1. Chapter 12 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. 12.1 At-Rest, Active, and Passive Pressure

4. 12.1 At-Rest, Active, and Passive Pressure
Active Pressure

5. 12.1 At-Rest, Active, and Passive Pressure

6. 12.1 At-Rest, Active, and Passive Pressure
: Variation of the magnitude of lateral earth pressure with wall tilt

7. 12.2 Earth Pressure at Rest - Assume frictionless wall
: No shear stress on the vertical &
Horizintal planes
Elastic equilibrium
: horizontal strain is ZERO
Fig Earth pressure at rest

8. 12.2 Earth Pressure at Rest h  = K0  v = K0 (  z),
K0= Coeffi. of earth pressure at Rest

9. 12.2 Earth Pressure at Rest
Elasticity

10. 12.2 Earth Pressure at Rest

11. 12.2 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. 12.2 Earth Pressure at Rest
where, d : actual compacted dry unit wt. of the sand
d,min : the loosest dry unit wt.

13. 12.2 Earth Pressure at Rest
(total force per unit length of the wall)
Fig Distribution of earth pressure at rest on a wall

14. 12.3 Earth Pressure at Rest for Partially Submerged Soil
- Partially submerged soil (ground W.T)
z < H1
z > H1

15. 12.3 Earth Pressure at Rest for Partially Submerged Soil

16. 12.3 Earth Pressure at Rest for Partially Submerged Soil
Fig Distribution of earth pressure at rest for partially submerged soil
Example 12.1

17. 12.4 Lateral Pressure on Retaining Walls from Surcharges  Based on Theory of Elasticity
- Point load Surcharge, Q
- Line load Surcharge, q (load/unit length)
- Strip load Surcharge, q/unit area
 Example 12.2

18. 12.5 & 12.6 Rankine’s Theory of Active & Passive Earth Pressures
- Rankine Theory(1857)-Limiting Plastic Equilibrium
- Assume : No Friction between wall and soil smooth wall H z

19. 12.5 & 12.6 Rankine’s Theory of Active & Passive Earth Pressures
Fig Rankine’s active earth pressure

20. 12.5 & 12.6 Rankine’s Theory of Active & Passive Earth Pressures
Rankine’s Active state
where,
(Rankine’s active earth pressure)

21. 12.5 & 12.6 Rankine’s Theory of Active & Passive Earth Pressures
If c=0 (for cohesionless soils)
: Coefficient of active earth Pressure.

22. 12.5 & 12.6 Rankine’s Theory of Active & Passive Earth Pressures
 Active state

23. 12.5 & 12.6 Rankine’s Theory of Active & Passive Earth Pressures
Fig.12.11(a) Rotation of frictionless wall about the bottom + = Z0 Pa H

24. 12.5 & 12.6 Rankine’s Theory of Active & Passive Earth Pressures
<tension crack depth>
Total active force

25. 12.5 & 12.6 Rankine’s Theory of Active & Passive Earth Pressures
Acting at

26. 12.5 & 12.6 Rankine’s Theory of Active & Passive Earth Pressures
- Rankine’s passive state

27. 12.5 & 12.6 Rankine’s Theory of Active & Passive Earth Pressures
(Rankine’s passive
earth pressure)
where,

28. 12.5 & 12.6 Rankine’s Theory of Active & Passive Earth Pressures
If, c=0
:coefficient of passive earth pressure

29. 12.5 & 12.6 Rankine’s Theory of Active & Passive Earth Pressures

30. 12.5 & 12.6 Rankine’s Theory of Active & Passive Earth Pressures
Fig.12.11(b) Rotation of frictionless wall about the bottom ＋ ＝ Ｈ

31. 12.7 Yielding of Wall of Limited Height

32. 12.7 Yielding of Wall of Limited Height
Figure Rotation of frictionless wall about the bottom

33. 12.8 Diagrams for Lateral Earth Pressure Distribution against Retaining Walls
Backfill-Cohesionless soil with Horizontal ground Surface
-Active case

34. 12.8 Diagrams for Lateral Earth Pressure Distribution against Retaining Walls
-Passive case
Fig Pressure distribution against a retaining wall for
cohesionless soil backfill with horizontal ground surface

35. 12.8 Diagrams for Lateral Earth Pressure Distribution against Retaining Walls
Backfill - Partially Submerged Cohensionless Soil Supporting a
Surcharge
- Active case

36. 12.8 Diagrams for Lateral Earth Pressure Distribution against Retaining Walls
Fig Rankine’s active earth pressure distribution

