Shear test similitude
Soil can only fail under conditions of local loading where the loading in the zone of in¯uence distorts the soil mass as a whole. Beneath a rapidly constructed embankment, for example, the previously horizontal ground surface de¯ects, the zone of in¯uence distorting without at the instant of loading changing its volume. The major principal stress direction is orientated vertically under the embankment. Towards the toe of the embankment the principal stress directions rotate until the major principal stress is horizontal. If model shear tests are to be used to predict the undrained shear strength in these varying stress and deformation zones the tests must, ideally, simulate the actual ®eld stress and deformation paths.
Excess pore pressure at failure for case B
Time Time P ore w ater pressure F actor of saf ety 1·0 A B
Fig. 1.10 The variation with time of overall stability and pore water pressure at a characteristic location for a cutting. Curve A indicates limiting equilibrium in the long term. Curve B indicates limiting equilibrium before the long term condition is reached
To approximate to similitude, Bjerrum (1972) suggests using different modes of shear test to evaluate the undrained shear strength for different areas of the distorted soil. Thus the triaxial compression test simulates distor- tion directly under the embankment where the shear surface is inclined near the major principal stress direction, the simple shear test simulates distortion where the shear surface is nearly horizontal, and the triaxial extension test simulates distortion near the toe of the embankment where the shear surface is inclined near the minor principal stress direction (Fig. 1.11).
The alternative to matching the shear test to the ®eld failure zone on the basis of like soil distortions is to adopt a purely empirical approach. Bjer- rum (1972) compared the stability of embankments that had failed with the predicted stabilities based on in situ shear vane measurements the results of which were used in limit analyses. Depending on the plasticity of the clay the shear vane overestimated the ®eld strength by up to 100%. There is a clear lack of geometrical similitude between the vane which shears the soil on an upright circumscribing cylinder, and the ®eld prototype which may fail along a circular arc in the vertical plane. The disparity between the strength measured by the shear vane and the ®eld strength is ascribed by Bjerrum to the combined effects of anisotropy, progressive failure and testing rate.
Analytical similitude
In the foregoing section the term `®eld strength' refers simply to a number that gives a realistic estimate of ®eld stability when used in a traditional
Embankment Slip surface Triaxial extension test Triaxial compression test Simple shear test
Fig. 1.11 Relevance of laboratory shear tests to shear strength mobilized in the ®eld (after Bjerrum, 1972)
limit analysis, assuming a rigid±plastic shear stress±displacement rela- tionship which does not vary with direction (Fig. 1.12). The limit analytical model does not therefore allow for the effect of progressive failure. Most naturally occurring soils are strain softening. Irrespective of whether the shear test is drained or undrained, they possess a shear stress± deformation relationship that is characterized by a peak followed by a reduction in strength to an ultimate or residual strength. In normally- consolidated clays, the reduction in strength from peak to residual may be slight while in over-consolidated clays the reduction is marked giving a brittle behaviour.
In a homogeneous soil which is not loaded, direct shear tests on speci- mens sampled along a circular arc through the soil will have different stress±deformation relationships due to the variation of strength with depth and with orientation. Distorting the soil in the area of the circular arc by a local surface loading will modify the stress±deformation relation- ships. During loading the stresses vary in the zone of in¯uence and hence the shear strength around the circular arc varies according to the effective stresses normal to the circular arc. If the circular arc becomes a slip surface some local regions will fail. Here the shear stresses tangential to the slip surface have exceeded the local soil strength generated by the local normal effective stresses. The strength in these failed regions reduces according to the strain softening behaviour. In the pre-failure regions, on the other hand, the shear stresses, increased by the load shedding of the strain softening zones will not have exceeded the soil strength available. Displacement Shear stress Rigid plastic Real soil sp s sr
Fig. 1.12 Shear stress±displacement relationship as measured in the direct shear test for an ideal rigid±plastic material and for a real soil. The Residual Factor (Skempton, 1964) is R spÿ s= spÿ sr
Astate of limiting equilibrium is attained when the reduction in strength of the elements of the slip surface in the failure zone just begins to exceed the increase in stress taken by those elements in the pre-failure zone (Bishop, 1971a, b). Some of the failed elements may have reduced in strength to the residual value but this is not strictly necessary for limiting equilibrium in a strain softening material. Indeed, the large displacements necessary to achieve ultimate residual strength in heavily over-consolidated clays (Bishop et al., 1971) would suggest that failure conditions in such a material, in the sense of a factor of safety of unity, might obtain without any element of the slip surface reaching the residual strength (Bishop, 1971b).
Conventional stability analyses do not model progressive failure. On the one hand, to take the peak strength as acting on all elements simulta- neously around the slip surface overestimates the factor of safety. On the other hand, to take the residual shear strength as acting simultaneously around the slip surface underestimates the factor of safety and is clearly inappropriate in any event as the residual shear strength holds on an established shear surface, i.e. after failure has occurred.
Summary
The applicability of shear test data to ®eld stability problems rests on the principle that similitude exists between the shear test model, the analyti- cal model and the ®eld prototype. Testing and analytical methods must be directed towards ensuring such similitude and assessing the effects of any lack of similitude that may arise. These points will be taken up in more detail in the following chapters.