Chapter 3 Effective stress 3.1 Figure Q3.1 Total vertical stress at 5 m depth: Pore water pressure: Therefore, effective vertical stress: Alternative method: Buoyant unit weight: Effective vertical stress: σ v = 51 The complete stress profile within the ground may be obtained using the spreadsheet tool Stress_CSM8 on the Companion Website. The complete solution is shown on the following page. The spreadsheet may then be used to query the stresses at 5 m depth, giving the values shown above.
Effective stress 23 3.2 Figure Q3.2 Total vertical stress at 5 m depth: Alternative method: Buoyant unit weight: σ v = 51 The complete stress profile within the ground may be obtained using the spreadsheet tool Stress_CSM8 on the Companion Website. The complete solution is shown on the following page. The spreadsheet may then be used to query the stresses at 5 m depth, giving the values shown above. Comparing the answer with Problem 3.1, it is clear that when the water table is above the ground surface, the depth of the overlying water has no effect on the effective stresses within the ground (though σ v and u do change). 3.3 Figure Q3.3
24 Effective stress At the top of the clay: σ v = 51.4 At the bottom of the clay: The elevation head (due to the location of the WT) is 6 m. Artesian pressure is additional to this, 4 m above ground level (AGL), i.e. 6 m above the WT: σ v = 33.3 (N.B. The alternative method of calculation shown in Problems 3.1 and 3.2 is not applicable under artesian ground water conditions.) The complete stress profile within the ground may be obtained using the spreadsheet tool Stress_CSM8 on the Companion Website. The complete solution is shown on the following page. The spreadsheet may then be used to query the stresses at 4 and 8 m depth, giving the values shown above. 3.4 Figure Q3.4 Total vertical stress at 8 m depth: σ v = 106 The complete stress profile within the ground may be obtained using the spreadsheet tool Stress_CSM8 on the Companion Website. The complete solution is shown on the following page. The spreadsheet may then be used to query the stresses at 8 m depth, giving the values shown above.
Effective stress 25 3.5 Figure Q3.5 (a) Immediately after WT rise: At 8 m depth, pore water pressure is governed by the new WT level because the permeability of the sand is high. σ v = 94 At 12 m depth, pore water pressure is governed by the old WT level because the permeability of the clay is very low. (However, there will be an increase in total stress of 9 due to the increase in unit weight from 16 to 19 kn/m 3 between 3 and 6 m depth: this is accompanied by an immediate increase of 9 in pore water pressure.) σ v = 154 (b) Several years after WT rise: At both depths, pore water pressure is governed by the new WT level, it being assumed that swelling of the clay is complete. Therefore, at 8 m depth: (as above) σ v = 94 At 12 m depth: σ v = 134
26 Effective stress The complete stress profile within the ground may be obtained using the spreadsheet tool Stress_CSM8 on the Companion Website. The complete solution is shown on the following page. The spreadsheet may then be used to query the stresses at 8 m depth, giving the values shown above. 3.6 Figure Q3.6 Total weight of the element = vector ab: Boundary water force on CD = vector bd. Referring to the figure above: Boundary water force on BC = vector de. Referring to the figure above: Resultant body force = vector ea. Measuring from the force diagram (drawn to scale) gives ea = 9.9 kn. This acts in a direction of 17 to the vertical. Resultant body force = 9.9 kn (@ 17 to vert.)
Effective stress 27 3.7 Figure Q3.7 (a) For case (1), h = 2 m at the top of the element and h = 0 m at the bottom. Seepage is therefore occurring from top to bottom and at the centre of the sample, h = 1 m: σ v = 30.2 For case (2), h = 2 m at the bottom of the element, h = 0 at the top and seepage is occurring from bottom to top. As before, h = 1 m at the centre of the sample: σ v = 10.6 (b) In both cases, Δh = 2 m across the sample, which has a length of Δs = 4 m. The hydraulic gradient is therefore i = Δh/Δs = 0.5 from top to bottom in (1) and from bottom to top in (2). This gives a seepage pressure j = iγ w = 4.9 kn/m 3 downwards in (1) and upwards in (2). For case (1): σ v = 30.2 For case (2): σ v = 10.6
28 Effective stress 3.8 The flownet for this problem was found in Problem 2.3, and is shown below: Figure Q3.8 The average exit hydraulic gradient may be found directly using the results of Flownet_ CSM8, or reading from the flownet sketch. In the latter case: Loss in total head between adjacent equipotentials: Δs 0.85 m, so exit hydraulic gradient: The critical hydraulic gradient is given by Equation 3.12: Therefore, factor of safety against boiling (Equation 3.14): F = 1.76 The pore pressures at C and D may also be found directly using the results of Flownet_ CSM8 to get the total head, or reading from the flownet sketch.
Effective stress 29 Total head at C: m Elevation head at C: m Pore water pressure at C: Therefore, effective vertical stress at C: σ vc = 14.5 For point D: m m σ vd = 93.4 3.9 The flow net for this problem was found in Problem 2.2, and is shown below: Figure Q3.9
30 Effective stress For a soil prism 1.50 3.00 m adjacent to the piling (shown shaded in the above figure): m (from Flownet_CSM8) Factor of safety against heaving (Equation 3.13): F = 1.88 With a filter: Depth of filter = 17.4/21 = 0.83 m (if above water level). Filter height = 0.83 m