37. 12.8 Diagrams for Lateral Earth Pressure Distribution against Retaining Walls
at z=0,
z=H1,
z=H,
at z=H, u=w · H2c

38. 12.8 Diagrams for Lateral Earth Pressure Distribution against Retaining Walls
- Passive case

39. 12.8 Diagrams for Lateral Earth Pressure Distribution against Retaining Walls
Fig Rankine’s passive earth pressure distribution

40. 12.8 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

41. 12.8 Diagrams for Lateral Earth Pressure Distribution against Retaining Walls
(undrained condition) ,

42. 12.8 Diagrams for Lateral Earth Pressure Distribution against Retaining Walls
for taking the tensile cracks into account

43. 12.8 Diagrams for Lateral Earth Pressure Distribution against Retaining Walls
- Passive case
at z=0
at z=H

44. 12.8 Diagrams for Lateral Earth Pressure Distribution against Retaining Walls
for =0, Kp=1
 Example 12.3~ 12.6

45. 12.9 Ranking Active and Passive Pressure with Sloping Backfill
where
Rankine’s active pressure coefficient

46. 12.9 Ranking Active and Passive Pressure with Sloping Backfill
where
12.57 참조
Soil
 Example 12.7

47. 12.10 & 12 & 14 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 dquilibrium, point of load application may not be
coincide)

48. 12.10 & 12 & 14 Coulomb’s Earth Pressure Theory
- Active case
Find maximum Pa

49. 12.10 & 12 & 14 Coulomb’s Earth Pressure Theory
Fig Coulomb’s active pressure: (a) trial failure wedge; (b) force polygon

50. 12.10 & 12 & 14 Coulomb’s Earth Pressure Theory
If  =  =  = 0
Same as Rankine’s earth PR. coeffi
-    Ka  (Table 12.5 참조)

51. 12.10 & 12 & 14 Coulomb’s Earth Pressure Theory
- Passive case

52. 12.10 & 12 & 14 Coulomb’s Earth Pressure Theory
Fig Coulomb’s passive pressure: (a) trial failure wedge; (b) force polygon

53. 12.10 & 12 & 14 Coulomb’s Earth Pressure Theory
Pp의 최소값
Where, Kp = <Eq >
Kp is increased with the wall friction
   Kp  (Table 12.7)
- Overestimates the passive resistance of walls, especially for  > /2

54. 12.11 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)

55. 12.11 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

56. 12.11 Graphic Solution for Coulomb’s Active Earth Pressure
Fig Culmann’s solution for active earth pressure

57. 12.11 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

58. 12.11 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

59. 12.11 Graphic Solution for Coulomb’s Active Earth Pressure
Fig Approximate method for finding the point of
application of the resultant active force

60. 12.12 Active Force on Retaining Walls with Earthquake Forces
FIGURE Active force on a retaining wall with earthquake forces

61. 12.12 Active Force on Retaining Walls with Earthquake Forces•
Where

62. 12.12 Active Force on Retaining Walls with Earthquake Forces
Force polygon
Mononobe-Okabe Eq.

63. 12.12 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

64. 12.12 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

65. 12.12 Active Force on Retaining Walls with Earthquake Forces

66. 12.12 Active Force on Retaining Walls with Earthquake Forces
FIGURE Location of the line of action of Pae

67. 12.12 Active Force on Retaining Walls with Earthquake Forces
Deign of Retaining Wall Based on Tolerable Lateral Displacement
- Richards & Elms (1979)
Proposed a procedure for designing gravity
retaining wall for E.Q conditions that allows
limited lateral displacement of the walls.

68. 12.12 Active Force on Retaining Walls with Earthquake Forces
where
(11.13)
(11.14)
• Determined the weight of the retaining wall ( )
1) Determine the tolerable displacement,

69. 12.12 Active Force on Retaining Walls with Earthquake Forces
2) Determine
: effective acceleration coefficients.
3) Determine based on = 0 ,
calculated in step 2
4) Determine by applying a S.F

70. FIGURE 12.35 Passive force on a retaining wall With earthquake forces
12.13 Pae for c’- Soil Backfill Passive Force on Retaining Walls with Earthquake Forces
FIGURE Passive force on a retaining wall
With earthquake forces

71. 12.15 Passive Force on Retaining Walls with Earthquake Forces
(12.91)
Where

72. 12.15 Passive Force on Retaining Walls with Earthquake Forces
FIGURE Variation of
with
for
(after Davies, Richards, and Chen, 1986)

73. 12.16 Summary and General Comments