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Passive earth pressure in sand on inclined walls with negative wall friction based on a statically admissible stress field

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Abstract

The problem considered in this paper is the passive earth pressure determination under various rigid wall geometry with negative wall-soil friction. An exact solution based on statically admissible stress fields is employed to evaluate the impact of negative wall-soil friction and inclined walls on the design of retaining walls. A series of conservative solutions for the passive earth pressure coefficient in the presence of negative wall friction is obtained, shedding light on the load transfer mechanism of cohesionless backfill materials. While focusing on the passive earth pressure coefficient, the study further identifies the stress state transition zone in the backfill and the influence of wall inclination on passive earth pressure. In addition, the load transfer mechanism of cohesionless backfill material in a retaining wall is affected by various factors, including wall geometry, degree of negative wall friction, the direction of the major principal stress at the wall, and the position and orientation of the transition zone and line of stress discontinuity. It follows that comprehensive design charts involving various rigid wall geometry and negative wall-soil friction are provided for engineering practice. Overall, the study successfully characterizes the admissible stress field in the backfill of rigid retaining walls, contributing to the knowledge of retaining wall design under the effect of negative wall friction.

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Data availability

All data, models, or code that support the findings of this study are available from the corresponding author upon reasonable request.

Abbreviations

x, z :

Coordinates in a rectangular coordinate system (xOz)

r, θ :

Radial and angular coordinates in a polar coordinate system (r, θ)

σ x, σ z, τ xz :

Horizontal, vertical, and shear stress components in rectangular coordinates (xOz) (compression taken as positive)

σ r, σ θ, τ r θ :

Radial, angular, and shear stress components in polar coordinates (r, θ) (compression taken as positive)

σ 1, σ 3 :

Major and minor principal stresses

p :

In-plane mean stress, p = (σr + σθ)/2; p1 is the mean stress on the line of stress discontinuity located on the side of the Rankine region (Ψ is constant) while p2 is that in the side of the non-Rankine region

±β (angle in degree):

Backfill inclination angle: (+) upward inclination, (−) downward inclination

±λ :

Inclination angle of wall rear face: (+) inclined inward to the backfill (counterclockwise direction from vertical axis) (−) inclined outward to the backfill (clockwise direction from vertical axis)

δ (angle in degree):

Wall-soil friction angle

γ :

Unit weight of the cohesionless material of backfill (assumed uniform and constant throughout the backfill of retaining walls)

ϕ (angle in degree):

The internal friction angle of the cohesionless material of backfill (value in design) which is assumed to be equal to the angle of repose of dry bulk solids in the backfills; ϕp (value at the peak state); ϕcv (value at the critical void state)

χ :

Non-dimensional mean stress

ψ (angle in degrees):

Relative angle of the major principal stress made by the direction of major principal stress with the radial direction (positive sign for counterclockwise direction)

ψ w (angle in degrees):

Relative angle of the major principal stress on the wall rear face (θ = λ)

θ D (angle in degrees):

Location (angular coordinate) of the stress discontinuity line in the polar coordinate system

Ψ (angle in degrees):

Angle of the major principal stress made by the direction of the major principal stress with the gravitational direction (positive sign for counterclockwise direction)

Ψ1 :

is the direction of the major principal stress on the line of stress discontinuity located on the side of the Rankine region (Ψ is constant) while Ψ2 is that on the side of the non-Rankine region

K p :

Passive earth pressure coefficient

ϒ (angle in degrees):

angle of dilatancy, ϒ = ϕ represented for an associated flow rule material

μ (angle in degrees):

Angle of the direction of stress characteristic to the direction of the major principal stress

C 1 and C 2 :

represent the Rankine and non-Rankine regions in the backfill separated by a line of stress discontinuity

A 1 and A 2 :

denote the Rankine and non-Rankine regions in the backfill separated by a line of stress characteristic

References

  1. Booker JR (1969) Application of theories of plasticity to cohesive frictional soils

  2. Budhu M (2010) Soil mechanics and foundations. Wiley, Hoboken

    Google Scholar 

  3. Clough G, Duncan J (1991) Earth pressures. Foundation engineering handbook. Springer, New York, pp 223–35

    Book  Google Scholar 

  4. Coulomb CA (1773) Essai sur une application des regles de maximis et minimis a quelques problemes de statique relatifs a 1'architecture. Mem Div Sav Acad

  5. Chu S-C (1991) Rankine’s analysis of active and passive pressures in dry sands. Soils Found 31(4):115–120. https://doi.org/10.3208/sandf1972.31.4_115

    Article  Google Scholar 

  6. Duncan JM, Mokwa RL (2001) Passive earth pressures: theories and tests. J Geotech Geoenviron Eng 127(3):248–257. https://doi.org/10.1061/(asce)1090-0241(2001)127:3(248)

