Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay

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2 Module 4: Lecture 6 on Stress-strain relationship and Shear strength of soils

3 Contents Stress state, Mohr s circle analysis and Pole, Principal stress space, Stress paths in p-q space; Mohr-Coulomb failure criteria and its limitations, correlation with p-q space; Stress-strain behavior; Isotropic compression and pressure dependency, confined compression, large stress compression, Definition of failure, Interlocking concept and its interpretations, Drainage conditions; Triaxial behaviour, stress state and analysis of UC, UU, CU, CD, and other special tests, Stress paths in triaxial and octahedral plane; Elastic modulus from triaxial tests.

4 Principal stress relations at failure: σ = σ tan (45 + φ / 2) + 2c tan(45 + φ / 2) τ σ 3 σ 1 X σ 3 2 = σ tan (45 φ / 2) 1 2c tan(45 φ / 2) σ 3 φ 90+φ σ 1 σ

5 Mohr-Coulomb Idealization of Geomaterials σ 1 -σ 3 σ 1 σ 2 = σ 3 E σ 3 σ 3 ε 1

6 Mohr diagram and failure envelopes Coulomb, in his investigations of retaining walls proposed a relationship: Where c is the inherent shear strength, also known as cohesion c and φ is angle of internal friction The criterion contains two material constants, c and φ, as opposed to one material constant for the Tresca criterion

7 Mohr Coulomb Yield/Failure Condition Yielding (and failure) takes place in the soil mass when mobilised (actual) shear stress at any plane (τ m ) becomes equal to shear strength (τ f ) which is given by: τ m = c + σ n tanφ = τ f where c and φ are strength parameters f (σ )= τ - σ n tanφ c = 0

8 Mohr-Coulomb Idealization of Geomaterials Note that the value of intermediate stress (σ 2 ) does not influence failure

9 Mohr-Coulomb Idealization of Geomaterials By constructing a Mohr circle tangent to the line (a stress state associated with failure) and using trigonometric relations, the alternative form of τ f = c + σ f tan φ in terms of principal stresses is obtained:

10 Relationship between K f line and Mohr-Coulomb failure envelope (in terms of principal stresses) q f = a + p f tanψ K f Ψ sinφ = tan Ψ a c = a/cosφ

11 Mohr-Coulomb(MC) failure criterion With no order implied by the principal stresses σ 1, σ 2, σ 3,the MC criterion can be written as: Where,

12 Mohr-Coulomb(MC) failure criterion T 0 is the theoretical MC uniaxial tensile strength that is not observed in experiments; rather, a much lower strength T is measured (σ 1 = 0, σ 2 = -T), with the failure plane being normal to σ 3. C 0 is the theoretical MC uniaxial compressive strength.

13 Mohr-Coulomb(MC) failure criterion The shape of the failure surface in principal stress space is dependent on the form of the failure criterion: linear functions map as planes and nonlinear functions as curvilinear surfaces. The following six equations below are represented by six planes that intersect one another along six edges, defining a hexagonal pyramid.

14 Pyramidal surface in principal stress space and cross-section in the equi-pressure plane This is the failure surface on the equipressure (σ 1 +σ 2 +σ 3 = constant) or π-plane perpendicular to thehydrostatic axis, where MC can be described as an irregular hexagon with sides of equal length.

15 Mohr-Coulomb in Principal stress space: Isotropy requires threefold symmetry because an interchange of σ 1, σ 2, σ 3 should not influence the failure surface for an isotropic material. Note that, the failure surface need only be given in any one of the 60 regions. Mohr-Coulomb failure surface is irregular hexagon in principal stress space.

16 Mohr-Coulomb in Principal stress space Consider the transformation from principal stress space (σ 1, σ 2, σ 3 ) to the Mohr diagram (σ, τ). Although the radial distance from the hydrostatic axis to the stress point is proportional to the deviatoric stress, a point in principal stress space does not directly indicate the value of shear stress on a plane. However, each point on the failure surface in principal stress space corresponds to a Mohr circle tangent to the failure envelope.

17 Mohr-Coulomb in Principal stress space For the particular case where σ 2 is the intermediate principal stress in the order σ 1 σ 2 σ 3, the failure surface is given by the side ACD of the hexagonal pyramid. The principal stresses at point D represent the stress state for a triaxial compression test (σ 1, σ 2 = σ 3 ) D, and point D is given by circle D in the Mohr diagram. Similarly, for point C with principal stresses (σ 3, σ 1 = σ 2 ) C associated with a triaxial extension test, Mohr circle C depicts the stress state. Points D and C can be viewed as the extremes of the intermediate stress variation, and the normal and shear stresses corresponding to failure are given by points D f and C f. Points lying on the line CD (on pyramid failure surface) will be represented by circles between C and D.

