Smooth 0 - 45 1-8

Vemeer & Sutjiadi(1985)

Elastoplastic FiniteElement/Upper bound

STRIP ? All 1-8

Tagaya et al (1988)Tagaya et al (1983)

Elastoplastic FiniteElement

CIRC/RECTL/B=2

? 31.6,35.142

0-30

Saeedy (1987) Limit Equilibrium CIRC ? 20-45 1-10Murray & Geddes

(1987)Limit Analysis &Limit Equilibrium

STRIPRECTCIRC

? All All

Koutsabeloulis &Griffiths (1989)

Finite Element -Initial Stress Method

STRIP/CIRC

? 20,30,40 1-8

Sarac (1989) Limit Equilibrium CIRC/SQR ? 0-50 1-4Basudhar & Singh

(1994)Limit Analysis -

Lower boundSTRIP Rough/

Smooth32 1-8

Kanakapura et al (1994) Method of Characteristics


Ghaly & Hanna (1994) Limit Equilibrium CIRC ? 30-46 1-10Smith (1998) Limit Analysis -

Lower boundSTRIP Rough ? 25 - 50 1-28

Sakai & Tanaka (1998) Elastoplastic FiniteElement

CIRC ? Dense 1-3

An approximate semi-empirical theory for the uplift capacity of horizontal strip, circular,

and rectangular anchors has been proposed by Meyerhof and Adams (1968). For a stripanchor, an expression for the ultimate capacity was obtained by considering the

equilibrium of the block of soil directly above the anchor (i.e. contained within the zonecreated when vertical planes are extended from the anchor edges). The cohesive force wasassumed to act along the vertical planes extending from the anchor edges, while the total

passive earth pressure was assumed to act at some angle to these vertical planes. This angle

was selected based on laboratory test results while the passive earth pressures were

evaluated from the results of Caquot and Kerisel (1949). For shallow anchors where thefailure surface extends to the soil surface, the ultimate capacity was determined by

considering equilibrium of the material between the anchor and soil surface. For a deep

anchor the equilibrium of a block of soil extending a vertical distance H above the anchor

26 Chapter 2

was considered, where H was less than the actual embedment depth of the anchor. The

magnitude of H was determined from the observed extent of the failure surface from

laboratory tests.

Table 2.4 Theoretical studies on vertical anchors in cohesionless soil.

Author Analysis Method Anchorshape

AnchorRoughness

FrictionAngles

H/B

Biarez et al (1965) Limit equilibrium STRIP ? All AllMeyerhof (1973) Limit Equilibrium -

Semi-analyticalSTRIP ? All All

Neely et al (1973) Limit equilibrium &Method of Character-

istics

STRIP Rough, /2 30 - 45 1-5.5

Rowe & Davis (1982b) Elastoplastic FiniteElement


Hanna et al (1988) Limiting Equilibrium STRIPInclined

? All All

Murray & Geddes (1989) Limit Analysis -Upper Bound

STRIPInclined

Smooth/Rough

43.6 1-8

Basudhar & Singh(1994)

Limit Analysis -Lower bound

STRIP Rough/Smooth

32, 38,35

1-5

The analysis of strip footings was extended by Meyerhof and Adams to include circular

anchors by using a semi-empirical shape factor to modify the passive earth pressure

obtained for the plane strain case. The failure surface was assumed to be a vertical

cylindrical surface through the anchor edge and extending to the soil surface. An

approximate analysis for the capacity of rectangular anchors was obtained as for downward

loads (Meyerhof 1951), by assuming the earth pressure along the circular perimeter of thetwo end portions of the failure surface is governed by the same shape factor adopted for

circular anchors.

The paper by Meyerhof and Adams (1968) is widely referenced when considering thecapacity of anchors. It is, however, based on two key assumptions; namely, the shape of

the failure surface and the distribution of stress along the failure surface. Even so, the

theory presented by Meyerhof and Adams (1968) has been found to give reasonableestimates for a wide range of anchor problems. It is one of only two methods available for

estimating the capacity of rectangular anchors.

Neely et al. (1973) used both a trial failure surface approach and the method ofcharacteristics to analyse a vertical strip anchor in a cohesionless material. In the first

method, a trial surface was adopted which consisted of a combination of straight lines and

27Chapter 2

logarithmic spirals. It was assumed initially that the soil above the level of the top of the

anchor would act as a simple surcharge. This was defined as the Surcharge method of

analysis. However, since this approach ignores the shearing resistance of the soil above

the anchor, the approach was modified by incorporating the strength of the soil above the

anchor through what was termed an equivalent free surface. This method was defined asthe Equivalent free surface method. It should be noted that although the analysis adopted

by Neely et al represents a more analytical attempt to predict the ultimate capacity of

vertical anchors than any preceding work, the proposed methods ignore the active stress

distribution behind the anchor and the kinematic behaviour of the material. Consequently

the results are considered as approximate only.

In a comprehensive paper, Rowe and Davis (1982b) described a theoretical investigationof anchors in cohesionless soils which considered the effect of anchor embedment, soil

friction angle, soil dilatancy, initial stress state and anchor roughness for both horizontal

and vertical anchors. Their theoretical solution was based on an elasto-plastic finite

element analysis using a soil structure interaction theory. The soil was assumed to have

a Mohr-Coulomb failure criterion and either an associated or non-associated flow rule. The

theoretical results were presented in the form of design charts which could be used in hand

calculations to obtain an estimate of anchor capacity for a wide range of anchor geometries

and soil types. To date this is by far the most rigorous attempt to study plate anchors in

cohesionless soil, and for that reason will be referenced throughout the thesis.

