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# Craig's Soil Mechanics 7th Edition

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of plane strain in which the strain in the direction of the intermediate principal stress is zero due to restraint imposed by virtue of the length of the structure in question. In the triaxial test, consolidation proceeds under equal all-round pressure (i.e. isotropic consolidation), whereas in-situ consolidation takes place under anisotropic stress conditions. Tests of a more complex nature, generally employing adaptions of triaxial equip- ment, have been devised to simulate the more complex states of stress encountered in practice but these are used principally in research. The plane strain test uses a prismatic specimen in which strain in one direction (that of the intermediate principal stress) is maintained at zero throughout the test by means of two rigid side plates tied together. The all-round pressure is the minor principal stress and the sum of the applied axial stress and the all-round pressure the major principal stress. A more sophisticated test, also using a prismatic specimen, enables the values of all three principal stresses to be controlled independently, two side pressure bags or jacks being used to apply the intermediate principal stress. Independent control of the three principal stresses can also be achieved by means of tests on soil specimens in the form of hollow cylinders in which different values of external and internal fluid pressure can be applied in addition to axial stress. Torsion applied to the hollow cylinders results in the rotation of the principal stress directions. Because of its relative simplicity it seems likely that the triaxial test will continue to be the main test for the determination of shear strength characteristics. If considered necessary, corrections can be applied to the results of triaxial tests to obtain the characteristics under more complex states of stress. 4.3 SHEAR STRENGTH OF SANDS The shear strength characteristics of a sand can be determined from the results of either direct shear tests or drained triaxial tests, only the drained strength of a sand normally being relevant in practice. The characteristics of dry and saturated sands are the same, provided there is zero excess pore water pressure in the case of saturated sands. Typical curves relating shear stress and shear strain for initially dense and loose sand specimens in direct shear tests are shown in Figure 4.8(a). Similar curves are obtained relating principal stress difference and axial strain in drained triaxial com- pression tests. In a dense sand there is a considerable degree of interlocking between particles. Before shear failure can take place, this interlocking must be overcome in addition to the frictional resistance at the points of contact. In general, the degree of interlocking is greatest in the case of very dense, well-graded sands consisting of angular particles. The characteristic stress–strain curve for an initially dense sand shows a peak stress at a relatively low strain and thereafter, as interlocking is 102 Shear strength progressively overcome, the stress decreases with increasing strain. The reduction in the degree of interlocking produces an increase in the volume of the specimen during shearing as characterized by the relationship, shown in Figure 4.8(c), between volumetric strain and shear strain in the direct shear test. In the drained Dense Dense Dense Loose Loose Loose Same σ′ Same e0 φcv σ′ Different σ′ φmax φcv ecv (φmax) (φcv) (φµ) σ′ (a) (b) (secant) (c) (e) (g) (d) Vo lu m e in cr ea se Vo lu m e de cr ea se dεvεv dγ τ τ τ/σ′ τ γ γ γ γ γ e e e0 (f ) A B C A B C A B C ′ ′ ′ ′ ′ ′ Figure 4.8 Shear strength characteristics of sand. Shear strength of sands 103 triaxial test a similar relationship would be obtained between volumetric strain and axial strain. The change in volume is also shown in terms of void ratio (e) in Figure 4.8(d). Eventually the specimen would become loose enough to allow particles to move over and around their neighbours without any further net volume change and the shear stress would reach an ultimate value. However, in the triaxial test non- uniform deformation of the specimen becomes excessive as strain is progressively increased and it is unlikely that the ultimate value of principal stress difference can be reached. The term dilatancy is used to describe the increase in volume of a dense sand during shearing and the rate of dilation can be represented by the gradient d"v/d�, the maximum rate corresponding to the peak stress. The angle of dilation ( ) is tan�1 (d"v/d�). The concept of dilatancy can be illustrated in the context of the direct shear test. During shearing of a dense sand the macroscopic shear plane is horizontal but sliding between individual particles takes place on numerous microscopic planes inclined at various angles above the horizontal, as the particles move up and over their neighbours. The angle of dilation represents an average value of this angle for the specimen as a whole. The loading plate of the apparatus is thus forced upwards, work being done against the normal stress. For a dense sand the maximum angle of shearing resistance (�0max) determined from peak stresses (Figure 4.8(b)) is significantly greater than the true angle of friction (�m) between the surfaces of individual particles, the difference representing the work required to overcome interlocking and rearrange the particles. In the case of initially loose sand there is no significant particle interlocking to be overcome and the shear stress increases gradually to an ultimate value without a prior peak, accompanied by a decrease in volume. The ultimate values of stress and void ratio for dense and loose specimens under the same values of normal stress in the direct shear test are essentially equal as indicated in Figures 4.8(a) and (d). Thus at the ultimate (or critical) state, shearing takes place at constant volume, the corresponding angle of shearing resistance being denoted �0cv (or � 0 crit). The difference between � 0 m and �0cv represents the work required to rearrange the particles. It may be difficult to determine the value of the parameter �0cv because of the relatively high strain required to reach the critical state. In general, the critical state is identified by extrapolation of the stress–strain curve to the point of constant stress, which should also correspond to the point of zero rate of dilation on the volumetric strain–shear strain curve. Stresses at the critical state define a straight line failure envelope intersecting the origin, the slope of which is �0cv. In practice the parameter �0max, which is a transient value, should only be used for situations in which it can be assumed that strain will remain significantly less than that corresponding to peak stress. If, however, strain is likely to exceed that corresponding to peak stress, a situation that may lead to progressive failure, then the critical-state parameter �0cv should be used. An alternative method of representing the results from direct shear tests is to plot the stress ratio �/�0 against shear strain. Plots of stress ratio against shear strain representing tests on three specimens of sand, each having the same initial void ratio, are shown in Figure 4.8(e), the values of effective normal stress (�0) being different in each test. The plots are labelled A, B and C, the effective normal stress being lowest in test A and highest in test C. Corresponding plots of void ratio 104 Shear strength against shear strain are shown in Figure 4.8(f ). Such results indicate that both the maximum stress ratio and the ultimate (or critical) void ratio decrease with increas- ing effective normal stress. The ultimate values of stress ratio,