Discussion: Passive and active earth pressures in the presence of groundwater flow Free

I was very interested to see the two papers in the April issue of Géotechnique on the issue of earth pressure coefficients in the presence of steady-state seepage. I studied this topic, using the method of characteristics, for my undergraduate project at Cambridge University, but apart from the project report (Houlsby, 1975) the results have not been published before. The results lend support to the useful study by the authors.

The method used a standard application of the method of characteristics for effective stresses, as described by Sokolovski (1965). However, when the effective stresses and pore pressures are substituted for the total stresses in the equilibrium equations, these are written as

The body force in the –y direction is therefore augmented by a term ∂u/∂y and there is also a body force in the –x direction ∂u/∂x As long as the spatial variation of pore pressure u is specified, the resulting problem can be treated simply as one involving a variable body force. The method was programmed in Fortran, and the pore pressure could be specified either through an analytical expression or by interpolation from a grid of points.

Of particular relevance to the paper by Benmebarek et al. are some calculations of the stability of a sheet pile wall with essentially the same geometry that they study (see their Fig. 1). For the pore pressure distribution I used the elegant analytical solution presented by Schofield & Wroth (1968: pp. 59–62). I carried out active and passive pressure calculations for ϕ = 30° and 40°; for angles of interface friction on the wall of zero, ϕ/2 and ϕ and for a variety of depths of water H (in the authors' notation). For simplicity I examined just the case γ′ = γ_w (the same as the authors).

My study was conducted in terms of the maximum hydraulic gradient at the soil surface i₀, which is related (using the analytical expression for the pore pressures) to the depth of water used by the authors by the expression H/f = πi₀, where f is the wall depth. The cases analysed in the two studies do not exactly correspond, but by interpolating linearly between the authors' figures for δ = ϕ/3 and δ = 2ϕ/3 it is possible to obtain the comparisons of active and passive earth pressure coefficients for the cases shown in Table 8 (which shows only a fraction of the possible comparisons). There is a very satisfactory agreement between the stress-characteristic solutions obtained by Houlsby (1975) and the figures quoted by the authors, which include FLAC analysis and limit equilibrium calculations by Soubra et al. (1999). The remarkably good comparison between the different methods lends confidence to both.

I took the study one stage further, and examined in two different ways the critical height of water on the upstream side of the sheet pile wall that would cause failure. The first method involved plotting against i₀ the horizontal components of the active and passive forces, including also terms from the water pressure both above and below the ground surface. The maximum allowable i₀ value was simply the point at which the two curves crossed (for lower i₀ values the active forces were calculated as less than the maximum passive resistance). This calculation would be relevant to a purely translational failure mechanism. The second calculation was the same in principle, but involved equating the moments about the toe of the sheet pile wall, thus representing a more realistic mechanism (although still not necessarily the most critical). The results of the two calculations, presented in terms of H/f are given in Table 9, which shows that the second mechanism is more critical.