Shear Strength of Soil. Hsin-yu Shan Dept. of Civil Engineering National Chiao Tung University

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1 Shear Strength of Soil Hsin-yu Shan Dept. of Civil Engineering National Chiao Tung University

2 Principal stress space a b c

3 Rendulic Plot a r r r

4 Rendulic Plot Axial Stress a ' Radial Stress r a We can plot stress paths of compression and extension tests without having them cross over one another r

5 Redulic Diagram. S-test: compression by increase of axial stress. R -test: compression by increase of axial stress a A f A f = > A f < 3 Af = 0 < 0 4 A f u d Compression test envelope (S.P.T.) Stress path for isotropic consolidation a = r Space diagonal 3. S-test: compression by decrease confining stress 4. Elastic material: on octahedral plane r

6 φ φ φ φ sin cos ] sin sin 3[ + + = c φ φ φ φ sin cos ] sin sin [ = c φ φ φ φ sin cos ] sin sin [ + + = c r a ψ tan d

7 Redulic Diagram. S-test: extension by increase of radial stress 3. R -test: extension by decrease of axial stress a Stress path for isotropic consolidation A f 3 > A f 4 = A f < Space diagonal = a r A f A f = 0 Extension test envelope (S.P.T.) < 0. S-test: extension by decrease axial stress 4. Elastic material: on octahedral plane r

8 φ φ φ φ sin cos ] sin sin [ + + = c a r φ φ φ φ sin cos ] sin sin [ + = c r a φ φ φ φ sin cos ] sin sin [ + = c r a ψ tan d

9 e.g. c = = 30 0 φ For compression For extension + sinφ tanψ = ψ = 65 sinφ sinφ tanψ = ψ = 3 + sinφ For space diagonal tan ( ) = 35

10 e.g. c = = 30 0 φ a 65 Space diagonal 35 = a r 3 r

11 Henkel (967) Compression Tests Extension Tests Tests φ Tests φ Undrained 3 Undrained.5 Drained, a Drained, a.5 Drained, r Drained, r.5 Drained, mean = const. Drained, mean = const.

12 N.C. Clay Henkel (960) Monitor the volume of water flowed into or out of the specimen during CD tests or trace effective stress path of CU tests Contour of constant water content (stress path of CU test) a w = 7% a = r w = 0% r

13 a For c=0 a = b = c b Octahedral plane m = ( a + b + c ) = 3 oct c

14 Stress is constant for any point on the octahedral plane oct = const. ) ( ) ( ) ( 3 a c c b b a r + + = r τ oct 3 = a r b c ) ( ) ( ) ( 3 a c c b b a oct τ + + = τ oct = octahedral shear stress = maximum shear stress on the octahedral plane

15 Failure Criteria Mohr-Coulomb Tresca (Extended Tresca) von Mises

16 Mohr-Coulomb For > > 3 ( 3) = ( + 3) sinφ Independent of T.C. a oct = ( a + b + c ) = 3 const. T.E. T.E. [( ) [( ) [( ) 3 3 ( + ) ( + ) ( + ) 3 3 sinφ] sinφ] sinφ] = 0 b T.C. T.E. T.C. c

17 a For c=0 b If c is not 0, but φ is, the shape would be a hexagonal column c

18 Tresca + + 3) = α = 3 ( 3 T.C. a α oct Often used for c=0 and φ=0 α is the Tresca parameter equivalent to φ Extended Tresca for φ > 0 T.E. T.E. a b c b c a α α α oct oct oct b T.C. T.E. T.C. c

19 Von Mises (φ = 0) a b c ( a b) + ( b c ) + ( c a ) = α ( T.C. a + + ) 3 Extended Tresca for φ > 0 T.E. T.E. c = 0 cone φ = 0 cyliner 9τ oct = α oct = const. Circle b T.C. T.E. T.C. c

20 If b = c + 3 a b ( a b) = α ( )

21 T.C. a T.E. T.E. b T.C. T.E. T.C. c

22 3 3 b = Bishop φ sin 3 3 = + Mohr-Coulomb ) 3 3 ( 3 3 b + = + α Extended Tresca ) 3 3 ( 3 3 b b b + + = + α Extended von Mises

23 φ f = sin 3 ( ) + 3 Extended von Mises Extended Tresca M-C T.E. M-C T.C. b

24 Triaxial compression tests: Mohr-Coulomb criteria underestimate strength in plane strain Von Mises criteria overestimate strength in plane strain

