Lecture 4 Effective Stress

Lecture 4 – Effective Stress Introduction to Stresses in Soil Total Stress Pore water pressure Effective Stress Eff Principle of Effective Stress Effective Vertical Stress

•Effect 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 Objectives Prepared by:[email protected]

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stress (kN/m2) y σH = horizontal stress (kN/m2) y γb = bulk unit weight (kN/m3) y γsat = saturated unit weight (kN/m3) y γw = water ater unit nit weight eight (kN/m3) y uw = pore water pressure (kN/m2) y z = depth d h off soil il y σv =vertical vertical

Remember these symbols!! Prepared by:[email protected]

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Introduction to Stresses in Soil 1.

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 be a solid material. ◦ for a small soil element at a depth z below ground level the g on the horizontal vertical stress,, σv would be the stress acting surface of the element (refer to Figure a) σH. ◦ Stresses in soil are not isotropic which is σv Bulk unit weight γb

Depth z

Bulk unit weight γb

z1

Water table

σV

Saturated unit weight γsat

σH

σV =γbz

z2 σV

σ V = γ b z1 + γ sat z 2

a) Above a water table *In this chapter, horizontal stress is neglected but always remembered this stress also act. Prepared by:[email protected]

a) Below a water table

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Pore water pressure , uw Pressure which is referring to pressure of the water filling the void space between the solid particles Water table = water pressure is the same as atmospheric pressure in the ground. 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.

2. y y y y y

If no seepage is occurring, only gravity forces are acting on the pore water so the hydrostatic pressure (pore water pressure) is given by:

u w = ρgz w or γ w z w Prepared by:[email protected]

Ground level Partially saturated zone

Water table Fully u y satu saturated ated zone o e

zw uw = γwzw Pore water Pressure in the ground

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The Principle of Effective Stress Terzaghi (1923) found that forces transmitted through soil skeleton can be presented t d iin principle i i l off effective ff ti stress, t based b d on experimental i t ld data. t The principle of effective stress only applicable to fully saturated soils P

T N' X X

A

P

Figure 1 External force or total stress, σ Internal resistance from water or pore water pressure Contact area Internal resistance from solids or effective stress, σ'

Let us consider an element of a saturated soil is subjected to a normal stress, σ= P/A, applied on the plane X-X as shown in Fi Figure 1. 1

*Effective stress will be denoted by a prime (').

The total normal stress, σ, q state must be in equilibrium (Newton’s 3rd law).

σ = σ' +uw

The equilibrium equation ti is: i

σ' = σ - uw

The resistance or reaction to σ is provided by combination of the stresses Principle of effective stress between inter-particles (effective stress, σ', and from pore water pressure, Prepared uw. by:[email protected] 6

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The principal of effective stress is the most important principle in soil mechanics 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 Stress Prepared by:[email protected]

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Effective stresses due to geostatic stress fields & water table The effective stress in a soil mass is subjected to unit weight of the soil & depth of groundwater. groundwater Let consider effective stress for a soil element in Figure 2: Ground level

γb

Total vertical stress is Water table

z1

Pore water pressure is

z2

γsat

z3

Figure 2

σ = γ b z1 + γ sat z 2 uw = γ w z2 Effective vertical stress is

σ ' = σ − u w = (γ b z1 + γ sat z 2 ) − γ w z 2 = γ b z1 + (γ sat − γ w ) z 2 = γ b z1 + γ ' z 2 Prepared by:[email protected]

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Work Examples 1 ( Effect of water table) A layer l off saturated t t d clay l 4m 4 thick thi k is i overlain l i by b sand d 5m 5 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 weight of the sand is 17kN/m3. Plot the values of total vertical stress & effective stress against depth. Solution: γ = 17kN/m3 W.T.

γsat = 20kN/m3

3 Sand 5

γsat = 19kN/m3

σ′

σ

Clay 9

0

50

100 kN/m2

150 Prepared by:[email protected]

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Calculation steps: Depth (m)

u (kN/m2)

σv (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) + (4 х 19)

= 167.0

6 х 9.8

= 58.8

108.2

Or.. Al O Also can bbe calculated l l t d as follows: f ll 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/m2 Prepared by:[email protected]

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

Figure 3

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y y

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;; byy summingg forces verticallyy (upward ( p forces are negative), g ), ΣFz = weight of water – tension forces from capillary action zc =

y

y

4T cos θ dγ w

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. p Since T = 0.073N/m, θ = 0, γw = 9.81kN/m3;

1 zcα d

Assumed as 0.1D10 Prepared by:[email protected]

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y Pore

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

Refer to Figure 3 Prepared by:[email protected]

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

If sand to a height of 1m above the water table is saturated with capillary p y water,, how are the 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 saturated above the water table, it does not contribute to hydrostatic pressure below the water table.

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γ = 17kN/m3 1m

WT W.T.

γsat = 20kN/m3

γsat = 19kN/m3

2

σv =91.0 91 0 + 3

3

Sand 5

σ'v =71.4 σ 71.4 + 3

σ′

σ

Clay 9

0

Depth (m)

50

100 2 kN/m

*At capillary level, σv < σ'v

150

u (kN/m2)

σv (kN/m2)

σ'v = σv – u (kN/m2)

0

0

0

0

0

0

2

2 x 17

= 34.0

-1x9.8

= -9.8

43.8

3

(2 x 17) +(1 20) +(1х

= 54.0

0

=0

54.0

5

54+ (2 х 20)

= 94.0

2 х 9.8

= 19.6

74.4

9

94.0 + (4 х 19)

= 170.0

6 х 9.8

= 58.8

111.2

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Plot o distribution d s bu o of o total o a stress, s ss, effective 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!! Prepared by:[email protected]

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y y y

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

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Thank you

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