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

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

13.1 At-Rest, Active, and Passive Pressure

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

13.1 At-Rest, Active, and Passive Pressure

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

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

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

13.2 Earth Pressure at Rest Elasticity

13.2 Earth Pressure at Rest

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)

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

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

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

13.3 Earth Pressure at Rest for Partially Submerged Soil

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

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

13.4 Rankine’s Theory of Active Pressure Fig. 13.9 Rankine’s active earth pressure

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

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

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

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

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

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

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

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

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

13.5 Rankine’s Theory of Passive Pressure

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

13.6 Yielding of Wall of Limited Height

13.6 Yielding of Wall of Limited Height

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

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

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

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

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

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

13.8 Diagrams for Lateral Earth Pressure Distribution against Retaining Walls

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

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

13.8 Diagrams for Lateral Earth Pressure Distribution against Retaining Walls

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

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

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

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

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

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

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

 

   

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)

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

13.10 & 12 Coulomb’s Earth Pressure Theory

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

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

13.10 & 12 Coulomb’s Earth Pressure Theory

13.10 & 12 Coulomb’s Earth Pressure Theory Pp의 최소값 Where, Kp = <Eq. 13.71> Kp is increased with the wall friction    Kp  (Table 13.9) - Overestimates the passive resistance of walls, especially for  > /2

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)

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

13.11 Graphic Solution for Coulomb’s Active Earth Pressure

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

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 11. load scale Culmann Solution : provides only the magnitude of the active force per unit length of the retaining wall

13.11 Graphic Solution for Coulomb’s Active Earth Pressure

13.13 Active Force on Retaining Walls with Earthquake Forces

13.13 Active Force on Retaining Walls with Earthquake Forces • Where

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

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

13.13 Active Force on Retaining Walls with Earthquake Forces 2. Calculate (Eq. 13.68) , 3. Calculate (Eq. 13.74) , 4. 5. from the base of the wall 6. Calculate the location of

13.13 Active Force on Retaining Walls with Earthquake Forces

13.13 Active Force on Retaining Walls with Earthquake Forces

Passive Force on Retaining Walls with Earthquake Forces FIGURE 13.35 Passive force on a retaining wall With earthquake forces

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

Passive Force on Retaining Walls with Earthquake Forces FIGURE 13.36 Variation of with for (after Davies, Richards, and Chen, 1986)

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

13.14 Common Types of Retaining Walls in the Field

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

13.14 Common Types of Retaining Walls in the Field

13.14 Common Types of Retaining Walls in the Field

13.16 Summary and General Comments