lecture 4 effective stress

Lecture 4 – Effective StressIntroduction to Stresses in Soil

Total StressPore water pressureEff Effective Stress

Principle of Effective StressEffective Vertical Stress

•Effect of water table Effect of water table •Effect of Capillary Rise

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Students should be able to:1. Determine values of total stress,

pore water pressure and effective stress.

2. Interpret the principle of effective stresses

Learning ObjectivesLearning ObjectivesLearning ObjectivesLearning ObjectivesPrepared by:[email protected] 2

σ =vertical stress (kN/m2)σv vertical stress (kN/m )σH = horizontal stress (kN/m2)γ = bulk unit weight (kN/m3)γb = bulk unit weight (kN/m )γsat = saturated unit weight (kN/m3)γ ater nit eight (kN/m3)γw = water unit weight (kN/m3)uw = pore water pressure (kN/m2)

d h f ilz = depth of soil

Remember these symbols!!Remember these symbols!!

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Introduction to Stresses in SoilIntroduction to Stresses in Soil1. Total Stress, σv

◦ Can be defined as stress = force per unit area transmitted in a normal direction acting on a plane assuming the soil to in a normal direction acting on a plane assuming the soil to be a solid material.

◦ for a small soil element at a depth z below ground level the vertical stress, σv would be the stress acting on the horizontal , v gsurface of the element (refer to Figure a)

◦ Stresses in soil are not isotropic which is σv σH.

Depth z σV

Bulk unit weight γb z1 Bulk unit weight γb

Saturated unit weight γsat

Water table

σH

σV =γbz

a) Above a water table

σV21 zz satbV γγσ +=

z2


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a) Above a water table

a) Below a water table

*In this chapter, horizontal stress is neglected but always remembered this stress also act.

2. Pore water pressure , uw

Pressure which is referring to pressure of the water filling the void space between the solid particlesWater table = water pressure is the same as atmospheric pressure in the groundpressure in the ground.water below the water table is known as phreatic water.Therefore, phreatic surface = water table.The pores in soil below the water table are fully saturated.The pores in soil below the water table are fully saturated.

Partially saturated zone

Ground levelIf no seepage is occurring, only gravity

Fully saturated zone

Water table

forces are acting on the pore water so the hydrostatic pressure (pore water pressure) u y satu ated o e

zw

uw = γwzw

(pore water pressure) is given by:

wwww zorgzu γρ=


5Pore water Pressure in the ground

The Principle of Effective StressThe Principle of Effective StressTerzaghi (1923) found that forces transmitted through soil skeleton can be

t d i i i l f ff ti t b d i t l d tpresented in principle of effective stress, based on experimental data.The principle of effective stress only applicable to fully saturated soils

P Let us consider an element of a saturated soil is

*Effective stress will be denoted

XN'

T

of a saturated soil is subjected to a normal stress, σ= P/A, applied on the plane X-X as shown in Fi 1

will be denoted by a prime (').

The equilibrium ti iX

A

P

Figure 1.

The total normal stress, σ,must be in equilibrium state

equation is:

σ = σ' +uwP

q(Newton’s 3rd law).

The resistance or reaction to σ is provided by

σ' = σ - uwFigure 1

External force or total stress, σ

σ is provided by combination of the stresses between inter-particles (effective stress, σ', and

Principle of effective stress

Contact area

Internal resistance from water or pore water pressure


from pore water pressure, uw.

Internal resistance from solids or effective stress, σ'

The principal of effective stress is the most important principle in soil mechanicsimportant principle in soil mechanics.Deformations of soils are a function of effective stresses not total stresses.The principle of effective stresses applies only to normal stresses σV(vertical stresses) not to shear stresses, τ.

The Principle of Effective StressThe Principle of Effective Stress


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Effective stresses due to geostatic stress fields & Effective stresses due to geostatic stress fields & water tablewater table

The effective stress in a soil mass is subjected to unit weight of the soil & depth of groundwaterthe soil & depth of groundwater. Let consider effective stress for a soil element in Figure 2: Ground level Total vertical stress is

Water tablez1

z2

γb

Total vertical stress is

21 zz satb γγσ +=

Pore water pressure is

z3

z2

γsat 2zu ww γ=

Effective vertical stress is

'21

221'

)()(

zz

zzzzzu

wsatb

wsatbw

γγ

γγγγγγσσ

+=

−+=−+=−=

Figure 2


21 zzb γγ +=

Work Examples 1 ( Effect of water table)Work Examples 1 ( Effect of water table)A l f t t d l 4 thi k i l i b d 5 A layer of saturated clay 4m thick is overlain by sand 5m deep, the water table being 3m below the surface. The saturated unit weights of the clay and sand are 19kN/m3 & 20kN/m3 respectively: above the water table the unit 20kN/m3 respectively: above the water table the unit weight of the sand is 17kN/m3. Plot the values of total vertical stress & effective stress against depth.

