GEOTECHNICAL ENGINEERING - II

Engr. Nauman Ijaz

LATERAL EARTH PRESSURE

Chapter # 04

UNIVERSITY OF SOUTH ASIA

LATERAL EARTH

PRESSURE

� Lateral Earth pressure is an important parameter for the design of bridge abutment, different types of retaining walls (Such as different types of retaining walls (Such as gravity retaining walls, cantilever walls, counterforts or buttresses), sheet piles and other retaining structures.

GRAVITY RETAINING

WALL

CANTILEVER WALL

COUNTERFORT OR BUTTRESS

RETAINING WALL

SHEET PILES

BRIDGE ABUTMENT

LATERAL EARTH PRESSURE

AND WALL MOVEMENT

� Lateral earth pressure are the direct result of horizontal stresses in the soil.

� In order to understand the lateral earth � In order to understand the lateral earth pressure we have to define the Coefficient of lateral earth pressure, K.

COEFFICIENT OF LATERAL EARTH PRESSURE “K”

� It is defined as the;“Ratio of the horizontal effective stress to the vertical effective stress at any point in a soil.”vertical effective stress at any point in a soil.”

K = σ’x / σ’z

K = Coefficient of lateral earth pressure.

σ’x = Horizontal effective stresses.

σ’z = Vertical effective stresses.

� K is important because it is an indicator of the lateral earth pressures acting on retaining wall.

� For purpose of describing lateral earth pressures, geotechnical engineers have defined three important soil conditions;defined three important soil conditions;

1. At – rest Condition2. The Active Condition3. Passive condition

� Two classic Earth pressure theories has been put forward in the eighteen and nineteen centuries by Coulomb and Rankine respectively.

1) Rankine (1857) Earth Pressure Theory1) Rankine (1857) Earth Pressure Theory2) Coulomb’s(1776) Earth Pressure Theory

� These two theories are still in use in their original form and in some modified forms to calculate the earth pressure.

� Consider an element of soil at depth z below the ground surface level (GSL) as shown in the figure.

� The vertical stress due to the self weight of soil, σ’z (also known as overburden pressure or gravitational stress) is given by;or gravitational stress) is given by;

σ’z = γz� Where;� γ = unit weight of in -situ soil

Figure # (a)

Figure # (b)

� When confined (as in general case below GSL due to the pressure of surrounding soil), this vertical stress,(σz) will tend to cause the expansion of soil element and in doing so a secondary lateral pressure is generated.

� These vertical (σz) and horizontal (σx) stresses are the major and minor principal stresses in this particular case respectively.

� The ratio of σx to σz is termed as the co-efficient of earth pressure at rest and denoted by Ko.

� Thus;

Ko = σx / σz = σ3 / σ1………..(a)

� Ko value in general is variable and depends upon t soil type and its history of deposition

� Numerous relations have been derived for its evaluation, but the following relationships given by Jaky (1948) is commonly used;

Ko = 1 – SinΦ’ under root(OCR) ……(b)

� Φ’ = Effective angle of internal friction.� OCR = Over-consolidation Ratio.

� For normally consolidated soils, the equation(b) is reduced to;

Ko = 1 – SinΦ’

TYPICAL VALUES OF KoSOIL TYPE Ko

LOOSE SAND 0.59

DENSE ASND 0.36DENSE ASND 0.36

NORMALLY CONSOLIDATED CLAY 0.56 – 0.80

PRECONSOLIDATED CLAY > 1

ACTIVE CASE� Consider the figure (b), if the wall moves

away from the backfill the soil expands and the confining stress, σx gradually decreases.

� If the movement is sufficiently large the σx will decrease to minimum value and the state of will decrease to minimum value and the state of equilibrium will then attained.

� As σz > σx in this case, σz is the major principal stress and σx is the minor principal stress.

� This condition of wall movement is said to generate an active stress condition and the ratio σx / σz is defined as the active earth pressure co-efficient, Ka.

� Thus: � Thus:

Ka = σx / σz = σ3 / σ1

PASSIVE CASE

� When the wall moves towards the backfill, and against the soil mass, the soil will be subjected to lateral compression.

� Under this condition, σx is the principal stress and σ becomes the minor principal stress and σz becomes the minor principal stress.

� this condition is known as Passive Earth Pressure condition and ratio is given by;

Kp = σx / σz = σ1 / σ3

� where,Kp = the coefficient of Passive Earth pressure.

� Thus soil can exist in any condition ranging from the active, through the at rest to the passive state.passive state.

DIAGRAMTIC RELATIONSHIP

BETWEEN THE LATERAL STRAIN AND

LATERAL EARTH PRESSURE

RANKINE THEORY (1857)

� In original form the theory was developed for purely non-cohesive soils (i.e. c = 0), but subsequently Bell (1915) extended this theory to c-Φ soil as well.

ASSUMPTIONS

1. Soil is non-cohesive (c = 0) dry, isotropic and homogenous.

2. Backfill is horizontal.3. Wall is vertical,4. Wall friction is neglected.5. Failure is a plain strain problem.

� Consider a unit length of an infinitely long wall.

Soil Element : Rankine Theory

Vertical Stress = σ’z = γz Major Principal stress.

At failure Horizontal stress , σ’x Minor Principal Stress.

� From ∆OAB,SinΦ = AB/OB = ½(σz – σx)

½(σz + σx)SinΦ(σz + σx) = (σz – σx)σz (1 – SinΦ) = σx (1 + SinΦ)Ka = σx / σz = (1 – SinΦ) = tan² (45 – Φ/2)

(1 + SinΦ)

and σx = active earth pressure = σz Ka

σa = γz tan² (45 – Φ/2)

� Similarly, passive earth pressure,Kp = tan² (45 – Φ/2) and,σp = γz tan² (45 + Φ/2)

m.irfan B-15952