SEEPAGE FORCESSEEPAGE FORCESConsider a random element of a flow Consider a random element of a flow net:net:

BB

DD

CC

the direction of flow is inclined at an angle of the direction of flow is inclined at an angle of θθ to the to the horizontalhorizontal

θθ

AA

lines AB and DC define the elemental flow lines AB and DC define the elemental flow channelchannellines AD and BC are equipotentials, with a drop in head of ∆h when lines AD and BC are equipotentials, with a drop in head of ∆h when water seeps from AD to BCwater seeps from AD to BC

Each side has the same Each side has the same length, blength, b

bb

bb

SEEPAGE FORCESSEEPAGE FORCESGeometrically:Geometrically:

BB

DD

CC

Each has an angle Each has an angle θθ as as shownshown

θθ

AA

Four congruent right angle triangles are formed from Four congruent right angle triangles are formed from vertical and horizontal lines projected inwards from the four vertical and horizontal lines projected inwards from the four corners of the flow net elementcorners of the flow net element

θθ

θθθθ

θθ

The difference in elevation between A and D is the same as The difference in elevation between A and D is the same as between B and C and is equal to bcosbetween B and C and is equal to bcosθθ..The difference in elevation between A and B is the same as The difference in elevation between A and B is the same as between D and C and is equal to bsinbetween D and C and is equal to bsinθθ..

SEEPAGE FORCESSEEPAGE FORCES

If the pore water If the pore water pressure at point A is pressure at point A is uuAA, and u = , and u = ww(h-z), and(h-z), and

The pore pressure distributions acting on The pore pressure distributions acting on each side of the element are shown below:each side of the element are shown below:

the change in pore the change in pore water pressure water pressure between point A and between point A and point D is due only to point D is due only to the elevation drop, the elevation drop, bcosbcosθθ,,

bcosbcosθθ

uD = uA + w w

bcosbcosθθ

The change in pore The change in pore water pressure from water pressure from point A to point B is point A to point B is due to a loss in total due to a loss in total head -∆h and the head -∆h and the elevation drop, bsinelevation drop, bsinθθ

bsinbsinθθ

uB = uA + ww(bsin(bsinθθ--∆h)∆h)

The change in pore The change in pore water pressure water pressure between point B and between point B and point C is due only to point C is due only to the elevation drop, the elevation drop, bcosbcosθθ,,

bcosbcosθθ

uC = uB + wwbcosbcosθθ oruC = uA + ww(bsin(bsinθθ-∆h) +-∆h) + wwbcosbcosθθ oruC = uA + ww(bsin(bsinθθ+bcos+bcosθθ--∆h)∆h)

-∆h-∆h

SEEPAGE FORCESSEEPAGE FORCESThe pore pressure distribution acting on AD The pore pressure distribution acting on AD

will be cancelled by that acting on BC, will be cancelled by that acting on BC, leaving:leaving:uD-uA = uC-uB = w w

bcosbcosθθ

The pore pressure distribution acting on AB The pore pressure distribution acting on AB will be cancelled by that acting on DC will be cancelled by that acting on DC

leaving:leaving:

uB-uA = uC-uD = ww(bsin(bsinθθ--∆h)∆h)

w w bcosbcosθθ

ww(bsin(bsinθθ--

∆h)∆h)

The equivalent point load (The equivalent point load (net boundary net boundary water forcewater force) acting on DC is: ) acting on DC is: b x b x wwbcosbcosθθ or or

wwbb22coscosθθ

w w bb22coscosθθ

b

The The net boundary water forcenet boundary water force acting on BC acting on BC is: is: b x b x ww(bsin(bsinθθ-∆h)-∆h) or or wwbb22sinsinθθ - ∆h - ∆hwwbb

b

wwbb

22 sin

sinθ

θ -

∆h

- ∆

hwwbb

uB = uA + ww(bsin(bsinθθ--∆h)∆h) uC = uA + ww(bsin(bsinθθ+bcos+bcosθθ--∆h)∆h)

uD = uA + w w

bcosbcosθθ

SEEPAGE FORCESSEEPAGE FORCESWhat would the boundary water What would the boundary water forces be if seepage stopped? (i.e., forces be if seepage stopped? (i.e., the static case)the static case)

w w bb22coscosθθ

wwbb

22 sin

sinθ

θ -

∆h

- ∆

hwwbb

the forces on DC and BC would be the forces on DC and BC would be ww b b22coscosθθ and and wwbb22sinsinθθ respectively, respectively, orthogonal vectors with a resultant orthogonal vectors with a resultant of of ww b b2 2 , (acting vertically), (acting vertically)If the average hydraulic gradient, i If the average hydraulic gradient, i across the element is:across the element is:

