Normal stress Fig. 9.13 Points of maximum shear stress.
Normal stress Fig. 9.13 Points of maximum shear stress.
If a Mohr circle diagram of stress is examined (Fig. 9.13) the point of maximum shear has the co-ordinates s and t where:
a 1 and 0-3 being the total principal stresses.
In terms of effective stresses, a\ and a'3, the point of maximum shear has the co-ordinates s' and t' where:
If a soil is subjected to a range of values of a[ and a\ the point of maximum shear stress can be obtained for each stress circle; the line joining these points, in the order that they occurred, is termed the stress path or stress vector of maximum shear. Any other point instead of maximum shear can be used to determine a stress path, e.g. the point of maximum obliquity, but
Lambe maintains that the stress paths of maximum shear are not only simple to use but also more applicable to consolidation work.
Typical effective stress paths obtained from a series of consolidated undrained triaxial tests on samples of normally consolidated clay together with the effective stress circles at failure are shown in Fig. 9.14.
If the results of a drained shear test on a soil are considered, the Mohr circle diagram is as shown in Fig. 9.14. The line tangential to the stress circles is the strength envelope, inclined at <f>' to the normal stress axis. If each Mohr circle is considered it is seen that the ratio cr\\a\ is a constant, to which the symbol Kf is applied.
If the points of maximum shear for each effective stress circle p'f and qf are joined together the stress path of maximum shear stress at failure is obtained. This line is called the Kf line and is inclined at angle a! to the normal stress axis; obviously tana' = sin^'.
For a soil undergoing one-dimensional consolidation the ratio <t'3/<t i is again constant and its value is given the symbol K0. Plotting the maximum shear stress points of these stress circles enables the stress path for one-dimensional consolidation, the Kc line, to be determined; this line is inclined at angle [3 to the normal stress axis.
K0 is the coefficient of earth pressure at rest. For consolidation work K0 may be defined for a soil with a history of one-dimensional strain as the ratio:
^ Lateral effective stress ° Vertical effective stress
Figure 9.15 shows the stress conditions that arise during and after the application of a pressure increment in the consolidation test. Initially the sample has been consolidated under a previous load and the pore pressure is zero; the Mohr circle is represented by (p, q) the point X, circle I. As soon as the vertical pressure increase, Actj, is applied, the total stresses move from X to Y (circle 1). As the soil is saturated Au = Aa\ and the effective stress circle is still represented by point X. As consolidation commences the pore water pressure, Au, begins to decrease and A£r'[ begins to increase. The consolidation is one-dimensional and therefore an increase in the major principal effective stress, Act',, will induce an increase in the minor principal effective stress Act'3 — K^Arr',. Hence the effective stress circles move steadily towards point Z (circles II, III and IV), where Z represents full consolidation.
The total stress circles can be determined from a study of the effective stress circles. For example the difference between Aen and Aa\ for circle III
represents the pore water pressure within the sample at that time: hence Acrj at this stage in the consolidation is Acr'3 for circle III plus the value of the pore water pressure. It can be seen therefore that Au decreases with consolidation and the size of the Mohr circle for total stress increases until the point Z is reached (circles 2, 3 and 4). Obviously circles 4 and IV are coincident.
The effective stress plot of Fig. 9.16 represents a typical case of general consolidation. The soil is normally consolidated and point A represents the initial Kc consolidation; AB is the effective stress path on the application of the foundation load and BC is the effective stress path during consolidation.
Skempton and Bjerrum's assumption that lateral strain effects during consolidation can be ignored presupposes that the strain due to the stress path BC is the same as that produced by the stress path DE. The fact that the method proposed by Skempton and Bjerrum gives reasonable results indicates that the effective stress path during the consolidation of soil in a typical foundation problem is indeed fairly close to the effective stress path DE of Fig. 9.16. There are occasions when this will not be so, however, and
the stress path method of analysis can give a more reasonable prediction of settlement values (see Lambe, 1964, 1967). The calculation of settlement in a soft soil layer under an embankment by this procedure has been discussed by Smith (1968a), and the method is also applicable to spoil heaps.
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