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# Craig's Soil Mechanics 7th Edition

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active state. If the deformation of the wall were not to satisfy the minimum deformation requirement, the soil adjacent to the wall would not reach a state of plastic equilibrium and the lateral pressure would be between the active and at-rest values. If the wall were to deform by rotation about its upper end (due, for example, to restraint by a prop), the conditions for the complete development of the active state would not be satisfied because of inadequate strain in the soil near the surface; consequently, the pressure near the top of the wall would approximate to the at-rest value. In the passive case the minimum deformation requirement is a rotational movement of the wall, about its lower end, into the soil. If this movement were of sufficient magnitude, the passive state would be developed within a wedge of soil between the wall and a failure plane at an angle of 45� þ �/2 to the vertical as shown in Figure 6.17(b). In practice, however, only part of the potential passive resistance would normally be mobilized. The relatively large deformation necessary for the full devel- opment of passive resistance would be unacceptable, with the result that the pressure under working conditions would be between the at-rest and passive values, as indi- cated in Figure 6.10 (and consequently providing a factor of safety against passive failure). Figure 6.17 Minimum deformation conditions. 184 Lateral earth pressure The selection of an appropriate value of �0 is of prime importance in the passive case. The difficulty is that strains vary significantly throughout the soil mass and in particular along the failure surface. The effect of strain, which is governed by the mode of wall deformation, is neglected both in the failure criterion and in analysis. In the earth pressure theories, a constant value of �0 is assumed throughout the soil above the failure surface whereas, in fact, the mobilized value of �0 varies. In the case of dense sands the average value of �0 along the failure surface, as the passive condition is approached, corresponds to a point beyond the peak on the stress–strain curve (e.g. Figure 4.8(a)); use of the peak value of �0 would result, therefore, in an overestimation of passive resistance. It should be noted, however, that peak values of �0 obtained from triaxial tests are normally less than the corresponding values in plane strain, the latter being relevant in most retaining wall problems. In the case of loose sands, the wall deformation required to mobilize the ultimate value of �0 would be unacceptably large in practice. Guidance on design values of �0 is given in codes of practice [5, 9]. The values of lateral strain required to mobilize active and passive pressures in a particular case depend on the value of K0, representing the initial state of stress, and on the subsequent stress path, which depends on the construction technique and, in particular, on whether backfilling or excavation is involved in construction. In general, the required deformation in the backfilled case is greater than that in the excavated case for a particular soil. It should be noted that for backfilled walls, the lateral strain at a given point is interpreted as that occurring after backfill has been placed and compacted to the level of that point. 6.5 DESIGN OF EARTH-RETAINING STRUCTURES There are two broad categories of retaining structures: (1) gravity, or freestanding walls, in which stability is due mainly to the weight of the structure; (2) embedded walls, in which stability is due to the passive resistance of the soil over the embedded depth and, in most cases, external support. According to the principles of limit state design, an earth-retaining structure must not (a) collapse or suffer major damage, (b) be subject to unacceptable deformations in relation to its location and function and (c) suffer minor damage which would necessitate excessive maintenance, render it unsightly or reduce its anticipated life. Ultimate limit states are those involving the collapse or instability of the structure as a whole or the failure of one of its compon- ents. Serviceability limit states are those involving excessive deformation, leading to damage or loss of function. Both ultimate and serviceability limit states must always be considered. The philosophy of limit states is the basis of Eurocode 7 [9], a standard specifying all the situations which must be considered in design. The design of retaining structures has traditionally been based on the specification of a factor of safety in terms of moments, i.e. the ratio of the resisting (or restoring) moment to the disturbing (or overturning) moment. This is known as a lumped factor of safety and is given a value high enough to allow for all the uncertainties in the analytical method and in the values of soil parameters. It must be recognized that relatively large deformations are required for the mobilization of available passive resistance and that a structure could be deemed to have failed due to excessive deformation before reaching a condition of collapse. The approach, therefore, is to Design of earth-retaining structures 185 base design on ultimate limit states with the incorporation of an appropriate factor of safety to satisfy the requirements of serviceability limit states. In general, the higher the factor of safety, the lower will be the deformation required to mobilize the proportion of passive resistance necessary for stability. The limit state approach is based on the application of partial factors to actions and soil properties. Partial factors are denoted by the symbol �, unfortunately the same symbol as is used for unit weight. Partial load factors are denoted by �F, material factors by �m and resistance factors by �R. In general, actions include loading, soil weight, in-situ stresses, pore water pressure, seepage pressure and ground movements. Actions are further classified as being either permanent or variable and as having either favourable or unfavourable effects in relation to limit states. Soil properties relevant to the design of retaining structures are c0, tan �0 and cu, as appropriate. Conservative values of the shear strength parameters, deduced from reliable ground investigation and soil test results, are referred to as the characteristic values: they may be either upper or lower values, whichever is the more unfavourable. Characteristic values of shear strength parameters are divided by an appropriate partial factor to give the design value. The design values of actions, on the other hand, are obtained by multiplying characteristic values by an appropriate factor. The subscripts k and d can be used to denote characteristic and design values, respectively. If the resisting moment in a particular design problem is greater than or equal to the disturbing moment, using design values of actions and soil properties, then the limit state in question will be satisfied. These principles form the basis of Eurocode 7 (EC7): Geotechnical Design [9], currently in the form of a prestandard prior to the publication of a final version. Three design cases (A, B and C) are specified in EC7 and are described in detail in Section 8.1. The geotechnical design of retaining structures is normally governed by Case C which is primarily concerned with uncertainties in soil properties. Partial factors currently recommended for this case are 1.25, 1.60 and 1.40 for tan �0, c0 and cu, respectively. The factor for both favourable and unfavourable permanent actions is 1.00 and for variable unfavourable actions 1.30. Variable favourable actions are not considered, i.e. the partial factor is zero. Case A is relevant to the overturning, and Case B to the structural design, of a retaining structure. In the limit state approach, the design values of active thrust and passive resist- ance are calculated from the design values of the relevant