25 Krey (97) Consolidated-Undrained Direct Shear Tests (Actually, it s difficult to control drainage in DS tests) Consolidate the specimen and then quickly reduce and then shear it τ φ φ c τ = c + tanφ f m c Stress path m max

26 Tiedemann (933) τ f = m tan φ + tanφ c r NC clay failure envelope τ tanφ r c m tanφ c φ r φ c Tiedemann failure envelope (only for < m ) If m changes, the envelope changes c = f( m ) m If truly undrained, φ r should be zero. τ f The maximum past pressure the soil was consolidated = tanφ m c

27 Gibson (953) Ph.D Dissertation φ r Bentonite, P.I. = P.I.

28 Hvorslev Drained direct shear tests c = K True cohesion e e It is a function of the void ratio c e e Virgin consolidation curve

29 φe True friction τ f = K e + tanφe τ f e τ f e = K + tanφe e This translates into a SET of envelopes one envelope for each e or e because c e = φ e K e K e

30 He took the soil out of shear zone and study the distribution of stress carefully Clay w L w PI < µ K φe Wiener Tegel % Kleil Belt Ton % As w L, w PI, < µ K and φe

31 Bjerrum (954) Ph.D Dissertation Basically unsuccessful in determining the Hvorslev coefficients τ f = c 0 = c0 + K e + tanφe e e c + K τ Intercept cohesion φ c φ e c 0 c e

32 Tests on specimens of the same void ratio tends to have same strength Hard to determine different τ Hard to determine c 0 and K e

33 Test by consolidating the specimens to different max, unload to the same e and shear 3 f e During shear: no drainage e = const. e Critical confining stress Virgin consolidation curve

34 Tests on specimens of the same void ratio tends to have same strength! 3 same 3 S.P.T. Hvorslev envelope Critical state line 3 f No change in effective stress for the specimen sheared at critical void ratio 3

35 Critical State Soil Mechanics p = I 3 I = first stress invariant = p = ( + + 3) = oct 3 For triaxial compression: = 3 MIT people: p p = = 3 ( + 3 ( + 3 ) )

36 3 J q = J = deviatoric component of the second stress invariant oct J + = 3 J τ oct = q τ oct 3 = 3 3 ) ( ) ( ) ( 3 τ + + = oct

37 For triaxial compression: = 3 τ oct = 3 ( 3) q = ( ) 3 p = q = 3 ( + 3 ( ) 3 )

38 For c=0 Space diagonal 3

39 Tension zone q 3 3 < 0 y z Space diagonal 3p y = ( + + 3) = 3 oct = 3p 3 z = 3 τ = oct 3 q 3

40 Tension zone 3 q 3 = 0 3 p

41 Tension zone q 3 State path: p, q, e (or other variables) path p

42 N.C. Clay Undrained Loading q Hvorslev s envelopes S.P.T. R Envelope of NC clay = Projection of critical void ratio (c.v.r.) line on p, q plane NC stress path p p p 3 p4 p

43 e Virgin isotropic consolidation curve p p p3 p4 p

44 Overconsolidate Clay Undrained loading p e Increase OCR e Virgin consolidation curve

45 3 same void ratio tends to have same strength common point (at large strain) S.P.T. same 3 Projection of critical void ratio (c.v.r.) line on p, q plane points on critical void ratio (c.v.r.) line NC 3 f No change in effective stress for the specimen sheared at critical void ratio 3

46 Drained tests p, q, e are changing throughout the test

47 Roscoe, S, and Wroth They did not distinguish between ( 3 ) max and ( ) max 3 Useful in pile driving where the strain is very large As a building block for developing other stress-strain models They adopted von Mises criteria in order to get from -D to 3-D

48 Line of critical overconsolidation ratio A line on the q = 0 plane that marks the boundary of soil tends to dilate or contract q On C.V.R. 3 Virgin consolidation curve Critical OCR line p Projection of C.V.R. line

49 e (w) Critical OCR line C.V.R. line Start from the left, dilate, e Start from the right, contract, e Drained test: no change in e Wet Dry Virgin consolidation curve Drained tests p

50 Dry of critical: tends to dilate during shear, heavily overconsolidated clay (on the left of the critical OCR line) Wet of critical: tends to contract during shear, normally consolidated or slightly overconsolidated clay (on the right of the critical OCR line)

51 Undrained Tests q u = 0 On C.V.R. 3 u Critical OCR line The only starting point with no pore water pressure change p

52 e Critical OCR line C.V.R. Line Drained path Projection of stress path on the 3: plane Undrained path p p p 3 p4 p

