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# Craig's Soil Mechanics 7th Edition

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The special case of a vertical wall and a horizontal soil surface will now be considered. For the undrained condition (�u ¼ 0), an expression for P can be obtained by resolving forces vertically and horizontally. The total active thrust is given by the maximum value of P, for which @P/@� ¼ 0. The resulting value is Pa ¼ 1 2 �ðH2 � z20Þ � 2cuðH � z0Þ ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1þ cw cu � �s For �u ¼ 0 the earth pressure coefficient Ka is unity and it is convenient to introduce a second coefficient Kac, where Kac ¼ 2 ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1þ cw cu � �s For the fully drained condition in terms of tangent parameters c0 and �0, it can be assumed that Kac ¼ 2 ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Ka 1þ cw c0 � h ir In general, the active pressure at depth z can be expressed as pa ¼ Ka�z� Kacc ð6:18Þ where Kac ¼ 2 ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Ka 1þ cw c � h ir ð6:19Þ Figure 6.14 Coulomb theory: active case with c > 0. Coulomb’s theory of earth pressure 179 the shear strength parameters being those appropriate to the drainage conditions of the problem. The depth of a dry tension crack (at which pa ¼ 0) is given by z0 ¼ 2c ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1þ cw c � r � ﬃﬃﬃﬃﬃﬃ Ka p ð6:20Þ The depth of a water-filled crack (z0w) is obtained from the condition pa ¼ �wz0w. Hydrostatic pressure in tension cracks can be eliminated by means of a horizontal filter. Passive case In the passive case, the reaction P acts at angle � above the normal to the wall surface (or � below the normal if the wall were to settle more than the adjacent soil) and the reaction R at angle � above the normal to the failure plane. In the triangle of forces, the angle between W and P is 180� � þ � and the angle between W and R is �þ �. The total passive resistance, equal to the minimum value of P, is given by Pp ¼ 1 2 Kp�H 2 ð6:21Þ where Kp ¼ sinð þ �Þ sin ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ½sinð � �Þ�p � ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ sinð�þ �Þ sinð�þ �Þ sinð � �Þ � s 0 BBBB@ 1 CCCCA 2 ð6:22Þ However, in the passive case it is not generally realistic to neglect the curvature of the failure surface and use of Equation 6.22 overestimates passive resistance, ser- iously so for the higher values of �, representing an error on the unsafe side. It is recommended that passive pressure coefficients derived by Caquot and Kerisel [8] should be used. Caquot and Kerisel derived both active and passive coefficients by integrating the differential equations of equilibrium, the failure surfaces being logarithmic spirals. Coefficients have also been obtained by Sokolovski [24] by numerical integration. In general, the passive pressure at depth z can be expressed as pp ¼ Kp�zþ Kpcc ð6:23Þ where Kpc ¼ 2 ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Kp 1þ cw c � h ir ð6:24Þ 180 Lateral earth pressure Kerisel and Absi [13] published tables of active and passive coefficients for a wide range of values of �, �, and �, the active coefficients being very close to those calculated from Equation 6.17. For ¼ 90� and � ¼ 0�, values of coefficients (denoted Kah and Kph) for horizontal components of pressure (i.e. Ka cos � and Kp cos �, respectively) are plotted in Figure 6.15. 30 20 10 5 4 3 2 1 0.5 0.4 0.3 0.2 0.1 10° 15° 20° 25° 30° 35° 40° 45° 0 0 0.67φ′ 0.33φ′ 0.67 φ′ φ′ δ δ φ′ Active Passive Kph Kah φ′ Figure 6.15 Coefficients for horizontal components of active and passive pressure. Coulomb’s theory of earth pressure 181 Compaction-induced pressure In the case of backfilled walls, lateral pressure is also influenced by the compaction process, an effect that is not considered in the earth pressure theories. During back- filling, the weight of the compaction plant produces additional lateral pressure on the wall. Pressures significantly in excess of the active value can result near the top of the wall, especially if it is restrained by propping during compaction. As each layer is compacted, the soil adjacent to the wall is pushed downwards against frictional resistance on the wall surface. When the compaction plant is removed the potential rebound of the soil is restricted by wall friction, thus inhibiting reduction of the additional lateral pressure. Also, the lateral strains induced by compaction have a significant plastic component which is irrecoverable. Thus, there is a residual lateral pressure on the wall. A simple analytical method of estimating the residual lateral pressure has been proposed by Ingold [10]. Compaction of backfill behind a retaining wall is normally effected by rolling. The compaction plant can be represented approximately by a line load equal to the weight of the roller. If a vibratory roller is employed, the centrifugal force due to the vibrating mechanism should be added to the static weight. The vertical stress immediately below a line load Q per unit length is derived from Equation 5.14: �z ¼ 2Q z Then the lateral pressure on the wall at depth z is given by pc ¼ Kað�zþ �zÞ When the stress �z is removed, the lateral stress may not revert to the original value (Ka�z). At shallow depth the residual lateral pressure could be high enough, relative to the vertical stress �z, to cause passive failure in the soil. Therefore, assuming there is no reduction in lateral stress on removal of the compaction plant, the maximum (or critical) depth (zc) to which failure could occur is given by pc ¼ Kp�zc Thus Kað�zc þ �zÞ ¼ 1 Ka �zc If it is assumed that �zc is negligible compared to �z, then zc ¼ K 2 a�z � ¼ K 2 a � 2Q zc 182 Lateral earth pressure Therefore zc ¼ Ka ﬃﬃﬃﬃﬃﬃﬃ 2Q � s The maximum value of lateral pressure ( pmax) occurs at the critical depth, therefore (again neglecting �zc): pmax ¼ 2QKa zc ¼ ﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2Q� r ð6:25Þ The fill is normally placed and compacted in layers. Assuming that the pressure pmax is reached, and remains, in each successive layer, a vertical line can be drawn as a pressure envelope below the critical depth. Thus the distribution shown in Figure 6.16 represents a conservative basis for design. However, at a depth za the active pressure will exceed the value pmax. The depth za, being the limiting depth of the vertical envelope, is obtained from the equation Ka�za ¼ ﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2Q� r Thus za ¼ 1 Ka ﬃﬃﬃﬃﬃﬃﬃ 2Q � s ð6:26Þ 0 pmax pc 0 pmax pc zc zc za z z pp pa σz Figure 6.16 Compaction-induced pressure. Coulomb’s theory of earth pressure 183 6.4 APPLICATION OF EARTH PRESSURE THEORY TO RETAINING WALLS In the Rankine theory the state of stress in a semi-infinite soil mass is considered, the entire mass being subjected to lateral expansion or compression. However, the move- ment of a retaining wall of finite dimensions cannot develop the active or passive state in the soil mass as a whole. The active state, for example, would be developed only within a wedge of soil between the wall and a failure plane passing through the lower end of the wall and at an angle of 45� þ �/2 to the horizontal, as shown in Figure 6.17(a); the remainder of the soil mass would not reach a state of plastic equilibrium. A specific (minimum) value of lateral strain would be necessary for the development of the active state within the above wedge. A uniform strain within the wedge would be produced by a rotational movement (A0B) of the wall, away from the soil, about its lower end and a deformation of this type, of sufficient magnitude, constitutes the minimum deformation requirement for the development of the active state. Any deformation configuration enveloping A0B, for example, a uniform translational movement A0B0, would also result in the development of the