Chapter 12 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

12.1 At-Rest, Active, and Passive Pressure

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

12.1 At-Rest, Active, and Passive Pressure

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

12.2 Earth Pressure at Rest - Assume frictionless wall : No shear stress on the vertical & Horizintal planes Elastic equilibrium : horizontal strain is ZERO Fig. 12.3 Earth pressure at rest

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

12.2 Earth Pressure at Rest Elasticity

12.2 Earth Pressure at Rest

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)

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

12.2 Earth Pressure at Rest (total force per unit length of the wall) Fig. 12.4 Distribution of earth pressure at rest on a wall

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

12.3 Earth Pressure at Rest for Partially Submerged Soil

12.3 Earth Pressure at Rest for Partially Submerged Soil Fig. 12.5 Distribution of earth pressure at rest for partially submerged soil Example 12.1

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

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

12.5 & 12.6 Rankine’s Theory of Active & Passive Earth Pressures Fig. 12.9 Rankine’s active earth pressure

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

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

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

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

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

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

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

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

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

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

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

12.7 Yielding of Wall of Limited Height

12.7 Yielding of Wall of Limited Height Figure 12.11 Rotation of frictionless wall about the bottom

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

12.8 Diagrams for Lateral Earth Pressure Distribution against Retaining Walls -Passive case Fig. 12.12 Pressure distribution against a retaining wall for cohesionless soil backfill with horizontal ground surface

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

12.8 Diagrams for Lateral Earth Pressure Distribution against Retaining Walls Fig. 12.13 Rankine’s active earth pressure distribution

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

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

12.8 Diagrams for Lateral Earth Pressure Distribution against Retaining Walls Fig. 12.14 Rankine’s passive earth pressure distribution

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

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

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

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

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

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

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

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)

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

12.10 & 12 & 14 Coulomb’s Earth Pressure Theory Fig. 12.22 Coulomb’s active pressure: (a) trial failure wedge; (b) force polygon

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

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

12.10 & 12 & 14 Coulomb’s Earth Pressure Theory Fig. 12.34 Coulomb’s passive pressure: (a) trial failure wedge; (b) force polygon

12.10 & 12 & 14 Coulomb’s Earth Pressure Theory Pp의 최소값 Where, Kp = <Eq. 12.89> Kp is increased with the wall friction    Kp  (Table 12.7) - Overestimates the passive resistance of walls, especially for  > /2

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)

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

12.11 Graphic Solution for Coulomb’s Active Earth Pressure Fig. 12.23 Culmann’s solution for active earth pressure

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

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

12.11 Graphic Solution for Coulomb’s Active Earth Pressure Fig. 12.24 Approximate method for finding the point of application of the resultant active force

12.12 Active Force on Retaining Walls with Earthquake Forces FIGURE 12.26 Active force on a retaining wall with earthquake forces

12.12 Active Force on Retaining Walls with Earthquake Forces • Where

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

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

12.12 Active Force on Retaining Walls with Earthquake Forces 2. Calculate (Eq. 12.68) , 3. Calculate (Eq. 12.72) , 4. 5. from the base of the wall 6. Calculate the location of

12.12 Active Force on Retaining Walls with Earthquake Forces

12.12 Active Force on Retaining Walls with Earthquake Forces FIGURE 12.28 Location of the line of action of Pae

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.

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,

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

FIGURE 12.35 Passive force on a retaining wall With earthquake forces 12.13 Pae for c’- Soil Backfill 12.15 Passive Force on Retaining Walls with Earthquake Forces FIGURE 12.35 Passive force on a retaining wall With earthquake forces

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

12.15 Passive Force on Retaining Walls with Earthquake Forces FIGURE 12.36 Variation of with for (after Davies, Richards, and Chen, 1986)

12.16 Summary and General Comments