55nptel.ac.in/courses/105101001/downloads/L55.pdfProf. B V S Viswanadham, Department of Civil...

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Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay 55

Transcript of 55nptel.ac.in/courses/105101001/downloads/L55.pdfProf. B V S Viswanadham, Department of Civil...

Page 1: 55nptel.ac.in/courses/105101001/downloads/L55.pdfProf. B V S Viswanadham, Department of Civil Engineering, IIT Bombay Mohr Coulomb Yield/Failure Condition Yielding (and failure) takes

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

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Page 2: 55nptel.ac.in/courses/105101001/downloads/L55.pdfProf. B V S Viswanadham, Department of Civil Engineering, IIT Bombay Mohr Coulomb Yield/Failure Condition Yielding (and failure) takes

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

Module 4: Lecture 6 on Stress-strain relationship

and Shear strength of soils

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

Stress state, Mohr’s circle analysis and Pole, Principalstress space, Stress paths in p-q space;

Mohr-Coulomb failure criteria and its limitations,correlation with p-q space;

Stress-strain behavior; Isotropic compression andpressure dependency, confined compression, large stresscompression, Definition of failure, Interlocking conceptand its interpretations, Drainage conditions;

Triaxial behaviour, stress state and analysis of UC, UU, CU,CD, and other special tests, Stress paths in triaxial andoctahedral plane; Elastic modulus from triaxial tests.

Contents

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

Principal stress relations at failure:

σ

τ

σ1σ3

φ 90+φ

Xσ3

σ1

)2/45tan(2)2/45(tan)2/45tan(2)2/45(tan

2

13

2

31

φφσσφφσσ

−−−=+++=

c

c

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

Mohr-Coulomb Idealization of Geomaterials

σ1′-σ3′

ε1

σ1′

σ2′ = σ3′

σ3′

E′σ3′

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

Mohr diagram and failure envelopesCoulomb, in his investigations of retaining walls proposeda relationship:

Where c is the inherentshear strength, alsoknown as cohesion cand φ is angle ofinternal frictionThe criterion contains twomaterial constants, c and φ, asopposed to one materialconstant for the Trescacriterion

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

Mohr Coulomb Yield/Failure Condition

Yielding (and failure) takes place in the soil mass whenmobilised (actual) shear stress at any plane (τm)becomes equal to shear strength (τf) which is given by:

τm = c′+ σ′ntanφ′ = τf

where c′ and φ′ are strength parameters

f (σ′ )= τ - σ′n tanφ′– c′= 0

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

Mohr-Coulomb Idealization of Geomaterials

Note that the value ofintermediate stress (σ2′)does not influencefailure

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

Mohr-Coulomb Idealization of GeomaterialsBy constructing a Mohr circletangent to the line (a stressstate associated with failure)and using trigonometricrelations, the alternative formof

in terms of principal stresses isobtained:

τf = c + σf tan φ

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

Relationship between Kf line and Mohr-Coulomb failure envelope (in terms of principal stresses)

ΨKf

a

qf = a + pf tanΨ

sinφ′ = tan Ψ

c′ = a/cosφ

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

Mohr-Coulomb(MC) failure criterion With no order implied by the principal stresses σ1, σ2, σ3,the MC criterion can be written as:

Where,

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

Mohr-Coulomb(MC) failure criterion T0 is the theoretical MC uniaxial tensile strength that is notobserved in experiments; rather, a much lower strength T ismeasured (σ1 = 0, σ2 = -T), with the failure plane being normalto σ3. C0 is the theoretical MC uniaxial compressive strength.

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

Mohr-Coulomb(MC) failure criterion

The shape of the failure surface in principal stressspace is dependent on the form of the failurecriterion: linear functions map as planes andnonlinear functions as curvilinear surfaces.

The following six equations below are represented bysix planes that intersect one another along six edges,defining a hexagonal pyramid.

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

Pyramidal surface in principal stress space andcross-section in the equi-pressure plane

This is the failure surface on the equipressure(σ1+σ2+σ3= constant) or π-plane perpendicular tothehydrostatic axis, where MC can be describedas an irregular hexagon with sides of equal length.

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

Mohr-Coulomb in Principal stress space:

Mohr-Coulomb failuresurface is irregularhexagon in principalstress space.

Isotropy requires threefoldsymmetry because aninterchange of σ1, σ2, σ3should not influence thefailure surface for anisotropic material. Notethat, the failure surfaceneed only be given in anyone of the 60° regions.

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

Consider the transformation from principal stress space(σ1, σ2, σ3) to the Mohr diagram (σ, τ).

