ENCE 461Foundation Anal ysis and

Design

Retaining WallsLateral Earth Pressure Theory

Retainin g Walls

� Necessary in situations where gradual transitions either take up too much space or are impractical for other reasons

� Retaining walls are analysed for both resistance to overturning and structural integrity

� Two categories of retaining walls

� Gravity Walls (Masonry, Stone, Gabion, etc.)

� In-Situ Walls (Sheet Piling, cast in-situ, etc.)

Lateral Earth Pressure Coefficient

� K = lateral earth pressure coefficient

� �x’ = horizontal effective stress

� ����������� ��������������

� ����������� ���������� ������������ �������� ������

� �� ����������������������� ��������� �

K��x '

�z'

Mohr’s Circle and Lateral

Earth Pressures

=�z'� x ' =

Development of Lateral Earth Pressure

Po

b��1 z1

2 K o

2

Note Pore Water Effect! subtract vertically add horizontally

Groundwater Effects

� Steps to properly compute horizontal stresses including groundwater effects:

� Compute total vertical stress

� Compute effective vertical stress by removing groundwater effect through submerged unit weight; plot on P

o diagram

� Compute effective horizontal stress by multiplying effective vertical stress by K

� Compute total horizontal stress by directly adding effect of groundwater unit weight to effective horizontal stress

Groundwater Effects

Conditions of Lateral Earth Pressure Coefficient

� At-Rest Condition

� Condition where wall movement is zero or “minimal”

� Ideal condition of wall, but seldom achieved in reality

� Active Condition

� Condition where wall moves away from the backfill

� The lower state of lateral earth pressure

� Passive Condition

� Condition where wall moves toward the backfill

� The higher state of lateral earth pressure

Effect of Wall Movement

Wall Movements Necessar y to Achieve Active or Passive

States

Estimates of At Rest Lateral Earth Pressure Coefficient

� Jaky’s Equation

� Modified for Overconsolidated Soils

� Applicable only when ground surface is level

� In spite of theoretical weaknesses, Jaky’s equation is as good an estimate of the coefficient of lateral earth pressure as we have

K o�1�sin� '

K o��1�sin� ' �OCRsin� '

Relationship of Poisson’s Ratio with Lateral Earth Pressure

Coefficient

K o��

1��

��2��1��1

��tan��1tan��2

(Normally Consolidated Soils)

Example of At Rest Wall Pressure

� Given

� Retaining Wall as Shown

� Find

� PA,

from At Rest Conditions

At Rest Pressure Example

� Compute at rest earth pressure coefficient

� Compute Effective Wall Force

K o�1�sin� 'K o�1�sin30º�0.5

Po

b��1 z1

2 K o

2Po

b�

1202020.52

Po

b�12000

lbsft�12

kipsft

hPA�203�6.67 ft.

(valid for all theories)

Development of Active Earth Pressure

Development of Passive Earth Pressure

Earth Pressure Theories

Rankine Earth Pressure EquationsLevel Backfills

Rankine Theor y with Inclined Backfills

Rankine Coefficients with Inclined Backfills

Inclined and level backfill equations are identical when � = 0

Example of Rankine Active Wall Pressure

� Given

� Retaining Wall as Shown

� Find

� PA,

from At Rest Conditions

Rankine Active Pressure Example

� Compute at rest earth pressure coefficient

� Compute Effective Wall Force

Po

b��1 z1

2 K a

2Po

b�

1202020.3332

Po

b�8000

lbsft�8

kipsft

K A�tan2�45º��

2�

K A�tan2�45�15��13

Rankine Passive Pressure Example

Rankine Passive Pressure Example

� Compute at rest earth pressure coefficient

� Compute Effective Wall Force

Po

b��1 z1

2 K p

2Po

b�

12020232

Po

b�72000

lbsft�72

kipsft

K P�tan2�45º�

2�

K P�tan2�4515��3

Summar y of Rankine and At Rest Wall Pressures

72,000 lbs.

12,000 lbs. 8000 lbs.

