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Fig. 6.28 Pressure distribution in strutted excavation (after BS 8002: 1994).

In Fig. 6.28b the shape A'B'C'D' represents, to an enlarged scale, the original form of the surface that has yielded to the position ABCD of Fig. 6.28a; the resulting pressure on the back of the wall is roughly parabolic and is indicated in Fig. 6.28c.

For design purposes a trapezoidal distribution is assumed, of the form recommended in BS8002: 1994 after the work of Terzaghi and Peck (1967), since revised (Terzaghi, Peck and Mesri, 1996). The design procedure for the struts is semi-empirical. For sands the pressure distribution is assumed to be uniform over the full depth of the excavation (Fig. 6.28d). For clays, the pressure distribution depends on the stability number, N:

If N is greater than 4, the distribution in Fig. 6.28e is used, provided that Ka is greater than 0.4. If N is less than 4, or if 0.2 < Ka < 0.4, the distribution in Fig. 6.28f is used. With respect to Fig. 6.28e, m is generally taken as 1.0. For soft clays however, m can reduce to «0.4.

6.12 Passive pressure in cohesionless soils

6.12.1 Rankine's theory (soil surface horizontal)

In this case the vertical pressure due to the weight of the soil, 7I1, is acting as a minor principal stress. Figure 6.29a shows the Mohr circle diagram representing these stress conditions and drawn in the usual position, i.e. with the

(a) Mohr diagram drawn in usual position (b) Diagram correctly orientated

Fig. 6.29 Passive earth pressure for a cohesionless soil with a horizontal upper surface.

(a) Mohr diagram drawn in usual position (b) Diagram correctly orientated

Fig. 6.29 Passive earth pressure for a cohesionless soil with a horizontal upper surface.

axis OX (the direction of the major principal plane) horizontal. Figure 6.29b shows the same diagram correctly orientated with the major principal stress, Kp7h, horizontal and the major principal plane vertical. The Mohr diagram, it will be seen, must be rotated through 90°. In the Mohr diagram:

<ti OB OC + DC 1 + sin 0 _ 2/ 0\ ~a-i OA ~ OC-DC ~ 1 -sin0 ~ tan ^ + 2)

hence

As with active pressure, there is a network of shear planes inclined at (45° - 0/2) to the direction of the major principal stress, but this time the soil is being compressed as opposed to expanded.

6.12.2 Rankine's theory (soil surface sloping at angle (3)

The directions of the principal stresses are not known, but we assume that the passive pressure acts parallel to the surface of the slope. The analysis gives:

Note The amount of friction developed between a retaining wall and the soil can be of a high magnitude (particularly in the case of passive pressure). The Rankine theory's assumption of a smooth wall with no frictional effects can therefore lead to a significant underestimation (up to about a half) of the true Kp value. The theory can obviously lead to conservative design which, although safe, might at times be over-safe and lead to an uneconomic structure.

6.12.3 The Coulomb theory

With the assumption of a plane failure surface leading to a wedge failure, Coulomb's expression for Kp for a granular soil is:

the symbols having the same meanings as previously. The expression reduces to:

With passive pressure, unfortunately, the failure surface only approximates to a plane surface when the angle of wall friction is small.

The situation arises because the behaviour of the soil is not only governed by its weight but also by the compression forces induced by the wall tending to push into the soil. These forces, unlike the active case, do not act on only one plane within the soil, resulting in a non-uniform strain pattern and the development of a curved failure surface (Fig. 6.30).

It is apparent that in most cases the assumption of a Coulomb wedge for a passive failure can lead to a serious overestimation of the resistance available. Terzaghi (1943) first analysed this problem and concluded that, provided the angle of friction developed between the soil and the wall is not more than 0/3, where 0 is the operative value of the angle of friction of the soil, the assumption of a plane failure surface generally gives reasonable results. For values of 8 greater than 0/3, the errors involved can be very large.

Adjusted values for Kp that allow for a curved failure surface are given in Table 6.2. These values apply to a vertical wall and a horizontal soil surface and include the multiplier cos 6 as the values in the table give the components of pressure that will act normally to the wall.

It is seen therefore that for a smooth wall where 5 = 0° the Rankine theory can be used for the evaluation of passive pressure. If wall friction is mobilised then 6 ^ 0° and the coefficients of Table 6.2 should be used (unless 6 < 0/3 in which case the Coulomb equation can be used directly).

Fig. 6.30 Departure of passive failure surface from a plane.

Table 6.2 Values of Kp for cohesionless soils (Kerisel and Absi, 1990).

Values of Values of <f>

0 0

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