CE 240Soil Mechanics & Foundations

Lecture 11.3

Lateral Earth Pressure I(Das, Ch. 12)

Class Outline

• Introduction: at-rest, active, and passive pressure for retaining structures

• Earth pressure at-rest• Rankine’s theory: Active and Passive• Retaining wall with limited height

Let us start to consider a question: What is the ratio of the horizontal pressure and the vertical pressure?

1) in water;2) in ice;3) any difference?4) which is larger?

Extend the discussion on the ratio of the horizontal stress and vertical stress in solids.

??h

v

K σσ

=

1) in water:

2) in ice:

3) in rock:

1h

v

K σσ

= =

1h

v

K σσ

= <

1 0.253

h

v

K ifσ νσ

= ≈ =

Basic Concepts on Earth Pressures

a'σ

o'σ)(' ahσ

)(' phσ

Earth Pressure at Rest• Coefficient of earth pressure at rest

whereσ’o = γ′zσ’h = Ko(γ′z)γ′ = γsat - γw

o

hoK

''

σσ

=

Earth Pressure at Rest (Cont.)• For coarse-grained soils

K0 = 1 – sin φ’ (Jaky, 1944)

• For correcting K0 due to over consolidationK0 = 1 – sin φ’ (OCR)sinφ’ (Mayne & Kulhawy 1982)

* For fine-grained, normally consolidated soils(Massarch, 1979)

• For overconsolidates claysWhere

⎥⎦⎤

⎢⎣⎡+=

100(%)42.044.0 PIKo

OCRKK NCoOCo )()( =

o

cPOCR'σ

=

1

2

3

Rankine’s Theory AssumptionsThe lateral earth pressure coefficients are valid for;

Vertical earth retaining wallSmooth wall in which the interface between the wall and soil is frictionlessThe supported soil is homogeneous and isotropicThe soil is loose and originally in an at-rest stateLateral earth pressures must be applied to effective stressCritical slip planes are oriented at 45o ± φ’/2

Figure 12.7

Rankine’s theory of active pressure

Compare these expressions with we’ve learned from equation 11.8 from Chapter 11, we can see the difference is the above equations is expressing the minimum principal stress by the maximum principal stress.

a'σ o'σ

Rankine’s Passive Pressures

• The coefficient of Rankine’s passive earth pressure, Kp

ap K

K 1=

⎟⎠⎞

⎜⎝⎛ +=

−+

==2'45tan

'sin1'sin1

'' 2)( φ

φφ

σσ

o

phpK

Lateral Stress due to Compaction• Additional lateral stresses are imposed by the

compaction equipment which should be considered while designing retaining walls

• Ingold (1979) used elastic theory to estimate these lateral stresses

• Some account for this by assuming that the resultant active lateral earth force acts at 0.4H0or 0.5H0 rather than H0/3 from the base of the wall

• Alternatively, active earth pressures can be multiplied by a factor(~1.20)

Reading Assignment:

Das, Ch. 12

HW: Problem 12.15