Soil Plasticity and the Structured Cam Clay Model

Transcription

1 Soil Plasticity an the Structure Cam Clay Moel Dr. Martin D. Liu School of Ciil Engineering Uniersity of New South Wales Syney, NSW 2052, Australia Tel.: Fax:

2 Soil Plasticity an the Structure Cam Clay Moel I: Soil lasticity Summary II: The Structure Cam Clay moel III: Further eeloment within the theoretical framework of the SCC moel

3 Soil Plasticity an the Structure Cam Clay Moel I: Soil lasticity : Knowlege an wisom from ast achieements 2: Constitutie moels an its imortance 3: General Elasticity 4: Original Cam Clay moel an soil lasticity 5: Moels of the Cam Clay family 6: Questions an iscussions at any time welcome! II: The Structure Cam Clay moel III: Further eeloment within the theoretical framework of the SCC moel

4 Soil Plasticity : Knowlege an wisom from ast achieements The earth, Mother Earth, mae u of rocks an soils the research of geotechnical engineering The great soil for human being: Homer: Greek oet, Saest of life: eath He cannot see the soil of his homelan again. He can neer smell the smells of home soil. He can touch the soil any more. The earth suorts human being hysically an siritually. Our ancestors knew soil well.

5 Our ancestors knew soil well. Three examles to see the knowlege an wisom: A agoa, a wall, an an merchant The agoa: the Pagoa of Phra Pathom Chei, Thailan The Wall: the Great Wall of China A merchant: a ritish merchant

6 Pagoa of Phra Pathom Chei, Thailan uilt in 300 C; Height: 5 m Diameter of the circular base: 58 m Weight: 500,000 ton

7 Pagoa of Phra Pathom Chei, Thailan uilt in 300 C; Size: 5 m 58 m 58 m; Weight: 0.5 MT After 2300 years Stan well with a uniform settlement of 2.5 m Engineering miracle. The loa concentration of the agoa: 0 MN/m 2 Euroean esign coe: 4-5 MN/M 2 for high builing;.5 4 MN/M2 for large silos.

8 Pagoa of Phra Pathom Chei Engineering miracle. Knowlege: () the non-uniform settlement, not the uniform settlement, that estroys our structure. (2) The bearing caacity of the groun increases with settlement. Moern numerical analysis gies the same conclusion. ut moern engineers are not esign such a builing.

9 The Great Wall of China uilt aroun 250 C as military efense roject

10 The Great Wall of China Techniques use for construction () Woo hammer to ensity the soil They knew: Strength of soil increase with ensity Stiffness of eformation A ery imortant research toic in 960s.

11 The Great Wall of China Techniques use for construction (2) Reinforcement by grass an tree They knew how to imroe the strength of soil an stiffness of eformation by geo-textile reinforcement

12 The Great Wall of China Techniques use for construction (3) Cementation by sticky rice cementing grael an rocks to gie strong suort

13 The Great Wall of China Reinforcement by strong materials an by cementation: ery oular methos of soil imroement in our moern worl since 970. Only 980, research in imroing soil strength an stiffness by reinforcement an cementation The work in Thailan is worl class (e.g., AIT, here by Dr. Horibulsuk)

14 A ritish corn merchant Darwin (883) an Renal (887) mae an imortant iscoery of soil roerty When sheare, loose soil will reuce in olume, but a ense soil will increase in olume. Characteristic of soils, calle ilatancy, funamental ifferent from other materials. A great challenge to geotechnical engineers. Rowe (962) erforme a ioneering stuy on ilatancy an roose a ilatancy law, a milestone in moern soil mechanics.

15 A ritish corn merchant Darwin (883) an Renal (887) notice. Corn merchants hae known ilatancy a ery long time ago. Corns bought an sol by olume. A common ractice: If you are a buyer, you will shake it. If corns ecrease in olume, it is loose. You shake it to make it small. If the olume increases, it is ense. You leae it alone, you o not touch it. A goo business sense.

16 Conclusion (a) Variation of the eak strength of soil with ensity an stress leel, (b) influence on the stiffness of soil to resist eformation, an (c) the ilatancy of soil. Three major challenges, inestigate in tests, moelling by theory. ()Our knowlege of soil mechanics comes from human ractice. (2)Our research on soil mechanics is to hel human ractice. Only by sering human nee, soil mechanics is a lie theory.

17 2: Constitutie moels an its imortance Constitutie moel: escribing the change in the strain state of an element of material to the change in the stress state acting on the element. ( ξ χ γ σ σ ) = f,,,, Strain increment Material roerties Stress history Strain history Stress increment Current stress state

18 Constitutie moel: ( ξ χ γ σ ) = f,,, Proiing information on (strength + eformation) of a material in an infinitesimal element Only through the knowlege of the strength an eformation of an element of material strength an eformation of an engineering structure No other way!!