53 Critical OCR line (R, S, & W) Depends on The nature of the tests Soil properties May not be parallel to the virgin consolidation curve

54 Partly Saturated Soils Q envelopes τ = + tan Q cq φq τ φ Q τ Concave downward c Q For partly saturated soils volume changes in Q tests

55 Q envelopes τ Low S r Intermediate S r High S r S r different effective stress is different

56 S envelopes after saturation τ Essentially one envelope regardless of S r at compaction

57 Capillary pressure Whenever there is a curved air-water interface, there s must be pressure drop

58 Capillary and the Capillary Fringe Capillary: water molecules at the water table subject to an upward attraction due to surface tension of the air-water interface and the molecular attraction of the liquid and the solid phases.

59 Tension If fluid pressures are measured above the water table, they will be found to be negative with respect to local atmospheric pressure

60 Air Pressure If air in the pores is connected, the air pressure is equal to the local atmospheric pressure If air in the pores is not connected and is in the forms of air bubbles such as in the capillary fringe, the air pressure is equal to the pressure of water, which is negative.

62 Capillary rise in a tube cosλ = T s i λ s f r R λ λ=0 R=r h c = cosλ ρ gr w h c u c = h c γ w = T s / γ w water

63 Fig. 6. IDEALIZED pore diameter in a sediment with cubic packing. The equivalent capillary tube has a radius of 0. the diameter of the grains

64 P atm z(m) P atm ψ p 0.4 ψ h (=0) ψ g water 0 y (J/kg) y r l (kj /cu.m = kp a) y /g (J/N = m)

65 z(m) water ψp ψh ψg =gz ψ (J/Kg)

66 z (m) ψ g = gz h heads (m)

67 Height of Capillary Rise Sediments Grain Diameter (cm) Pore Radius (cm) Capillary Rise (cm) Fine silt Coarse silt Very fine sand Fine sand Medium sand Coarse sand Very coarse sand Fine gravel

68 Water Potential The potential energy, or force potential of ground water consists of two parts: elevation and pressure (velocity related kinetic energy is neglected)

69 Suction of Water in Soils Fluid pressures in the vadose zone are negative, owing to tension of the soilsurface-water contact The negative pressure head is measured in the field with a tensiometer

70 d b z Soil Surface z c a Suction Calibrate Before Use

71 Suction Matric suction Elevation head Pressure head Osmotic suction

72 Head (Water Potential) Gravity potential, Z Elevation head Moisture potential, ψ Suction head Can be several orders of magnitude greater than the gravity potential bar 0 m of water column

73 Soil Water Characteristics Defines the relationship between water content and water potential (suction)

74 θ h m (m) SWCC of a Coarse Sand

75 Soil Water Characteristic Curve

76 SWCC of various soils

77 Absorption and desorption characteristics with primary scanning curves

78 Poorly sorted -0 4 P ressure head (cm) h b W ell sorted Entry pressure h b Water content,θ

79 Drying scanning curve ψ Main drying curve Main wetting curve Wetting scanning curve Volumetric water content, θ Absorption and desorption characteristics with primary scanning curves

80 Hysteresis Ink bottle effect Trapping of air Advancing and receding contact angle

81 Determination of SWCC Laboratory Pressure plate apparatus Filter paper Thermocouple Psychrometer Centrifuge Field Tensiometer Water content measurement

82 soil Ceramic plate water z (m) Effluent port that can vary elevation ψ g = gz h heads (m)

83 Pychrometers Measure relative humidity in close proximity to air-water interface h c uc ( γ w ) = RT Mg H ln( ) 00% H = relative humidity R = the gas constant T = absolute temp. M = molecular wt. of water vapor g = acceleration due to gravity

84 Typical values of u c (psi) H (%)

85 Amount of NaCl in water (g/l) Osmotic suction at 0 C (bars)

86 Most negative pore pressure is measured from: Highly plastic fine-grained soil

88 (A) (B) (C) (D)

89 (a) (b) (c)

90 Effective stress Terzaghi (93) : = u

91 Skempton 960 = ku k= (-a tanϕ/tanφ) for shear strength k = - C s /C for compressibility a: contact area of solids on a cross section ϕ: friction angle of solids φ: friction angle of soil C s : compressibility of solids C: compressibility of soil skeleton

92 Stress/Pressure u a pore air pressure u w pore water pressure u c = u a - u w

93 Partly saturated soil Bishop s chi theory (955) = χ ( u ) + ( u u ) a a w Oven-dried soil S r = 0% = u a χ = 0 Saturated soil S r = 00% = uw χ = 0