Although the radial distance from the hydrostaticaxis to the stress point is proportional to thedeviatoric stress, a point in principal stress spacedoes not directly indicate the value of shear stress ona plane.

However, each point on the failure surface inprincipal stress space corresponds to a Mohr circletangent to the failure envelope.

Mohr-Coulomb in Principal stress space

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

For the particular case where σ2 is the intermediate principalstress in the order σ1 ≥ σ2 ≥ σ3, the failure surface is given by theside ACD of the hexagonal pyramid. The principal stresses atpoint D represent the stress state for a triaxial compression test(σ1, σ2 = σ3)D, and point D is given by circle D in the Mohrdiagram.

Similarly, for point C with principal stresses (σ3, σ1 = σ2)Cassociated with a triaxial extension test, Mohr circle C depicts thestress state. Points D and C can be viewed as the extremes of theintermediate stress variation, and the normal and shear stressescorresponding to failure are given by points Df and Cf.

Points lying on the line CD (on pyramid failure surface) will berepresented by circles between C and D.

Mohr-Coulomb in Principal stress space

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

Mohr-Coulomb

σ1’

σ3’ σ2’

It has corners that maysometime create problems incomputations

Mohr-Coulomb in Principal stress space:

However, this particulardifficulty is quite easily overcomeby introducing a local roundingof the corners (Griffiths, 1990)

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

Mohr-Coulomb model in p-q space

0'cos'2'sin)''()''( 3131 =−+−−= φφσσσσ cFAs described earlier,

'2'3' 31 σσ −= p q−= '' 13 σσ

32'32'2'3' 11

qpqp +=+−= σσ

3'3'2'3' 33

qpqp −=−−= σσ

3'6)''( 31

qp +=+σσ

Page 20: 55nptel.ac.in/courses/105101001/downloads/L55.pdfProf. B V S Viswanadham, Department of Civil Engineering, IIT Bombay Mohr Coulomb Yield/Failure Condition Yielding (and failure) takes

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

Mohr-Coulomb model in p-q space

Or,φφ cos2

3'6sin cqpq +

+

=

φφφ cos6sinsin'63 cqpq ++=

)'sin3('cos6'

)'sin3('sin6

φφ

φφ

−+

−=

cpq

*' cpq +=η

where,,

)'sin3('sin6 and

φφη

−= )'sin3(

'cos6*

φφ

−=

cc

So, Formulation for Mohr-Coulomb model in p-q space is,

0' * =−−= cpqF η

Page 21: 55nptel.ac.in/courses/105101001/downloads/L55.pdfProf. B V S Viswanadham, Department of Civil Engineering, IIT Bombay Mohr Coulomb Yield/Failure Condition Yielding (and failure) takes

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

Usual experimental range in the laboratory

Limitations of Mohr-Coulomb theory:1. Linearization of the limit stress envelope

φ, c

• Possible overestimation of the safety factor in slope stability calculations,• Difficulties in calibration because of linearization

τ

σ

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

Mohr circles for three dimensional state of stress

2. Effect of intermediate principal stress σ2 on condition at failure.

It is obvious that σ2 can have no influence on theconditions at failure for the Mohr failure criterion, nomatter what magnitude it has.

The intermediate principal stress σ2 probably doeshave an influence in real soil, but theMohr‐Coulomb failure theory does not consider it.

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

3. Mohr-Coulomb failure criterion is well proven for mostof the geomaterials, but data for clays is stillcontradictory.

4. Soils on shearing exhibit variable volume changecharacteristics depending on pre-consolidationpressure which cannot be accounted with Mohr-Coulomb theory.

5. In soft soils volumetric plastic strains on shearing arecompressive (negative dilation) whilst the Mohr-Coulomb model will predict continuous dilation.

Page 24: 55nptel.ac.in/courses/105101001/downloads/L55.pdfProf. B V S Viswanadham, Department of Civil Engineering, IIT Bombay Mohr Coulomb Yield/Failure Condition Yielding (and failure) takes

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

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)

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

Definition of failure :

A

B

σ

τ = f(σ)τ = c + σtanφ

Stress state cannot exist

Safe against failure

Shear failure occurs

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

Flow Rule for Mohr Coulomb

For Mohr-Coulomb flow ruleis defined through the‘dilatancy angle’ of the soil.

G(σ′)= τ - σ′n tanψ′ – const.= 0

where ψ′ is the dilatancyangle and ψ′ ≤ φ′

Page 27: 55nptel.ac.in/courses/105101001/downloads/L55.pdfProf. B V S Viswanadham, Department of Civil Engineering, IIT Bombay Mohr Coulomb Yield/Failure Condition Yielding (and failure) takes

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

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= φu + β3. where, φ is angle of sliding friction between mineral surfaces

and β is the effect of interlocking.

