SHEARING STRENGTH OF SOILS SYMBOLS Notation Dimensional Analysis b = a length L c = the cohesion intercept M L-1 T-2 Ip = the plasticity index - P = a force M L T-2 u = the pore fluid pressure ML-1 T-2 θ = an angle Angle µ1 = shear strength reduction factor related to time effects - µ2 = shear strength reduction factor related to fissuring - σ = the normal stress on a plane M L-1 T-2 τ = the shearing stress on a plane M L-1 T-2 φ = the angle of internal friction Angle ψ = an angle related to cohesion Angle Subscripts etc. where not identified above ´ = parameter measured in terms of intergranular or effective quantities c = preconsolidation values cu = consolidated undrained parameters d = drained test parameters i.e. dissipated pore pressures e = so-called true parameters f = failure values max = maximum values n = normal to plane values nc = normally consolidated values o = overburden values u = unconsolidated undrained test parameters 1 = major principal values 2 = intermediate principal values 3 = minor principal values 1. INTRODUCTION Stability analysis in geotechnical engineering includes all studies which attempt to determine whether or not the average shearing strength of soil over the assumed failure surface has a sufficient factor of safety against failure. Basically such studies consist of comparisons between all the forces which are or may act to cause failure and the resisting forces provided by the soil's shearing strength. The shearing strength of a soil sample is generally defined as its maximum resistance to shearing forces. In special cases an ultimate, Figure 1. Stress-strain curves for sand. residual or post peak value is used. The peak and ultimate values are shown on the normal stress-strain plot of test results in Figure 1. The residual value (Skempton, 1964) may be taken as the value recorded on a presheared sample or on an intact sample which is subject to excessive shearing, such as in a ring shear apparatus (Bishop et al, 1971), where the Coulomb's original shearing strength equation or Law is shown on soil particles can rearrange themselves into preferred orientations. Figure 3 and was expressed in terms of total stresses by Normally stability problems are solved by approximating the stress-strain behaviour by an ideal rigid-plastic material as shown in Figure 2. Where such an assumption is made the stability can be expressed in terms of τf ' c % σn tan φ (1) some definition of the shearing strength τf alone (this need not be a maximum value of τ). In actual fact the failure strength varies with the normal stress on the failure plane. Coulomb is credited with being the first person to express this variable It should be noted that unless φ = 0 the value of τ is not equal to the failure strength in terms of two engineering properties or parameters, f maximum shearing strength stress τ . This relationship expresses both namely: max the variation of strength of a single sample to changing external stresses (1) cohesion (c), or the resistance due to the forces tending to bond or or the locus of test results using different samples of the same type soil. hold the soil particles together in a solid mass; Another point worth noting is that the equation must have some limitation (2) internal friction (φ), or the rate of change of the resistance due to an otherwise soil slopes with slope angles less than φ could theoretically be increase of normal stress (σ ) on the failure plane. n infinitely high. Particle crushing will obviously occur first. Shearing Strength of Soils -- GEOTECHNICAL ENGINEERING-1997 -- Prof. G.P. Raymond© 99 remoulded soils, showed that the cohesion intercept was dependent on the moisture or water content of the failing soil sample. This led to what has become termed the true cohesion (ce) and true angle of internal friction (φe) which are determined as shown in Figure 4 and a restatement of equation (1) by τf ' ce % σ´n tan φe (4) ce ' σ´nc tan ψe (5) where σ´nc is the effective intergranular pressure which would have to be applied normally to the failure plane of a normally consolidated sample of the same soil to give a failure water content of the same value as the failing sample, and ψe is an angle which relates the true cohesion to σ´nc. Hvorslev's proposals are useful as an academic model for soil strength but are normally not used in practice. Figure 2. Stress-strain curve for ideal rigid-plastic material. Figure 3. Coulomb's law. 2. EFFECTIVE STRESS CONCEPT Unfortunately the values of c and φ were not found to be constant values. Their usefulness was not universally accepted until Terzaghi's concept of effective intergranular stress (σ´) was postulated. This related the soil's behaviour to the difference between the total stresses (σ) and soil fluid pressure (u) by σ´ ' σ & u (2) Figure 4. Hvorslev's concept of true cohesion and friction angle (e.g. Gibson, 1953) In accordance with Terzaghi's concept the shearing strength of a soil may be expressed as 4. PRACTICAL CONSIDERATIONS τ ' c´ % (σ & u)tanφ´ ' In nature soil has been affected by weathering and other phenomenon f n (3) c´ % σ´n tan φ´ so that fundamental relationships, such as suggested by Hvorslev, have not found general acceptance in practical engineering. In addition the where the primes signify that the soil parameters are determined using shearing strength of soil is, like most other engineering material, effective stresses or equivalent intergranular stresses. dependent on factors such as creep and fatigue. For these and other more complex reasons the shearing strength parameters used in practice are 3. TRUE PARAMETER CONCEPT generally based on simplifying conditions of determining the stability Hvorslev (e.g. Gibson, 1953) working with multiple samples of against immediate failure and (or) long term failure depending on the Shearing Strength of Soils -- GEOTECHNICAL ENGINEERING-1997 -- Prof. G.P. Raymond© 100 nature of the problem. expression for shearing strength (equation (1)) and then apply the result to practical problems in terms of short term undrained or total stress In the analysis of immediate stability it is assumed that the soil has a parameters (equation (6)) or long term drained or effective stress very low permeability and the moisture content of the soil will remain parameters (equation (13)). unchanged during the course of the engineering works. For such conditions the soil is tested rapidly enough to ensure undrained conditions. 6. RELATIONSHIP BETWEEN PRINCIPAL STRESSES AND Interpretation of the test results is then based on considering the soil as a FAILURE SHEARING STRESS single phase material much as most other engineering materials (i.e. steel, The relationship between the principal major and minor stress and concrete and the like). Analysis is then performed by working in total failure shearing stress using the Coulomb failure criteria is illustrated by stresses. The shearing strength equation for immediate, total stress or resolution of the forces on the element shown in Figure 5. (Note that the undrained stability analysis for a soil at a given moisture content intermediate principal stress has no theoretical effect on the Coulomb failure criteria). τf ' cu % σn tan φu (6) where cu = the undrained cohesion φu = the undrained angle of internal friction It has been shown by Skempton (1948a) and Bishop and Eden (1951) that when soil samples at the same moisture content and same stress history are fully saturated and the pore fluid and soil solids are incompressible in comparison with the soil skeleton φu ' 0 (7) This result is extensively used when dealing with immediate stability problems involving saturated or near saturated clays and relatively impermeable saturated silts for which, very simply (further corrections Figure 5. Elemental stresses on random plane. may be required prior to application) for the identical saturated samples Resolving forces (P) normal to any assumed plane making an angle θ τf ' cu (8) with the plane on which the major principal stress acts For saturated samples with different moisture contents cu would, of course, Pn ' P1 cos θ % P3 sin θ (14) vary. Thus in a natural soil deposit cu may vary with depth. b On the other hand where the permeability of the soil is high and rapid σn ' σ1 b cos θ % σ3 b tan θ sin (15) dissipation of pore fluid pressures occur, such as with clean sands and cos θ gravel, or in low permeability soils where a change in moisture content is likely during loading, the shearing strength is expressed in terms of 2 2 ˆ σn ' σ1 cos θ % σ3 sin θ (16) effective intergranular stresses as given by equation (3). The parameters are sometimes referred to as drained (more logically dissipated pore fluid where subscripts 1 and 3 refer to major and minor principal values, and b pressure) parameters is a length over which P1 acts. Resolving forces parallel to any assumed plane to find the shear force τf ' cd % σn tan φd (9) Pτ on the plane In a drained test the pore fluid pressures are zero (or used as the zero Pτ ' P1 sin θ & P3 cos θ (17) datum when not zero) so that c ' c´ b d (10) τ ' cos θ (18) σ1 b sin θ & σ3 b tan θ cos θ σn ' σ´n (u ' 0) (11) ˆ τ ' (σ1 & σ3)sinθ cos θ (19) φd ' φ´ (12) Thus equation (9) becomes identical to equation (3) or If the assumed plane is the failure plane and Coulomb's relationship is τf ' c´ % (σn & u)tanφ´ (13) taken as being valid on the plane It should be clearly understood that the term "drained" refers to the τ ' τf ' c % σn tan φ (20) dissipation of pore fluid pressures and not to the drainage under gravity of pore fluid from the soil.