Coulomb Theor y

K p�cos2����

cos2�cos������1�sin����sin�� �

cos�����cos� ����

2

K a�cos2�����

cos2�cos�����1sin����sin��� �

cos����cos��� ��

2�

Typical Values of Wall Friction

Example of Coulomb Theor y

� Given

� Wall as shown above

� Find

� KA, K

P, P

A

Solution for Coulomb Active Pressures

� Compute Coulomb Active Pressure

� KA = 0.3465

� Compute Total Wall Force

� PA = 8316 lb/ft of wall

Solution for Coulomb Passive Pressures

� Compute Coulomb Passive Pressure

� KP = 4.0196

� Compute Total Wall Force

� PA = 96,470 lb/ft of wall

Walls with Cohesive Backfill

� Retaining walls should generally have cohesionless backfill, but in some cases cohesive backfill is unavoidable

� Cohesive soils present the following weaknesses as backfill:

� Poor drainage

� Creep

� Expansiveness

� Most lateral earth pressure theory was first developed for purely cohesionless soils (c = 0) and has been extended to cohesive soils afterward

Theor y of Cohesive

Soils

Active Case(Overburdendriving)

Passive Case(Wall Driving)

1sin�1�sin�

� tan2��

4�

2�

Rankine Pressures with Cohesion (Level Backfill)

�3��1 tan2��

4��

2��2c tan�

�

4��

2�

�1�� H

K A��3

�1

�tan2��

4��

2��

2c� H

tan��

4��

2�

� Active

� Passive

�1��3 tan2��

4�

2�2c tan�

�

4�

2�

�3�� H

K P��1

�3

�tan2��

4�

2�

2c� H

tan��

4�

2�

Overburden Driving

Wall Driving

Comments on Rankine

Equations

� Valid if wall-soil friction is not taken in to account

� Do not take into consideration soil above critical height

� Do not take into consideration sloping walls

� For practical problems, should use equations as they appear in the book

H c�2c

� K a

Equivalent Fluid Method

� Simplification used to guide the calculations of lateral earth pressures on retaining walls

� Can be used for Rankine and Coulomb lateral earth pressures

� Can be used for at rest, active and passive earth pressures

� Transforms the soil acting on the retaining wall into an equivalent fluid

Example of Equivalent Fluid Method

� Given

� Wall as shown above

� KA = 0.3465

� KP = 4.0196

� �w

= 3 degrees

� Find

� Forces acting on the wall (both horizontal and vertical)

Example of Equivalent Fluid� Compute Equivalent Fluid Unit Weights (Active

Case)Gh��K acos�w

Gh�1200.3465cos3ºGh�41.52pcfGv��K asin�w

Gv�1200.3465sin3ºGv�2.18pcf

Example of Equivalent Fluid� Compute Wall Load (Active Case)

Pa

b�

Gh H 2

2Pa

b�

41.52202

2�8304 lb/ft

V a

b�

Gv H 2

2V a

b�

2.18202

2�436 lb/ft

Example of Equivalent Fluid� Compute Equivalent Fluid Unit Weights (Passive

Case)Gh��K pcos�w

Gh�1204.0196cos3ºGh�481.69pcfGv��K psin�w

Gv�1204.0196sin3ºGv�25.24pcf

Example of Equivalent Fluid� Compute Wall Load (Passive Case)

Pp

b�

Gh H 2

2Pp

b�

481.69202

2�96338 lb/ft

V p

b�

Gv H 2

2V p

b�

25.24202

2�5048 lb/ft

Terzaghi Model

� Assumes log spiral failure surface behind wall

� Requires use of suitable chart for K

A

and KP

� Not directly used in this course, but option in SPW 911

Presumptive Lateral Earth Pressures

� Based on Terzaghi theory

� Suitable for relatively simple retaining walls in homogeneous soils

� Classifies soils into five types:

1. “Clean” coarse grained soils

2. Coarse grained soils of low permeability; mixed with fine grained soils

3. Residual soils with granular materials and clay content

4. Very soft clay, organic silts, or silty clays

5. Medium or stiff clay, very low permeability

Presumptive Lateral Earth Pressures

Presumptive Lateral Earth

Pressures

Effects of Surface Loadin g

Surchar ge and Groundwater Loads

Homework Set 5

� Reading

� McCarthy: Chapter 16

� Coduto: Chapters 22, 23, 24 & 25

� Homework Problems

� McCarthy: 16-1, 16-8, 16-12a, 16-17

� Coduto: 25.3 (Hand and Chart Solutions); 25.5 (SPW 911)

� Due Date: 17 April 2002

Questions