19 Constitutie moel: The strength an eformation of a structure. That is all of An engineer s work, no more. This is the imortant an the reason for constitutie moelling

20 3: General Elasticity Terzaghi (936): Effectie stress rincile σ = σ -u Pore ressure Effectie stress Total stress It is the effectie stress that etermines the eformation of soil!

21 3: General Elasticity Terzaghi (936): Effectie stress rincile σ = σ -u The rincile of effectie stress: founation of moern soil mechanics; Terzaghi: founer of moern soil mechanics. ecause it enables us make a sensible link between the eformation of soil an the stress acting on it. Again for constitutie moelling

22 3: General Elasticity With the effectie stress rincile, theory of elasticity introuce into soil mechanics. Elastic eformation roose by Hook (678) Deformation is roortional to force. At the element leel = σ E Effectie stress alie to soil Strain increment Stiffness, a material constant

23 Original Hook s law Elastic resonse of soil to loaing eenent on mechanical roerties of the soil. material constant, Young s moulus. = σ E Stiffness, a material constant

24 Original Hook s law: = σ E Exerimental obseration: eformation also foun at irection ertical to the alie stress 2 3 Alying stress in irection strain in irections, 2, an 3

25 Lateral eformation ae 3 2 Coorinate Hence for stress σ, 2 3 = = ν 2 σ E E σ σ Young s moulus E in irection Poisson s ratio ν for eformation in irection 2 by stressing in irection = ν Poisson s ratio ν 3 3 E

26 Elastic eformation 3 2 Coorinate Hence for stress σ, = σ E 2 = ν Deformation with three rincial stresses σ, σ 2, an σ 3, Elastic, linear an small eformation eformation suerimose. 2 σ E 3 = ν 3 σ E

27 Elastic eformation General elastic equation with σ, σ 2, an σ 3 Via rincile of irtual work, we know ν 2 = ν 2, ν 23 = ν 32, ν 3 = ν E E E σ ν σ ν σ = E E E σ ν σ σ ν = E E E σ σ ν σ ν =

28 General anisotroic elastic equation General anisotroic elasticity: because ()Young s moulus in irection, E, can be ifferent from E 2 an E 3. (2)Poisson s ratio for ν 2, eformation in irection 2 by loaing in irection, can be ifferent from ν 23 an ν E E E σ ν σ ν σ = E E E σ ν σ σ ν = E E E σ σ ν σ ν = ν 2 = ν 2, ν 23 = ν 32, ν 3 = ν 3.

29 General isotroic elastic equation Isotroy: Definition: material has no referre irection. E=E2=E3=E, Therefore ν 2 = ν 2 = ν 23 = ν 32 = ν 3 = ν 3 =ν E E E 3 2 σ ν σ ν σ = E E E σ ν σ σ ν = E E E σ σ ν σ ν =

30 Elastic eformation Deformation comletely recoerable Elastic eformation eenent on the alue of the stress, ineenent of the stress ath, e.g., how soil loae to the stress A ( E E E ν ν ν σ σ ) =,,,,,, A Deformation from A to Shear stress q A Mean effectie stress '

31 4: Original Cam Clay moel an soil lasticity Original Cam Clay moel roose by Roscoe, Schofiel an Wroth (958) reolutional contribution to moern soil mechanics. Soil mechanics: elegant & consistent science

32 4: Original Cam Clay moel an soil lasticity Mathematical moel: Object too large, too comlicate for us to hanle, we iealize it, simlify it into an mathematical moel. Moel: reresent the features of an object of first imortance. Of secon imortance, ignore. Moel: human inention for unerstaning an analyzing a reality. Necessarily simle an iealize.

33 Original Cam Clay moel forme by stuy the eformation of soils. To simlify the work, the eformation of soils in secial states: reconstitute clay, not the soil you fin in nature. reconstitute clay: soils taken from nature, ry an breaking to ower, then mixing with water carefully. Examining exerimental result Simlest one: isotroic comression test σ = σ 2 = σ 3. Material: reconstitute Kaolin clay

34 Original Cam Clay moel Isotroic comression test on reconstitute Kaolin e: ois ratio : the mean Effectie stress = σ + σ 2 + σ 3 3 ( ) Isotroic test: = σ. Vois ratio e C E A Mean effectie stress ' (kpa) D

35 Isotroic comression test Two scales: e- scale an e-ln scale Vois ratio e A C 2. E D Mean effectie stress ' (kpa) Vois ratio e A C E D Mean effectie stress ' (kpa) The e - ln scale The e- scale aroximately linear essentially non-linear We loe linearality, the e ln sace