    Article  Google Scholar 

  7. Gutberlet C, Katzenbach R, Hutter K (2013) Experimental investigation into the influence of stratification on the passive earth pressure. Acta Geotech 8(5):497–507. https://doi.org/10.1007/s11440-013-0270-3

    Article  Google Scholar 

  8. James RG, Bransby PL (1970) Experimental and theoretical Investigations of a passive earth pressure problem. Géotechnique 20(1):17–37. https://doi.org/10.1680/geot.1970.20.1.17

    Article  Google Scholar 

  9. Kérisel J, Absi E (1990) Tables for the calculation of passive pressure, active pressure and bearing capacity of foundations. Gauthier-Villard, Paris

    Google Scholar 

  10. Kumar J, Rao KSS (1997) Passive pressure coefficients, critical failure surface and its kinematic admissibility. Géotechnique 47(1):185–192. https://doi.org/10.1680/geot.1997.47.1.185

    Article  Google Scholar 

  11. Lee E (1950) On stress discontinuities in plane plastic flow. In: Elasticity: proceedings of the third symposium in applied mathematics of the American mathematical society: McGraw-Hill: New York, p 213

  12. Liu S, Xia Y, Liang L (2018) A modified logarithmic spiral method for determining passive earth pressure. J Rock Mech Geotech Eng 10(6):1171–1182. https://doi.org/10.1016/j.jrmge.2018.03.011

    Article  Google Scholar 

  13. Nguyen T (2018) Load transfer mechanisms and seismic stability of embankments subjected to basal subsidence

  14. Nguyen T (2022) Passive earth pressures with sloping backfill based on a statically admissible stress field. Comput Geotech 149:104857. https://doi.org/10.1016/j.compgeo.2022.104857

    Article  Google Scholar 

  15. Nguyen T (2023) An exact solution of active earth pressures based on a statically admissible stress field. Comput Geotech 153:105066. https://doi.org/10.1016/j.compgeo.2022.105066

    Article  Google Scholar 

  16. Nguyen T (2023) Statically admissible stress fields in conical sand valleys and heaps: a validation of Haar–von Kármán Hypothesis. Int J Geomech. https://doi.org/10.1061/ijgnai.gmeng-7863

    Article  Google Scholar 

  17. Nguyen T, Pipatpongsa T (2020) Plastic behaviors of asymmetric prismatic sand heaps on the verge of failure. Mech Mater 151:103624. https://doi.org/10.1016/j.mechmat.2020.103624

    Article  Google Scholar 

  18. Nguyen T, Pipatpongsa T, Kitaoka T, Ohtsu H (2019) Stress distribution in conical sand heaps at incipient failure under active and passive conditions. Int J Solids Struct 168:1–12. https://doi.org/10.1016/j.ijsolstr.2018.04.001

    Article  CAS  Google Scholar 

  19. Poncelet J (1840) Memoire sur la stabilite des revetments er de loure foundations. Note addition elle sur les relatins analytiques qui lient entre ells la poussee et la butee de la terre. Memorial de l’officiere de genie, Paris

  20. Rankine WJM (1857) On the stability of loose earth. Philosophical transactions of the Royal Society of London. Phil Trans R Soc 1857(147):9–27

    Google Scholar 

  21. Savage SB, Yong RN, McInnes D (1969) Stress discontinuities in cohesionless particulate materials. Int J Mech Sci 11(7):595–602. https://doi.org/10.1016/0020-7403(69)90058-7

    Article  Google Scholar 

  22. Schmüdderich C, Tschuchnigg F, Schweiger HF (2021) Significance of flow rule for the passive earth pressure problem. Acta Geotech 17(1):81–92. https://doi.org/10.1007/s11440-021-01193-x

    Article  Google Scholar 

  23. Schofield AN (2006) Interlocking, and peak and design strengths. Géotechnique 56(5):357–358. https://doi.org/10.1680/geot.2006.56.5.357

    Article  Google Scholar 

  24. Shiau JS, Augarde CE, Lyamin AV, Sloan SW (2008) Finite element limit analysis of passive earth resistance in cohesionless soils. Soils Found 48(6):843–850. https://doi.org/10.3208/sandf.48.843

    Article  Google Scholar 

  25. Shield R (1953) Mixed boundary value problems in soil mechanics. Q Appl Math 11(1):61–75

    Article  MathSciNet  Google Scholar 

  26. Shields DH, Tolunay ZA (1972) Passive pressure coefficients for sand. Can Geotech J 9(4):501–503. https://doi.org/10.1139/t72-048