Page 28: 55nptel.ac.in/courses/105101001/downloads/L55.pdfProf. B V S Viswanadham, Department of Civil Engineering, IIT Bombay Mohr Coulomb Yield/Failure Condition Yielding (and failure) takes

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

Interlocking concept and its interpretations :

φ = φu + β

• φ varies with the nature of packing of the soil.

• Denser the packing, higher is the value of φ .

• If φu 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.

Page 29: 55nptel.ac.in/courses/105101001/downloads/L55.pdfProf. B V S Viswanadham, Department of Civil Engineering, IIT Bombay Mohr Coulomb Yield/Failure Condition Yielding (and failure) takes

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

Interlocking concept and its interpretations :

Effect of angularity of soil particles:

• Soil possessing angular soil particles will showhigher friction angle than that of rounded soilparticles.

• Because angular soil particles will showgreater degree of interlocking and highervalue of β.

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

Interlocking concept and its interpretations : Dilation

φ= φu + Ψ

• β is the function of dilatancy of the soil.∆x

∆x

∆y

∆y

∆x

+∆y

dense

loose

−∆y

Page 31: 55nptel.ac.in/courses/105101001/downloads/L55.pdfProf. B V S Viswanadham, Department of Civil Engineering, IIT Bombay Mohr Coulomb Yield/Failure Condition Yielding (and failure) takes

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

Interlocking concept and its interpretations : Dilation

• For loose sand the volume decreases with shearing• For dense sand the volume increases with shearing

∆x

∆y

dense

loose

Page 32: 55nptel.ac.in/courses/105101001/downloads/L55.pdfProf. B V S Viswanadham, Department of Civil Engineering, IIT Bombay Mohr Coulomb Yield/Failure Condition Yielding (and failure) takes

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

Interlocking concept and its interpretations : Dilation/ direct shear response

y

x

P

Q

Q/P

x

x

y

loose

loose

dense

dense

Page 33: 55nptel.ac.in/courses/105101001/downloads/L55.pdfProf. B V S Viswanadham, Department of Civil Engineering, IIT Bombay Mohr Coulomb Yield/Failure Condition Yielding (and failure) takes

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

Interlocking concept and its interpretations : Dilation/ direct shear response

y

x

P

Q

Total work done,

dW = Pδy + Qδx

Pδy + Qδx = µPδx

δy/δx = µ – Q/P = - tanψ

tanψ = tanφm − tanφc , where, φc = tan-1µ

Alternatively, φm = φc + ψ

Page 34: 55nptel.ac.in/courses/105101001/downloads/L55.pdfProf. B V S Viswanadham, Department of Civil Engineering, IIT Bombay Mohr Coulomb Yield/Failure Condition Yielding (and failure) takes

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

How to understand dilatancyi.e., why do we get volume changes when applying shear stresses?

φ = ψ + φi

The apparent externally mobilized angle of friction on horizontal planes (φ)is larger than the angle of friction resisting sliding on the inclined planes (φi).

strength = friction + dilatancy

Page 35: 55nptel.ac.in/courses/105101001/downloads/L55.pdfProf. B V S Viswanadham, Department of Civil Engineering, IIT Bombay Mohr Coulomb Yield/Failure Condition Yielding (and failure) takes

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

How to understand dilatancy

Bolton, 1991

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

When soil is initially denser than thecritical state which it must achieve, thenas the particles slide past each otherowing to the imposed shear strain theywill, on average separate.

The particle movements will be spreadabout mean angle of dilation Ψ

How to understand dilatancy

See the orientation

Page 37: 55nptel.ac.in/courses/105101001/downloads/L55.pdfProf. B V S Viswanadham, Department of Civil Engineering, IIT Bombay Mohr Coulomb Yield/Failure Condition Yielding (and failure) takes

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

When soil is initially looser than thefinal critical state, then particles willtend to get closer together as thesoil is disturbed, and the averageangle of dilation will be negative,indicating a contraction.

How to understand dilatancy

See the orientation

Page 38: 55nptel.ac.in/courses/105101001/downloads/L55.pdfProf. B V S Viswanadham, Department of Civil Engineering, IIT Bombay Mohr Coulomb Yield/Failure Condition Yielding (and failure) takes

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

If the density of the soil does not have tochange in order to reach a critical statethen there is zero dilatancy as the soilshears at constant volume.

It is important to realise that a criticalstate is only reached when the particleshave had full opportunity to jugglearound and come into newconfigurations. If the confining pressure isincreased while the particles are beingmoved around then they will tend tofinish up in a more compact state.

How to understand dilatancy