94 Bishop(960) and Aitchison(960) = k u k u w a k =- k, if k = χ = ( ua ) + χ( ua uw) χ depends on S r and void ratio (e)

95 Fredlund&Morgenstern(977) Control s, u a, u w and observe the volume change of soils (fix two and let one of them change at a time) If any one of them were kept constant, volume remained the same

96 Mongiovi Tarantino Keep the volume of the soil specimen constant observe the change of stress variables Very small changes of the stress variables The effective stress of unsaturated soil is controlled by two stress variables: ( u a ) and (u a u w )

97 Allam & Sridharan (987) = ( u + χ u u + γt ) ( ) a a w π γ = perimeter of referenced air-water interface T = air-water interfacial tension make solids get closer to each other π = tension of dissolved material

98 Sr

99 Allam & Sridharan (987) Kaolinite S r = 0.3, γt accounts for 3 36% of the suction Effect of π was not significant

100 Fredlund(979) Regina clay: Water content: 4% 30% π accounts for 40% 80% of the total suction

101 Tensor expression a z zy zx yz a y yx xz xy a x u u u τ τ τ τ τ τ w a w a w a u u u u u u

102 When S r 00%, u w u a, (u a u w ) 0 w z zy zx yz w y yx xz xy W x u u u τ τ τ τ τ τ

103

104 Pore pressure coefficient χ Bishop, Blight, and Donald(960): χ is between 0 and Sparks(963) based on force equilibrium: χ = π sinθ + cosθ 4 cosθ πτ rp sinθ + cosθ cosθ

105 p = pore water pressure = T = surface tension of water r = radius of soil grains Τcosθ r ( cosθ ) ( sinθ + cosθ ) θ = angle between surface of water and center of soil grains, When S r 00% and q is 45 χ =.57

106 Blight(967) Get χ by CD and CU tests When close to saturation, c may become > Blight thought it might due to lack of lab data, and this was not real However, he did not consider the effect of air-water interface If consider γt in the formulation c = S r ; but not experimental evidence to support this χ = + 3 ( u u ) a ( + ) w 3 u a

107 Shear strength of unsaturated soil According to Bishop, the generalized Mohr- Coulomb failure criteria is: τ f = c + ( u ) tan φ + [ χ ( u u )] tan φ n a a w This equation is difficult to verify, because it is difficult to determine χ

108 Fredlund Suggests that the shear strength is controlled by two independent variables: ( -u a ) and (u a -u w ) He and Morgenstern suggest: τ f = c + ( u ) tanφ + ( u u ) tanφ n a a w b ( n u a ) and (u a u w ) is the normal stress and matric suction on the failure surface φ b is the friction angle associated with the matric suction; generally it is not a constant, but decreases as the matric tension increases

109 Relationship between φ b and matric suction (Gan and Fredlund,988)

110 Lu (99) Suggests the following criteria for expansive soils: τ f = ( u ) tan φ P tan φ c + + n a s P s : expansive pressure of soil

111 Xu (998) Based on the concept of fractal dimension on expansive soil: τ f = c + ( ) φ ( ) n mn+ u tan + k u u tan φ n a a w k: fitting coefficient m, n: fractal dimension parameters

112 The introduced theories and criteria have a common problem: difficult to measure the parameters Vanapalli And Fredlund(996) developed an empirical equation relating the shear strength and the soil water characteristic curve

113 Fredlund et al. (978 The shear strength provided by the matric suction, τ us, is: τ us = ( u a u w ) tan φ b

114 φ b varies with the water content of soil, i.e., it varies with the matric suction of soil tanφ = b d = ( u u ) [( )( ) w Θ ] k τ tanφ us dτ a ( u u ) a w = ( Θ) + ( u a u w d ) d( u ( Θ) a k u Θ is the normalize volumetric water content θ s is the volumetric water content at saturation k is a fitting parameter has to be determined by experiments k w ) Θ tanφ = θ θ s

115 Vanapalli et al. (996) Obtain shear strength based on SWCC ( ) ( ) φ θ θ θ θ φ τ tan tan + + = r s r w a a n f u u u c

Stress and Strains in Soil and Rock. Hsin-yu Shan Dept. of Civil Engineering National Chiao Tung University

Stress and Strains in Soil and Rock Hsin-yu Shan Dept. of Civil Engineering National Chiao Tung University Stress and Strain ε 1 1 2 ε 2 ε Dimension 1 2 0 ε ε ε 0 1 2 ε 1 1 2 ε 2 ε Plane Strain = 0 1 2