    Article  Google Scholar 

  27. Sokolovskii VV (1965) Statics of granular media. Pergamon

  28. Taylor DW (1948) Fundamentals of soil mechanics. vol 2

  29. Terzaghi K (1943) Theoretical Soil Mechanics. Wiley, New York, p 314

    Book  Google Scholar 

  30. Xu S-Y, Lawal AI, Shamsabadi A, Taciroglu E (2018) Estimation of static earth pressures for a sloping cohesive backfill using extended Rankine theory with a composite log-spiral failure surface. Acta Geotech 14(2):579–594. https://doi.org/10.1007/s11440-018-0673-2

    Article  Google Scholar 

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Acknowledgements

This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant Number NCUD.02-2022.08. This funding is gratefully acknowledged. Gratitude is extended to the anonymous reviewers for your time and effort thus giving us immensely valuable comments. Finally, the insightful discussions from Prof. Thirapong Pipatpongsa (National Yang Ming Chiao Tung University, Taiwan) is gratefully acknowledged. 

Author information

Authors and Affiliations

  1. Smart Computing in Civil Engineering Research Group, Ton Duc Thang University, Ho Chi Minh City, Vietnam

    Tan Nguyen

  2. Faculty of Civil Engineering, Ton Duc Thang University, Ho Chi Minh City, Vietnam

    Tan Nguyen

  3. School of Engineering, University of Southern Queensland, Toowoomba, QLD, 4350, Australia

    Jim Shiau

Authors
  1. Tan Nguyen
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  2. Jim Shiau
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Contributions

TN contributed to conceptualization, methodology, formal analysis, investigation, data curation, writing—original draft, writing—review and editing, and fund acquisition. JS contributed to conceptualization, investigation, data curation, writing—original draft, writing—review and editing.

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Correspondence to Tan Nguyen.

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Appendices

Appendix

1.1 Appendix A1: Numerical frameworks

Given that Ψ is constant and unaffected by θ in the Rankine region, the differentiation dΨ/dθ = 0. Consequently, equating the differential Eq. (10) to zero results in the value of mean stress (χ) in the Rankine region as follows:

$$\chi \left( \theta \right) = \frac{{\cos \theta - \sin \phi \cos \left( {2\psi \left( \theta \right) + \theta } \right)}}{\cos^2 \phi }$$
(A1.1)

The orientation of the major principal stress at the traction-free surface remains undefined; thus, the computational process initiates by ascertaining the value of ψ at the traction-free surface through the resolution of Eq. (A1.2).

$$\cos \theta - \sin \phi \cos \left( {2\psi \left( \theta \right) + \theta } \right) = 0$$
(A1.2)

The flowchart to solve the case 1 (ref. Fig. 2a) is given as below (See Fig. 

Fig. 31
figure 31

Flowchart illustrating the numerical frameworks employed to achieve the solution for case 1

Full size image

31).

The flowchart to solve the case 4 (ref. Fig. 2d) is given as below (See Fig. 

Fig. 32
figure 32

Flowchart illustrating the numerical frameworks employed to achieve the solution for case 4 (ref. Fig. 2d)

Full size image

32).

For case 3 (ref. Figure 2c) the numerical framework could be stepwise as follows:

Step 1: Identifying the location of the stress characteristic line.

Step 2: Applying the 5th order adaptive Runge–Kutta method to integrate the governing equations [Eqs. (9)−(10)] within the Rankine region (A1), as illustrated in Fig. 2c.

Step 3: Shooting the pair of ordinary differential equations (ODEs) along the characteristic line to the wall rear face (θ = λ).

For case 2 (ref. Fig. 2b), the solution is obtained by integrating the governing equations [Eqs. (9)−(10)] through the utilization of the 5th order adaptive Runge–Kutta method. Alternatively, the solution can be achieved using Eq. (A1.1).

Appendix A2

2.1 Passive earth coefficient for inclined wall λ = −30°

λ = −30°

β (°)

δ°

− 30

− 20

− 10

0

10

20

30

0

1.0185

2.3994

3.6130

5.0547

6.7167

8.6316

10.6986

− 5

1.0038

2.2510

3.2785

4.5140

5.8852

7.4281

9.0605

− 10

0.9849

2.0945

2.9597

4.0026

5.0789

6.2389

7.4802

− 15

0.9631

1.9340

2.6658

3.4261

4.2440

5.0998

5.9738

− 20

0.9375

1.7650

2.3415

2.9161

3.4639

4.0336

4.5838

− 25

0.9109

1.5833

1.9972

2.3800

2.7089

3.0364

3.2961

− 30

0.9109

1.3766

1.5590

1.6436

1.6481

1.6200

1.6400

2.2 Passive earth coefficient for inclined wall λ = −20°

λ = −20

β (°)

