1. Chapter 13 Lateral Earth Pressure : At-Rest, Rankine, and Coulomb
연세대학교 지반공학연구실

2. Introduction - 토압의 크기 : 배면지반의 강도정수와 관련(cu, u or c, ), 배수조건
- Retaining Structures
: retaining walls, basement walls, bulkhead, temporary retaining wall
- 구조물에 작용하는 토압의 분포는 구조물과 흙의 상대적인 변위에 따라 달라짐
변위 토압
변위 토압
변위 토압
변위 토압
- 토압의 크기 : 배면지반의 강도정수와 관련(cu, u or c, ), 배수조건

3. 13.1 At-Rest, Active, and Passive Pressure

4. 13.1 At-Rest, Active, and Passive Pressure
Active Pressure

5. 13.1 At-Rest, Active, and Passive Pressure

6. 13.1 At-Rest, Active, and Passive Pressure
: Variation of the magnitude of lateral earth pressure with wall tilt

7. 13.2 Earth Pressure at Rest - Assume frictionless wall
: No shear stress on the vertical &
Horizintal planes
Elastic equilibrium
: horizontal strain is ZERO

8. 13.2 Earth Pressure at Rest h  = K0  o = K0 (  z),
K0= Coeffi. of earth pressure at Rest

9. 13.2 Earth Pressure at Rest
Elasticity

10. 13.2 Earth Pressure at Rest

11. 13.2 Earth Pressure at Rest - Soil is elasto-plastic behavior
( Jaky, 1944) : 사질토, NC clay
(OC clay)
Comments on Earth PR. Increase caused by Compaction
Jaky’s eq. : good results for loose sand backfill
- For a dense sand backfill (Sherif, Fang, 1984)

12. 13.2 Earth Pressure at Rest
where, d : actual compacted dry unit wt. of the sand
d,min : the loosest dry unit wt.

13. 13.2 Earth Pressure at Rest
(total force per unit length of the wall)

14. 13.3 Earth Pressure at Rest for Partially Submerged Soil
- Partially submerged soil (ground W.T)
z < H1
z > H1

15. 13.3 Earth Pressure at Rest for Partially Submerged Soil

16. 13.3 Earth Pressure at Rest for Partially Submerged Soil
Example 13.1

17. 13.4 Rankine’s Theory of Active Pressure
- Rankine Theory(1857)-Limiting Plastic Equilibrium
- Assume : No Friction between wall and soil smooth wall H z

18. 13.4 Rankine’s Theory of Active Pressure
Fig Rankine’s active earth pressure

19. 13.4 Rankine’s Theory of Active Pressure
Rankine’s Active state
where,
(Rankine’s active earth pressure)

20. 13.4 Rankine’s Theory of Active Pressure
If c=0 (for cohesionless soils)
: Coefficient of active earth Pressure.

21. 13.4 Rankine’s Theory of Active Pressure
 Active state
책에 내용 없음

22. 13.4 Rankine’s Theory of Active Pressure
Rotation of frictionless wall about the bottom + = Z0 Pa H
책에 내용 없음

23. 13.4 Rankine’s Theory of Active Pressure
<tension crack depth>
책에 내용 없음
Total active force

24. 13.4 Rankine’s Theory of Active Pressure
책에 내용 없음
Acting at

25. 13.5 Rankine’s Theory of Passive Pressure
- Rankine’s passive state

26. 13.5 Rankine’s Theory of Passive Pressure
(Rankine’s passive
earth pressure)
where,

27. 13.5 Rankine’s Theory of Passive Pressure
If, c=0
:coefficient of passive earth pressure

28. 13.5 Rankine’s Theory of Passive Pressure

29. 13.5 Rankine’s Theory of Passive Pressure
Rotation of frictionless wall about the bottom ＋ ＝ Ｈ

30. 13.6 Yielding of Wall of Limited Height

31. 13.6 Yielding of Wall of Limited Height

32. 13.7 A Generalized Case for Rankine Active and Passive Pressure – Granular Backfill

33. 13.7 A Generalized Case for Rankine Active and Passive Pressure – Granular Backfill
where
Rankine’s active pressure coefficient
- For a vertical backface of the wall

34. 13.7 A Generalized Case for Rankine Active and Passive Pressure – Granular Backfill
Frictionless vertical retaining
where

35. 13.8 Diagrams for Lateral Earth Pressure Distribution against Retaining Walls
Backfill-Cohesionless soil with Horizontal ground Surface
-Active case

36. 13.8 Diagrams for Lateral Earth Pressure Distribution against Retaining Walls
-Passive case

37. 13.8 Diagrams for Lateral Earth Pressure Distribution against Retaining Walls
Backfill - Partially Submerged Cohensionless Soil Supporting a
Surcharge
- Active case

38. 13.8 Diagrams for Lateral Earth Pressure Distribution against Retaining Walls

39. 13.8 Diagrams for Lateral Earth Pressure Distribution against Retaining Walls
at z=0,
z=H1,
z=H,
at z=H, u = w · H2

40. 13.8 Diagrams for Lateral Earth Pressure Distribution against Retaining Walls
- Passive case

41. 13.8 Diagrams for Lateral Earth Pressure Distribution against Retaining Walls

42. 13.8 Diagrams for Lateral Earth Pressure Distribution against Retaining Walls
Backfill-Cohesive Soil with Horizontal Backfill
- Active case
a is negative
in the upper part of retaining wall because of the cohesion effect

