1. SEEPAGE FORCES Consider a random element of a flow net: B D C the direction of flow is inclined at an angle of θ to the horizontal θ A lines AB and DC define the elemental flow channel lines AD and BC are equipotentials, with a drop in head of ∆h when water seeps from AD to BC Each side has the same length, b b b

2. Geometrically: B D C Each has an angle θ as shown θ A Four congruent right angle triangles are formed from vertical and horizontal lines projected inwards from the four corners of the flow net element θ θ θ θ The difference in elevation between A and D is the same as between B and C and is equal to bcosθ. The difference in elevation between A and B is the same as between D and C and is equal to bsinθ.

3. If the pore water pressure at point A is uA, and u = w(h-z), and The pore pressure distributions acting on each side of the element are shown below: the change in pore water pressure between point A and point D is due only to the elevation drop, bcosθ, bcosθ u D = u A +   w bcosθ The change in pore water pressure from point A to point B is due to a loss in total head -∆h and the elevation drop, bsinθ bsinθ u B = u A +   w(bsinθ-∆h) The change in pore water pressure between point B and point C is due only to the elevation drop, bcosθ, bcosθ u C = u B +   wbcosθ oru C = u A +   w(bsinθ-∆h) +   wbcosθ oru C = u A +   w(bsinθ+bcosθ-∆h) -∆h

4. SEEPAGE FORCES The pore pressure distribution acting on AD will be cancelled by that acting on BC, leaving: u D -u A = u C -u B =   w bcosθ The pore pressure distribution acting on AB will be cancelled by that acting on DC leaving: u B -u A = u C -u D =   w(bsinθ-∆h) w bcosθ w(bsinθ-∆h) The equivalent point load (net boundary water force) acting on DC is: b x wbcosθ or wb2cosθ w b2cosθ b The net boundary water force acting on BC is: b x w(bsinθ-∆h) or wb2sinθ - ∆hwb b  w b 2 s i n θ - ∆ h  w b  w (bsinθ-∆h) u B = u A +  w (bsinθ-∆h)  w (bsinθ+bcosθ-∆h) u C = u A +  w (bsinθ+bcosθ-∆h)  w bcosθ u D = u A +  w bcosθ

5. SEEPAGE FORCES What would the boundary water forces be if seepage stopped? (i.e., the static case) w b2cosθ  w b 2 s i n θ - ∆ h  w b the forces on DC and BC would be w b2cosθ and wb2sinθ r rr respectively, orthogonal vectors with a resultant of w b2, (acting vertically) If the average hydraulic gradient, i across the element is: The only difference between the static and seepage cases is the force ∆hwb called the seepage force, J Then: ∆h would be 0, and If b2 x 1 m is the volume of the element, V then the seepage pressure, j is defined as the seepage force per unit volume: j = iw

6. The concern is with the support conditions of the soil.How will seepage affect the effective stress at any point in the soil mass? w b2cosθ  w b 2 s i n θ - ∆ h  w b If the effective stress is reduced too much by upward seepage, then the soil will lose its ability to support loads. In the extremes: if the seepage direction is downward, the effective stress will be increased or if upward the effective stress will be decreased Therefore, let’s consider all the gravitational and seepage forces acting on the soil element à la a vector diagram. First, the SEEPAGE case: the total weight of the element = satb2 = vector ab Boundary water force on CD = wb2cosθ = vector bd Boundary water force on BC = wb2sinθ-∆hwb = vector de SEEPAGE CASE Resultant boundary water force = vector be Resultant body force = vector ae = Effective Stress, σ’

7. w b2cosθ  w b 2 s i n θ - ∆ h  w b Now consider the STATIC case:the total weight of the element = satb2 = vector ab Boundary water force on CD = wb2cosθ = vector bd Boundary water force on BC = wb2sinθ = vector dc STATIC CASE Resultant boundary water force = wb2 = v vv vector bc Resultant body force =  ’b2 v vv vector ac = Effective Stress, σ’

8. This brings up an alternative solution to the seepage case: Effective weight of the element =  ’b2 = vector ac Seepage force = ∆hwb = vector ce SEEPAGE CASE (reprise) Resultant body force vector ac = Effective Stress, σ’ To summarize, the resultant body force (effective stress) can be obtained by considering: A)t he equilibrium of the whole soil mass, add the total saturated weight of the soil mass (ab) to the resultant boundary water force (ce) to find effective stress (ae) OR B) the equilibrium of the soil skeleton, add the effective weight of the soil mass (ac) to the seepage force (be) to find effective stress (ae)