δ°

− 30

− 20

− 10

0

10

20

30

0

0.8809

2.0582

3.0546

4.3340

5.6762

7.2785

8.9973

− 5

0.8560

1.9054

2.8004

3.8170

4.9026

6.1763

7.5170

− 10

0.8277

1.7519

2.5005

3.2967

4.1564

5.1255

6.2389

− 15

0.7958

1.5899

2.1912

2.8267

3.4505

4.1430

4.8354

− 20

0.7620

1.4251

1.8918

2.3536

2.7855

3.2342

3.6828

− 25

0.7244

1.2592

1.5990

1.8968

2.1567

2.4166

2.6764

− 30

0.6882

1.0518

1.2046

1.2818

1.3104

1.2645

1.3100

2.3 Passive earth coefficient for inclined wall λ = −10°

λ = −10

β (°)

δ°

− 30

− 20

− 10

0

10

20

30

0

0.7430

1.7266

2.6149

3.5989

4.7663

6.0478

7.5257

− 5

0.7104

1.5809

2.3253

3.1030

4.0659

5.0667

6.1710

− 10

0.6765

1.4256

2.0352

2.6936

3.3949

4.1623

4.9652

− 15

0.6418

1.2800

1.7733

2.2669

2.7733

3.3252

3.8768

− 20

0.6035

1.1285

1.5032

1.8711

2.2204

2.5687

2.9163

− 25

0.5611

0.9771

1.2380

1.4788

1.7041

1.8864

2.0595

− 30

0.5113

0.7853

0.9045

0.9737

1.0005

0.9976

1.0000

2.4 Passive earth coefficient for inclined wall λ = 10°

λ = 10

β (°)

δ°

− 30

− 20

− 10

0

10

20

30

0

0.4818

1.1238

1.7284

2.4301

3.2620

4.0884

5.0174

− 5

0.4498

0.9968

1.4913

2.0427

2.6516

3.2779

3.9830

− 10

0.4174

0.8756

1.2631

1.6868

2.1449

2.6463

3.0947

− 15

0.3837

0.7598

1.0650

1.3744

1.7075

2.0179

2.3282

− 20

0.3507

0.6481

0.8782

1.1009

1.3073

1.5229

1.7084

− 25

0.3165

0.5453

0.7038

0.8404

0.9662

1.0702

1.1742

− 30

0.2813

0.4292

0.5008

0.5389

0.5514

0.5583

0.5653

2.5 Passive earth coefficient for inclined wall λ = 20°

λ = 20

β (°)

δ°

− 30

− 20

− 10

0

10

20

30

0

0.4750

0.8597

1.3455

1.9126

2.5425

3.2656

3.9968

− 5

0.3426

0.7514

1.1437

1.5754

2.0715

2.5655

3.1218

− 10

0.3144

0.6509

0.9590

1.2941

1.6461

2.0144

2.3827

− 15

0.2864

0.5603

0.7945

1.0351

1.2800

1.5322

1.7845

− 20

0.2598

0.4752

0.6459

0.8157

0.9745

1.1341

1.2937

− 25

0.2345

0.3975

0.5173

0.6203

0.7131

0.7923

0.8715

− 30

0.2813

0.3347

0.3874

0.4112

0.4143

0.4174

0.4205

2.6 Passive earth coefficient for inclined wall λ = 30°

λ = 30

β (°)

δ°

− 30

− 20

− 10

0

10

20

30

0

Infeasible

0.6339

1.0130

1.4685

1.9798

2.5626

3.1555

− 5

Infeasible

0.5471

0.8492

1.2022

1.5804

1.9934

2.4065

− 10

Infeasible

0.4701

0.7040

0.9605

1.2319

1.5242

1.8164

− 15

Infeasible

0.4010

0.5792

0.7638

0.9550

1.1352

1.3154

− 20

Infeasible

0.3412

0.4729

0.6031

0.7226

0.8409

0.9592

− 25

Infeasible

0.2930

0.3859

0.4670

0.5373

0.5959

0.6544

− 30

Infeasible

0.2844

0.3290

0.3406

0.3406

0.3406

0.3406

Appendix A3: Poncelet 1840’s solution

Poncelet derived expressions for Kp within Coulomb's limit equilibrium framework in 1840s, considering scenarios involving wall friction (± δ), an inclined wall face at an angle ± λ to the vertical, and sloping backfill at an angle ± β as follows:

$$K_{\text{P}} = \frac{{\cos^2 \left( {\phi + \lambda } \right)}}{{\cos^2 \lambda \cos \left( {\lambda - \delta } \right)\left[ {1 - \sqrt {{\left\{ {\frac{{\sin \left( {\phi + \delta } \right)\sin \left( {\phi + \beta } \right)}}{{\cos \left( {\lambda - \delta } \right)\cos \left( {\lambda - \beta } \right)}}} \right\}}} } \right]^2 }}$$
(A3.1)

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Nguyen, T., Shiau, J. Passive earth pressure in sand on inclined walls with negative wall friction based on a statically admissible stress field. Acta Geotech. (2024). https://doi.org/10.1007/s11440-024-02278-z

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