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and is shown on Figure 7.8.

A design angle of shearing resistance obtained from:

A design angle of shearing resistance obtained from:

is an entirely meaningless value, falling at an arbitrary position (shown by the cross) within the probability density function for 9. Instead, the calculation of the superior design angle of shearing resistance 9ksup should be based on:

Returning now to Figure 7.7, the curve labelled 'superior design' assumes that 9 = 9dsup, with ^ksup = ^k/inf + 10 °. (The difference of 10 ° selected here for illustration only.) This curve gives the most pessimistic values of P when ^k,inf

When establishing the resistance provided by retaining walls against uplift, it is not obvious whether a high or a low characteristic angle of shearing resistance will give the most conservative result. Instinctively, we might expect that a weaker soil would give lower resistance but, because of the interaction between strength and earth pressures, this is not always true.

7.5 Hydraulic failure

Ultimate limit state HYD is defined in Eurocode 7 as:

hydraulic heave, internal erosion and piping in the ground caused by hydraulic gradients [en 1997-1 §2.4.7.1(1)P]

The following sub-sections consider each of these phenomena in turn.

7.5.1 Hydraulic heave

Verification of stability against hydraulic heave is expressed in Eurocode 7 by two different (but supposedly equivalent) inequalities. One is given in terms of forces and weights:

in which Sd,dst = the design seepage force destabilizing a column of soil and G'dstb = the design submerged weight of that soil column.

The other is expressed in terms of stresses and pressures:

in which ud,dst = the design total pore water pressure that is destabilizing the soil column and odstb = the (stabilizing) design total stress that resists that pore pressure. Regrettably, Eurocode 7 does not specify how partial factors should be applied in the verification of HYD, which can lead to an apparent disparity between these two equations.

Applying Terzaghi's principle of effective stress3, we can rearrange the latter equation as follows:

which merely states that the design effective stress at the base of the soil column must not become negative.

Consider the embedded retaining wall shown in Figure 7.9, under which water flows owing to a difference in water level across the wall. Experiments have shown4 that a block of soil (shaded) of width d/2 is susceptible to piping failure if the hydraulic gradient over the depth of embedment d exceeds a critical value icrit.

Datum level

which merely states that the design effective stress at the base of the soil column must not become negative.

Datum level

Flow of water

Figure 7.9. Embedded retaining wall subject to piping due to heave

Flow of water

Figure 7.9. Embedded retaining wall subject to piping due to heave

For the purposes of this example, the datum has been taken at formation level on the left hand side of the wall. With this assumption, the total head h is given by Bernouilli's equation (total head = elevation + pressure + kinetic heads):

Yw 2g where z is the elevation above the datum level; u the pore water pressure; v the water velocity; and g the acceleration due to gravity. In situations involving groundwater, the kinetic head is usually negligible in comparison with the other heads and can be ignored. Hence:

The total head acting over the base of the shaded soil column shown in Figure 7.9 may be approximated by:

where H is the height of the retained water above formation level. This assumes that the head loss caused by seepage into the excavation is equal on both sides of the wall.

With the above assumptions, the characteristic hydraulic gradient ik through the shaded region is:

The characteristic seepage force destabilizing the soil column (per unit run of wall) is given by:

v 2 y where yw is the weight density of water. Since this is a permanent destabilizing action, its design value is:

v 2 y where Yc,dst is the partial factor (= 1.35) on destabilizing permanent actions.

The characteristic submerged weight of the soil column (per unit run of wall) is given by:

v 2 y where Yk' is the soil's characteristic submerged weight density and Yk its characteristic total weight density. Since this is a permanent stabilizing action, its design value is:

v 2 y where YG,stb is the partial factor (= 0.9) on stabilizing permanent actions.

Substituting these expressions into Sd,dst < G'dstb and simplifying produces:

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