The shear strength characteristics of coarse-grained soils such as sands and gravels can be deter- mined from the results of either direct shear tests or drained triaxial tests, only the drained strength of such soils normally being relevant in practice. The strength and stiffness characteris- tics of dry and saturated sands or gravels are the same, provided there is zero excess pore water pressure generated in the case of saturated soils, as these characteristics are dependent on effec- tive stress. Typical curves relating shear stress and shear strain for initially dense and loose sand specimens in direct shear tests are shown in Figure 5.17(a). Similar curves are obtained relating principal stress difference (q) and axial strain (εa) in drained triaxial compression tests (CD).
In dense deposits (high relative density, ID, see Chapter 1) there is a considerable degree of interlocking between particles. Before shear failure can take place, this interlocking must be over- come in addition to the frictional resistance at the points of inter-granular contact. In general, the degree of interlocking is greatest in the case of very dense, well-graded soils consisting of angular particles. The characteristic stress–strain curve for initially dense sand shows a peak stress at a relatively low strain, and thereafter, as interlocking is 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 characterised by the relationship between volumetric strain and shear strain in the direct shear test, shown in Figure 5.17(c). In the drained 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 5.17(d). Eventually the speci- men becomes loose enough to allow particles to move over and around their neighbours without any further net volume change, and the shear stress reduces to 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.
Figure 5.16 Unconfined compression test interpretation.
The term dilatancy is used to describe the increase in volume of a dense coarse-grained soil during shearing, and the rate of dilation can be represented by the gradient dεv/dγ, the maxi- mum rate corresponding to the peak stress (Figure 5.17(c)). The angle of dilation (ψ) is defined as tan–1(dεv/dγ). The concept of dilatancy can be illustrated in the context of the direct shear test by considering the shearing of dense and loosely packed spheres (idealised soil particles) as shown in Figure 5.18. During shearing of a dense soil (Figure 5.18(a)), 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 these angles for the specimen as a whole. The loading plate of the apparatus is thus forced upwards, work being done against the normal stress on the shear plane. For a dense soil, the maximum (or peak) angle of shearing resistance (fmax¢ ) determined from peak stresses (Figure 5.17(b)) is significantly greater than the true angle of friction (ϕμ) 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 soil (Figure 5.18(b)), 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 (Figure 5.17(c)). The ultimate values of shear stress and void ratio for dense and loose specimens of the same soil under the same values of normal stress in the direct shear test are essentially equal, as indicated in Figure 5.17(a) and (d). The ultimate resistance occurs when there is no further change in volume or shear stress (Figure 5.17(a) and (c)), which is known as the Critical State. Stresses at the critical state define a straight line (Mohr–Coulomb) Figure 5.17 Shear strength characteristics of coarse-grained soils.
Soil behaviour in shear
failure envelope intersecting the origin, known as the Critical State Line (CSL), the slope of which is tanfcv¢ (Figure 5.17(b)). The corresponding angle of shearing resistance at Critical State (also called the critical state angle of shearing resistance) is usually denoted fcv¢ or fcrit¢ . The subscript
‘cv’ indicates ‘constant volume’ to imply that shearing is occurring with no further change in vol- ume. The difference between f¢m and fcv¢ represents the work required to rearrange the particles.
The friction angles fcv¢ and fmax¢ are related to ψ after the relationship given by Bolton (1986):
¢ = ¢ +
fmax fcv 0 8. y (5.38)
Equation 5.38 applies for conditions of plane strain within soil, such as those induced within the DSA or SSA. Under triaxial conditions, the final term becomes approximately 0.5ψ.
It can be difficult to determine the value of the parameter f¢cv from laboratory tests 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 cor- respond to the point of zero rate of dilation (dεv/dγ = 0) on the volumetric strain–shear strain curve.
An alternative method of representing the results from laboratory shear tests is to plot the stress ratio (τ/σ′ in direct shear) against shear strain. Plots of stress ratio against shear strain represent- ing tests on three specimens of sand in a direct shear test, each having the same initial void ratio, are shown in Figure 5.19(a), the values of effective normal stress (σ′) 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 against shear strain are also shown in Figure 5.19(b).