The finite element method has also been used by Vemeer & Sutjiadi (1985), Tagaya et al(1983,1988), and Sakai and Tanaka (1998). Unfortunately, only limited results werepresented in these studies.

Tagaya et al (1983,1988) conducted two-dimensional plane strain and axi-symmetric finiteelement analyses using the constitutive law of Lade and Duncan (1975). Scale effects forcircular anchors in dense sand were investigated by Sakai and Tanaka (1998) using aconstitutive model for a non-associated strain hardening-softening elasto-plastic material.

The effect of shear band thickness was also introduced.

Koutsabeloulis and Griffiths (1989) investigated the trapdoor problem using the initialstress finite element method. Both plane strain and axi-symmetric studies were conducted.

The Authors concluded that an associated flow rule has little effect on the collapse load for

28 Chapter 2

strip anchors but a significant effect (30%) for circular anchors. Large displacements wereobserved for circular anchors prior to collapse.

The remaining numerical studies shown in Table 2.3 and Table 2.4 estimate the anchor

capacity using either the Limit Equilibrium Method (LEM) or method of Limit Analysis.

In the LEM, an arbitrary failure surface is assumed along with a distribution of stress along

the assumed surface. Equilibrium conditions are then considered for the failing soil mass

and an estimate of the collapse load is obtained. In the study of horizontal anchor capacity,

the failure mechanism is generally assumed to be log spiral in shape (Saeedy (1987), Sarac(1989), Murray and Geddes (1987), Ghaly and Hanna (1994)) and the distribution of stressis obtained by using either Kotters equation (Balla (1961)), or by making an assumptionregarding the orientation of the resultant force acting on the failure plane.

Upper and lower bound limit analysis techniques have been used used by Murray and

Geddes (1987,1989), Basudhar and Singh (1994) and Smith (1998) to estimate the capacityof horizontal and vertical strip anchors. Basudhar and Singh (1994) obtained estimatesusing a generalized lower bound procedure based on finite elements and non-linear

programming similar to that of Sloan (1988a). The solutions of Murray and Geddes(1987,1989) were obtained by manually constructing kinematically admissible failuremechanisms (upper bound), while Smith (1998) presented a novel rigorous limiting stressfield (lower bound) solution for the trapdoor problem.

2.3.2.1 Other investigations

A number of Authors have conducted laboratory research into other aspects of plate anchor

behaviour, including the effects of initial stress state (Hanna and Carr (1971), Hanna et al(1972), Hanna & Ghaly (1992)) and plate flexibility (Rahman et al (1992), Othman et al(1993)) .

Hanna and Carr (1971) and Hanna et al. (1971) demonstrated that the stress history of asand can significantly affect the load carrying capacity of an anchor. Laboratory tests were

conducted on sands with an overconsolidation ratio (OCR) as high as 14, but no attemptwas made to mathematically quantify the effect of OCR on the uplift capacity. This

shortcoming was addressed by Hanna and Ghaly (1992) who conducted experimentalpullout tests on circular screw anchors in sand to determine the effects of the coefficient

29Chapter 2

of earth pressure at rest (Ko) and OCR. Although this study was limited to helical screwtype anchors, a number of interesting observations were reported. Firstly, the Authors

revealed it was inherently difficult to reconstruct a residual stress profile in the laboratory

that closely models what is observed in the field. It was discovered that by compacting the

sand in layers inside a sand chamber, a deposit is created in which the OCR increases with

depth. This is not what is observed in a naturally occurring deposit. Secondly, it was

suggested that the anchor capacity increases with increasing OCR.

Rahman et al (1992) conducted a numerical and experimental investigation into theinfluence of plate flexibility on the behaviour of shallow circular anchors in sand. They

concluded that plate flexibility has a significant effect on the observed failure mechanism,

displacement field within the soil mass, and the load-displacement relationship. The

capacity of more flexible anchors was found to be marginally higher than that for thick rigid

plate anchors.

2.4 SUMMARY

A number of conclusions can be drawn from the forgoing review into plate anchor research:

(1) The majority of past research has been experimentally based, as evidenced by the largenumber of studies shown in Table 2.1 and Table 2.2. Unfortunately, results obtained

from laboratory testing are typically problem specific and are difficult to extend to

field problems with different material or geometric parameters. Moreover, a lack of

reported experimental data often makes comparisons with theory difficult.

(2) Very few rigorous numerical studies have been undertaken to determine anchorbehaviour. It is generally agreed that existing theories do not adequately describe the

behaviour of anchor plates (Sutherland (1988)). Most methods of analysis are basedupon the initial assumption of a particular failure mode (limit equilibrium method andupper bound limit analysis). Given that few attempts have been made to accuratelymonitor internal soil deformations under laboratory conditions, the validity of the

assumed failure mechanisms remain largely unproven. A rigorous numerical study of

soil anchors using advanced numerical methods is clearly needed.

(3) No attempt has been made to determine the capacity of anchors in inhomogeneouspurely cohesive soil, even though this is a common field characteristic.

30 Chapter 2

(4) Most anchor studies have been concerned with either vertical or horizontal re

numerical modelling of soil anchors

Documents

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plate anchors

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engineering structures

rigorous numerical analyses

comprehensive numerical