The only difference between the The only difference between the static and seepage cases is the static and seepage cases is the force force ∆h∆hwwb b called the seepage called the seepage force, force, JJ

bh

i

2w

2ww bib

bΔh

bΔhJ γγγ Then:Then:

∆∆h would be 0, andh would be 0, and

If bIf b22 x 1 m is the volume of the x 1 m is the volume of the element, V then the element, V then the seepage seepage pressurepressure, , jj is defined as the is defined as the seepage force per unit volume:seepage force per unit volume:j = ij = iww

SEEPAGE FORCESSEEPAGE FORCESThe concern is with the support conditions of the soil.The concern is with the support conditions of the soil.How will seepage affect the effective stress at any point in the soil How will seepage affect the effective stress at any point in the soil

mass?mass?

w w bb22coscosθθ

wwbb

22 sin

sinθ

θ -

∆h

- ∆

hwwbb

If the effective stress is reduced too much by upward seepage, then If the effective stress is reduced too much by upward seepage, then the soil will lose its ability to support loads.the soil will lose its ability to support loads.In the extremes: if the seepage direction is downward, the effective In the extremes: if the seepage direction is downward, the effective stress will be increased or if upward the effective stress will be stress will be increased or if upward the effective stress will be decreaseddecreased

Therefore, let’s consider all the gravitational and seepage forces Therefore, let’s consider all the gravitational and seepage forces acting on the soil element à la a vector diagram. First, the SEEPAGE acting on the soil element à la a vector diagram. First, the SEEPAGE case:case:

the total weight of the element = the total weight of the element = satsatbb22 = vector = vector ababBoundary water force on CD = Boundary water force on CD = wwbb22coscosθθ = vector = vector bdbdBoundary water force on BC = Boundary water force on BC = wwbb22sinsinθθ-∆h-∆hwwbb = vector = vector dede

SEEPAGE SEEPAGE CASECASE

Resultant boundary water force = vector Resultant boundary water force = vector bebeResultant body force = vector Resultant body force = vector ae = Effective ae = Effective Stress, Stress, σσ’’

SEEPAGE FORCESSEEPAGE FORCES

w w bb22coscosθθ

wwbb

22 sin

sinθ

θ -

∆h

- ∆

hwwbb

Now consider the STATIC case:Now consider the STATIC case:the total weight of the element = the total weight of the element = satsatbb22 = vector = vector ababBoundary water force on CD = Boundary water force on CD = wwbb22coscosθθ = vector = vector bdbdBoundary water force on BC = Boundary water force on BC = wwbb22sinsinθθ = vector = vector dcdc

STATIC CASESTATIC CASE

Resultant boundary water force = Resultant boundary water force = wwbb22 = vector vector bcbc

Resultant body force = Resultant body force = ’b ’b22 vector vector ac = Effective ac = Effective StressStress, , σσ’’

SEEPAGE FORCESSEEPAGE FORCESThis brings up an alternative solution to the seepage This brings up an alternative solution to the seepage case:case:Effective weight of the element = Effective weight of the element = ’b ’b22 = vector = vector acac

Seepage force = Seepage force = ∆h∆hwwb b = vector = vector cece

SEEPAGE CASE SEEPAGE CASE (reprise)(reprise)

Resultant body force vector Resultant body force vector ac = Effective Stressac = Effective Stress, , σσ’’

To summarize, the resultant body force (effective To summarize, the resultant body force (effective stress) can be obtained by considering:stress) can be obtained by considering:

A)A) the equilibrium of the whole soil mass,the equilibrium of the whole soil mass,

add add the total saturated weight of the soil mass the total saturated weight of the soil mass ((abab))

to to the resultant boundary water forcethe resultant boundary water force ( (cece))

to find to find effective stresseffective stress ( (aeae))OROR

B) the equilibrium of the soil skeleton,B) the equilibrium of the soil skeleton,

add add the effective weight of the soil mass the effective weight of the soil mass ((acac))

to to the seepage forcethe seepage force ( (bebe))

to find to find effective stresseffective stress ( (aeae))