Such results indicate that both the maximum stress ratio and the ultimate (or critical) void ratio decrease with increasing effective normal stress. Thus, dilation is suppressed by increasing mean effective stress (normal effective stress σ′ in direct shear). This is described in greater detail by Bolton (1986). The ultimate values of stress ratio (= tanfcv¢, i.e. at Critical State), however, are the same. From Figure 5.19(a) it is apparent that the difference between peak and ultimate stress decreases with increasing effective normal stress; therefore, if the maximum shear stress is plotted against effective normal stress for each individual test, the plotted points will lie on an envelope which is slightly curved, as shown in Figure 5.19(c). Figure 5.19(c) also shows the stress paths for Figure 5.18 Mechanics of dilatancy in coarse-grained soils: (a) initially dense soil, exhib-
iting dilation, (b) initially loose soil, showing contraction.
each of the three specimens leading up to failure. For any type of shear test, two stress paths may be plotted: the total stress path (TSP) plots the variation of σ and τ through the test; the effec- tive stress path (ESP) plots the variation of σ′ and τ. If both stress paths are plotted on the same axis, the horizontal distance between the two paths at a given value of τ (i.e. σ–σ′) represents the pore water pressure in the sample from Terzaghi’s Principle (Equation 3.1). In direct shear tests, the pore water pressure is approximately zero such that the TSP and ESP lie on the same line, as shown in Figure 5.19(c). Remembering that it is the effective (not total) stresses that govern soil shear strength (Equation 5.12), failure occurs when the ESP reaches the failure envelope.
The value of fmax¢ for each test can then be represented by a secant parameter: in the shearbox test, fmax¢ =tan (-1tmax / )s¢. The value of fmax¢ decreases with increasing effective normal stress until it becomes equal to fcv¢. The reduction in the difference between peak and ultimate shear stress with increasing normal stress is mainly due to the corresponding decrease in ultimate void ratio.
The lower the ultimate void ratio, the less scope there is for dilation. In addition, at high stress levels some fracturing or crushing of particles may occur, with the consequence that there will be less particle interlocking to be overcome. Crushing thus also causes suppression of dilatancy and contributes to the reduced value of fmax¢ .
In the absence of any cementation or bonding between particles, the curved peak failure enve- lopes for coarse-grained soils would show zero shear strength at zero normal effective stress.
Mathematical representations of the curved envelopes may be expressed in terms of power laws, i.e. of the form τf = AγB. These are not compatible with many standard analyses for geotechnical Figure 5.19 Determination of peak strengths from direct shear test data.
Soil behaviour in shear structures which require soil strength to be defined in terms of a straight line (Mohr–Coulomb model, Equation 5.12). It is common, therefore, in practice to fit a straight line to the peak failure points to define the peak strength in terms of an angle of shearing resistance ϕ′ and a cohesion intercept c′. It should be noted that the parameter c′ is only a mathematical line-fitting constant used to model the peak states at τ> 0, and should not be used to imply that the soil has shear strength at zero normal effective stress. This parameter is therefore also commonly referred to as the apparent cohesion of the soil. In soils which do have natural cementation/bonding, the cohe- sion intercept will represent the combined effects of any apparent cohesion and the true cohesion due to the interparticle bonding.
Once soil has been sheared to the Critical State (ultimate conditions), the effects of any true or apparent cohesion are destroyed. This is important when selecting strength properties for use in design, particularly where soil has been tested under its in-situ condition (where line-fitting may suggest c′> 0), then sheared during excavation and subsequently placed to support a foundation or used to backfill behind a retaining structure. In such circumstances the excavation/placement imposes large shear strains within the soil such that critical state conditions (with c′ = 0) should be assumed for design calculations.