43. 13.8 Diagrams for Lateral Earth Pressure Distribution against Retaining Walls
Backfill-Cohesive Soil with Horizontal Backfill
Active case

44. 13.8 Diagrams for Lateral Earth Pressure Distribution against Retaining Walls
(undrained condition) ,

45. 13.8 Diagrams for Lateral Earth Pressure Distribution against Retaining Walls
for taking the tensile cracks into account
 for =0

46. 13.8 Diagrams for Lateral Earth Pressure Distribution against Retaining Walls
- Passive case
at z=0
at z=H

47. 13.8 Diagrams for Lateral Earth Pressure Distribution against Retaining Walls
- Passive case

48. 13.8 Diagrams for Lateral Earth Pressure Distribution against Retaining Walls
for =0, Kp=1
 Example 13.2~ 13.5

51. 13.10 & 12 Coulomb’s Earth Pressure Theory
- Coulomb’s Theory(1776) : Stability of soil wedge
· Consider wall friction
· Coulomb assumes failure lines is straight
· Force equilibrium only considered
(Not moment equilibrium, point of load application may not be
coincide)

52. 13.10 & 12 Coulomb’s Earth Pressure Theory
- Active case
Find maximum Pa

53. 13.10 & 12 Coulomb’s Earth Pressure Theory

54. 13.10 & 12 Coulomb’s Earth Pressure Theory
If  =  =  = 0
Same as Rankine’s earth PR. coeffi
-    Ka  (Table 12.6 참조)

55. 13.10 & 12 Coulomb’s Earth Pressure Theory
- Passive case

56. 13.10 & 12 Coulomb’s Earth Pressure Theory

57. 13.10 & 12 Coulomb’s Earth Pressure Theory
Pp의 최소값
Where, Kp = <Eq >
Kp is increased with the wall friction
   Kp  (Table 13.9)
- Overestimates the passive resistance of walls, especially for  > /2

58. 13.11 Graphic Solution for Coulomb’s Active Earth Pressure
Culmann’s Solution(1875) :
 Graphic Solution of Coulomb’s Earth PR. Theory
 Consider for ant Wall friction, regardless of irregularity of backfill and surcharge
 Active Earth PR. with granular backfill(c=0)

59. 13.11 Graphic Solution for Coulomb’s Active Earth Pressure
Steps
1. Draw retaining wall and backfill to a convenient scale
2. Determine  = 90 -  -  ,  , 
3. Draw a line BD ( with the horizontal)
4. Draw a line BE ( with line BD)
5. Draw lines BC1, BC2, BC3, … BCn

60. 13.11 Graphic Solution for Coulomb’s Active Earth Pressure

61. 13.11 Graphic Solution for Coulomb’s Active Earth Pressure
6. Determine the weight of soil, W
W1=area(ABC1)    1 
Wn=area(ABCn)    1
7. Adopt a convenient load scale and plot the weight
W1=BC1, W2=BC2, … Wn=BCn
8. Draw C1C1, … CnCn parallel to the line BE

62. 13.11 Graphic Solution for Coulomb’s Active Earth Pressure
9. Draw a smooth curve through points c1, c2, c3, … cn called
the “Culmann line”
10. Draw a tangent BD parallel to line BD
load scale
Culmann Solution
: provides only the magnitude of the active force
per unit length of the retaining wall

63. 13.11 Graphic Solution for Coulomb’s Active Earth Pressure

64. 13.13 Active Force on Retaining Walls with Earthquake Forces

65. 13.13 Active Force on Retaining Walls with Earthquake Forces•
Where

66. 13.13 Active Force on Retaining Walls with Earthquake Forces
Force polygon
Mononobe-Okabe Eq.

67. 13.13 Active Force on Retaining Walls with Earthquake Forces
Where
If no inertia force from E.Q ,
Location of Line of Action of Resultant Force, Pae
- Seed & Whitman (1970) : Location of the
1. Let •
= E.Q effect

68. 13.13 Active Force on Retaining Walls with Earthquake Forces
2. Calculate (Eq ) ,
3. Calculate (Eq ) , 4.
from the base of the wall
6. Calculate the location of

69. 13.13 Active Force on Retaining Walls with Earthquake Forces

70. 13.13 Active Force on Retaining Walls with Earthquake Forces

71. Passive Force on Retaining Walls with Earthquake Forces
FIGURE Passive force on a retaining wall
With earthquake forces

72. Passive Force on Retaining Walls with Earthquake Forces
(13.91)
Where

73. Passive Force on Retaining Walls with Earthquake Forces
FIGURE Variation of
with
for
(after Davies, Richards, and Chen, 1986)

74. 13.14 Common Types of Retaining Walls in the Field
Rigid Retaining Walls
Gravity retaining walls
Semigravity retaining walls
Cantilever retaining walls
Counterfort retaning walls

75. 13.14 Common Types of Retaining Walls in the Field

76. 13.14 Common Types of Retaining Walls in the Field
Mechanically Stabilized Earth (MSE) Walls
Backfill-which is granular soil
Reinforcement in the backfill
A cover (or skin) on the front face

77. 13.14 Common Types of Retaining Walls in the Field

78. 13.14 Common Types of Retaining Walls in the Field

79. 13.16 Summary and General Comments