Solution:

γ = 17kN/m3

3

5

W.T.

Sandγsat = 20kN/m3

9Clay

σ′ σγsat = 19kN/m3


0 10050 150kN/m2

Calculation steps:Depth (m)

σv(kN/m2)

u (kN/m2) σ'v = σv – u(kN/m2)

3 3 х 17 = 51.0 - 0 51.0

5 (3 х 17) + (2 х 20)

= 91.0 2 х 9.8 = 19.6 71.4

9 (3 х 17) + (2 х 20)

= 167.0 6 х 9.8 = 58.8 108.2

O Al b l l t d f ll

х 20)+ (4 х 19)

Or.. Also can be calculated as follows:Effective unit weight of sand = 20 – 9.8 = 10.2 kN/m3

Effective unit weight of clay = 19 – 9.8 = 9.2 kN/m3

At 5m depth: σ'v = (3 x 17) +( 2 x 10.2) = 71.4 kN/m2

At 9m depth: σ'v = (3 x 17) +( 2 x 10.2) + (4 x 9.2) = 108.2 kN/m2kN/m


Effect of Capillary Rise to Effective StressesEffect of Capillary Rise to Effective StressesIn silts and fine sands, the soil above the groundwater can be saturated by capillary action.The illustration of capillarity in soils can be idealized as in Figure 3.


Figure 3

From Figure 3, continuous void spaces can be idealized as capillary tubes. Consider a single idealized tube as shown in the figure. The height at which water will rise in the tube can be found from statics; by summing forces vertically (upward forces are negative),; y g y ( p g ),

ΣFz = weight of water – tension forces from capillary action

wc d

Tzγ

θcos4=

Where T is the surface tension (force per unit length), θ is the contact angle, zc is the height of capillary rise, and d is the diameter of the void space. pSince T = 0.073N/m, θ = 0, γw = 9.81kN/m3;

z 1α Assumed as 0.1D10


dzcα 10

Pore water pressure due to capillarity isPore water pressure due to capillarity is negative (a.k.a suction) & is a function of the size of the soil pores and water content.size of the soil pores and water content.Pore water pressure =0 (at ground water level) & decreases (-ve sign) as move up the& decreases ( ve sign) as move up the capillary zone.The effective stress increase because the poreThe effective stress increase because the pore water pressure is –ve. i e effective stress; σ' = σ ( z γ ) = σ + z γ

Refer to Figure 3Refer to Figure 3

i.e effective stress; σ = σ – (-zcγw) = σ + zcγw

Refer to Figure 3Refer to Figure 3Prepared by:[email protected] 13

Work Examples 1(Effect of capillary rise to effective Work Examples 1(Effect of capillary rise to effective t )t )stress)stress)

If sand to a height of 1m above the water table is saturated with capillary water, how are the p y ,above stresses?

The water table is level at which pore water pressure is atmospheric (i.e. u=0)

Above the water table, water is held under negative pressure and even if the soil is negative pressure and even if the soil is saturated above the water table, it does not contribute to hydrostatic pressure below the water table.


1m 291 0 + 3

γ = 17kN/m3

W T

γsat = 20kN/m3

1m σv =91.0 + 3

σ' =71.4 + 3

3

5

W.T.

Sand

σ ′ σγsat = 19kN/m3 σ v 71.4 + 3

0 10050 150

9Clay

*At capillary level, σv < σ'v

Depth (m)

σv(kN/m2)

u (kN/m2) σ'v = σv – u(kN/m2)

0 10050 150kN/m 2

0 0 0 0 0 0

2 2 x 17 = 34.0 -1x9.8 = -9.8 43.8

3 (2 x 17)+(1 20)

= 54.0 0 = 0 54.0+(1х 20)

5 54+ (2 х 20) = 94.0 2 х 9.8 = 19.6 74.4

9 94.0 + (4 х = 170.0 6 х 9.8 = 58.8 111.2


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Plot distribution of total stress, effective o d s bu o o o a s ss,stress, and pore water pressure with depth for the soil profile as given & neglect capillary action:

4.5 m e0 = 0.7, S = 0.85 Water table

5.0 m w = 28%

Now u try it!!Now u try it!!Now u try it!!Now u try it!!Prepared by:[email protected] 16

Barnes G E (2000) Soil Mechanics Principles andBarnes, G.E. (2000), Soil Mechanics Principles and Practice, Antony Rowe Ltd, Edition 2.Craig, R.F. (1992), Soil Mechanics, Chapman & g, ( ), , pHall, Edition 5Muni Budhu (2007), Soil Mechanics and Foundations, John Wiley & Sons, Inc., Edition 2.

ReferencesReferencesPrepared by:[email protected] 17

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lecture 4 effective stress

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