Figure 5.20 shows the behaviour of soils A, B and C as would be observed in a drained tri- axial test (CD). The main differences compared to the behaviour in direct shear (Figure 5.19) lie
Figure 5.20 Determination of peak strengths from drained triaxial test data.
in the appearance of the stress paths and failure envelope shown in Figure 5.20(c). In standard triaxial compression, radial stress is held constant (Δσr = 0) while axial stress is increased (by Δσa). From Equations 5.23 and 5.24, this gives Δp = Δσa/3 and Δq = Δσa. The gradient of the TSP is therefore Δq/Δp = 3. In a drained test there is no change in pore water pressure, so the ESP is parallel to the TSP. If the sample is dry, the TSP and ESP lie along the same line (as in Figure 5.19); if the sample is saturated and a back pressure of u0 applied, the TSP and ESP are parallel, maintaining a constant horizontal separation of u0 throughout the test, as shown in Figure 5.20(c). As before, failure occurs when the ESP meets the failure envelope. The value of fmax¢ for each test is determined by finding M (= q/p′) at failure and using Equation 5.29, with the resulting value of f¢ = ¢fmax.
In practice, the routine laboratory testing of coarse-grained soils is difficult because of the problem of obtaining undisturbed specimens and setting them up, still undisturbed, in the test apparatus. If required, tests can be undertaken on specimens reconstituted in the apparatus at appropriate densities, but the in-situ structure is then unlikely to be reproduced. Frozen specimens can be collected with the aim of preserving the in-situ structure, but this is expensive. As a result, in-situ tests (Chapter 7) are often preferred to laboratory tests for coarse-grained soils.
Example 5.1
The results shown in Figure 5.21 were obtained from direct shear tests on reconstituted specimens of sand taken from loose and dense deposits and compacted to the in-situ den- sity in each case. The raw data from the tests and the use of a spreadsheet to process the test data may be found on the Companion Website. Plot the failure envelopes of each sand for both peak and ultimate states, and hence determine the critical state friction angle fcv¢.
Solution
The values of shear stress at peak and ultimate states are read from the curves in Figure 5.21 and are plotted against the corresponding values of normal stress, as shown in Figure 5.22.
The failure envelope is the line having the best fit to the plotted points; for ultimate con- ditions a straight line through the origin is appropriate (CSL). From the gradients of the failure envelopes, fcv ¢ = 33.4° for the dense sand and 32.6° for the loose sand. These are within 1° of each other, and demonstrate that the critical state friction angle is an intrinsic Figure 5.21 Example 5.1.
Soil behaviour in shear
soil property which is independent of state (i.e. density). The loose sand does not exhibit peak behaviour, while the peak failure envelope for the dense sand may be characterised by c′ = 15.4 kPa (apparent cohesion) and ϕ′ = 38.0° or by secant values as given in Table 5.1.
Liquefaction
Liquefaction is a phenomenon in which loose saturated sand loses a large percentage of its shear strength due to high excess pore water pressures, and develops characteristics similar to those of a liquid. It is usually induced by cyclic loading over a very short period of time (usually seconds), resulting in undrained conditions in the sand despite the high permeability. Cyclic loading may be caused, for example, by vibrations from machinery and, more seriously, by earthquakes.
Loose sand tends to compact under cyclic loading. The decrease in volume causes an increase in pore water pressure which cannot dissipate under undrained conditions. Indeed, there may be a cumulative increase in pore water pressure under successive cycles of loading. If the pore water pressure becomes equal to the maximum total stress, normally the overburden pressure, σv, the value of the effective stress will be zero by Terzaghi’s Principle, as described in Section 3.7 – i.e.
interparticle forces will be zero, and the sand will exist in a quasi-liquid state with negligible shear strength. Even if the effective stress does not fall to zero, the reduction in shear strength may be sufficient to cause failure.
Liquefaction may develop at any depth in a sand deposit where a critical combination of in-situ density and cyclic deformation occurs. The higher the void ratio of the sand and the lower the
Figure 5.22 Example 5.1: Failure envelopes for (a) loose, and (b) dense sand samples.
Table 5.1 Example 5.1
Normal effective stress (kPa) Secant peak friction angle, fmax¢ (°)
50 46.8
100 43.7
181 40.8
confining pressure, the more readily liquefaction will occur. The larger the strains produced by the cyclic loading, the lower the number of cycles required for liquefaction.
Liquefaction may also be induced under static conditions where pore pressures are increased as a result of seepage. The techniques described in Chapter 2 and Section 3.7 may be used to deter- mine the pore water pressures in such cases and, by Terzaghi’s Principle, the effective stresses in the soil for a given seepage event. The shear strength at these low effective stresses is then approxi- mated by the Mohr–Coulomb criterion.