18 Mohr-Coulomb in Principal stress space: Mohr-Coulomb σ 1 It has corners that may sometime create problems in computations However, this particular difficulty is quite easily overcome by introducing a local rounding of the corners (Griffiths, 1990) σ 3 σ 2

19 Mohr-Coulomb model in p-q space As described earlier, F = ( σ ' σ ') ( σ ' + σ ')sinφ' 2c'cosφ' = σ = 3p' 2 ' σ 3 = σ ' q ' σ 1 3 σ = 3p' 2σ ' + 2q σ 1 ' 1 = 3p' 2σ ' q 3 ' 3 ' 1 3p' + 2q = 3 3p' q = 3 6 p ' +q σ ' +σ ') = 1 3 ( 3

20 Mohr-Coulomb model in p-q space Or, 6 p' q q sinφ + = + 2ccosφ 3 3q = 6 p'sinφ + qsinφ + 6ccosφ 6sinφ' 6ccosφ' q = p' + (3 sinφ') (3 sinφ') So, Formulation for Mohr-Coulomb model in p-q space is, F = q ηp' c * = 0 q =ηp' + c * where, η = 6sinφ' and, (3 sinφ') c * = 6ccosφ' (3 sinφ')

21 Limitations of Mohr-Coulomb theory: 1. Linearization of the limit stress envelope τ Usual experimental range in the laboratory φ, c σ Possible overestimation of the safety factor in slope stability calculations, Difficulties in calibration because of linearization

22 Mohr circles for three dimensional state of stress 2. Effect of intermediate principal stress σ 2 on condition at failure. It is obvious that σ 2 can have no influence on the conditions at failure for the Mohr failure criterion, no matter what magnitude it has. The intermediate principal stress σ 2 probably does have an influence in real soil, but the Mohr Coulomb failure theory does not consider it.

23 3. Mohr-Coulomb failure criterion is well proven for most of the geomaterials, but data for clays is still contradictory. 4. Soils on shearing exhibit variable volume change characteristics depending on pre-consolidation pressure which cannot be accounted with Mohr- Coulomb theory. 5. In soft soils volumetric plastic strains on shearing are compressive (negative dilation) whilst the Mohr- Coulomb model will predict continuous dilation.

24 Definition of failure : Mohr-Coulomb failure criteria: Failure along a plane in a material occurs by critical combination of normal and shear stress. τ = f(σ) τ = c + σtanφ Shear stress is function of material cohesion (c) and angle of internal friction (φ)

25 Definition of failure : Stress state cannot exist τ C τ = f(σ) B τ = c + σtanφ A Shear failure occurs Safe against failure σ

26 Flow Rule for Mohr Coulomb For Mohr-Coulomb flow rule is defined through the dilatancy angle of the soil. G(σ )= τ - σ n tanψ const.= 0 where ψ is the dilatancy angle and ψ φ

27 Interlocking concept and its interpretations : Frictional soil behaviour is mainly influenced by two factors: 1. Frictional resistance between the soil particles. 2. To expand the soil against confining pressure (Dilatancy). So, angular friction can be defined as: φ= φ u + β 3. where, φ is angle of sliding friction between mineral surfaces and β is the effect of interlocking.

28 Interlocking concept and its interpretations : φ = φ u + β φ varies with the nature of packing of the soil. Denser the packing, higher is the value of φ. If φ u for a given soil is constant, β must change with the denseness of the soil packing. β increases with increasing denseness of the soil, because more work to be done to overcome the effect of interlocking.

29 Interlocking concept and its interpretations : Effect of angularity of soil particles: Soil possessing angular soil particles will show higher friction angle than that of rounded soil particles. Because angular soil particles will show greater degree of interlocking and higher value of β.

30 Interlocking concept and its interpretations : Dilation φ= φ u + Ψ β is the function of dilatancy of the soil. x y dense y x x y + y loose

31 Interlocking concept and its interpretations : Dilation dense x y loose For loose sand the volume decreases with shearing For dense sand the volume increases with shearing

32 Interlocking concept and its interpretations : Dilation/ direct shear response Q/P loose dense y P x Q y dense x loose x

33 Interlocking concept and its interpretations : Dilation/ direct shear response Total work done, dw = Pδy + Qδx Pδy + Qδx = µpδx δy/δx = µ Q/P = - tanψ tanψ = tanφ m tanφ c, where, φ c = tan -1 µ x y P Q Alternatively, φ m = φ c + ψ

34 How to understand dilatancy i.e., why do we get volume changes when applying shear stresses? φ = ψ + φ i The apparent externally mobilized angle of friction on horizontal planes (φ) is larger than the angle of friction resisting sliding on the inclined planes (φ i ). strength = friction + dilatancy

35 How to understand dilatancy Bolton, 1991

36 How to understand dilatancy When soil is initially denser than the critical state which it must achieve, then as the particles slide past each other owing to the imposed shear strain they will, on average separate. The particle movements will be spread about mean angle of dilation Ψ See the orientation

37 How to understand dilatancy When soil is initially looser than the final critical state, then particles will tend to get closer together as the soil is disturbed, and the average angle of dilation will be negative, indicating a contraction. See the orientation

38 How to understand dilatancy If the density of the soil does not have to change in order to reach a critical state then there is zero dilatancy as the soil shears at constant volume. It is important to realise that a critical state is only reached when the particles have had full opportunity to juggle around and come into new configurations. If the confining pressure is increased while the particles are being moved around then they will tend to finish up in a more compact state.