36 Isotroic comression test in the e-ln sace Vois ratio e A C 2. E D Mean effectie stress ' (kpa) To form a moel: Aroximation Simlification Iealization To reresent features of first imortance To form a moel for the isotroic comression behaiour of soil

37 A moel for the comression behaiour of soil Vois ratio e E C Mean effectie stress ' (kpa) Soil behaiour for loaing AD Iealize as linear in the e-ln sace A λ Tye of loaing: current stress is the maximum stress the soil eer exerience irgin loaing D

38 2.5 A Vois ratio e 2.3 C 2. E.9 λ κ D Mean effectie stress ' (kpa) Soil behaiour for unloaing C an DE an reloaing C: also Iealize as linear. ()aroximately linear for unloaing an reloaing C, an unloaing DE. (2) hysteretic loo, of secon imortance, ignore (3)The swelling an recomression inex are the same.

39 2.5 A Vois ratio e 2.3 C 2. E.9 λ κ D Mean effectie stress ' (kpa) Soil behaiour for unloaing C an DE an reloaing C: also Iealize as linear. Tye of loaing: current stress is less than the maximum stress the soil exerience, subsequent loaing

40 A Vois ratio e C E D F Mean effectie stress ' (kpa) During subsequent loaing, soil behaiour for unloaing an reloaing C, D E D, eformation comletely recoerable elastic, escribe by Hook s law

41 A Vois ratio e C E D F Mean effectie stress ' (kpa) Soil behaiour for reloaing C, elastic When stress goes beyon, the current stress the maximum stress the soil eer exerience Soil eformation () much greater than elastic Deformation;

42 A Vois ratio e C E D F Mean effectie stress ' (kpa) Loaing from D (2) if loae from to D, then unloaing from D to, the stress is less than the maximum stress Elastic eformation, along line DE. The eformation along first D loaing not comletely recoerable, not elastic.

43 Vois ratio e A C E ICL D F Mean effectie stress ' (kpa) The eformation along first D loaing not elastic oth elastic an lastic eformation occurre for loaing D!

44 Vois ratio e A C E ICL D F Mean effectie stress ' (kpa) For loaing from CD, at oint, there is a shar change in the stress an strain cure: Yieling oint The start oint for soil from elastic eformation to lastic eformation

45 Vois ratio e A C E ICL A moel for the comression behaiour of soil D F Mean effectie stress ' (kpa) For irgin comression, current stress is maximum the soil exerience, soil behaes linearly with graient λ. For subsequent loaing, current stress less than maximum the soil exerience, e.g., unloaing an reloaing; soil behaes linearly with graient κ.

46 Vois ratio e A C E ICL A moel for the comression behaiour of soil D F Mean effectie stress ' (kpa) For soil behaiour with κ, the eformation is elastic; For soil behaiour with λ, the eformation is lastic.

47 Shear stress q 6 a 4 Exerimental ata: soil behaiour for other stress aths Mean effectie stress ' 3 2 Yiel oint Stress ath a: isotroic comression a2, a3, a4: with shear stress Soil behaiour similar to that for ICL Soil behaiour into two regions: ure elastic eformation an lastic eformation

48 Shear stress q 6 4 a Mean effectie stress ' Exerimental ata: soil behaiour for other stress aths 3 2 There is a yiel oint for a gien soil for tests along stress aths: a, at y a2, at y2 a3, at y

49 Yiel surface Shear stress q 6 a Mean effectie stress ' All the yieling oints make u a bounary in the -q sace. This bounary is the yiel surface. () Loaing insie it ure elastic eformation () Loaing outsie it lastic eformation.

50 Yiel surface Collection of all the yieling oints in stress sace Shear stress q a Mean effectie stress ' A yiel surface iies the behaiour of soil in the stress sace into two regions: ure elastic behaiour region an lastic behaiour region

51 A feature of the yiel surface Shear stress q a Mean effectie stress ' If a soil has no cohesion, e.g., non cohesie soil, tensile force cannot be alie yiel surface only for 0. Soil a frictional material: stress state (=0, q=0) on the yiel surface Yiel surface asses through the origin of the stress coorinates.

52 Yiel surface: f (,, ) = 0 q o Shear stress q o Size of the yiel surface; usually o= for the isotroic stress state on the surface. Mean effectie stress '

53 Flow rule or lastic otential t Shear stress q f (, q, ) = 0 o A t Mean effectie stress ' For a stress state on yiel surface an with stress increment ointing outsie the surface, lastic eformation occurs.

54 Flow rule irection of lastic strain increment. Plastic olumetric strain increment Plastic shear strain increment Exeriment & theory stuy in metal eform. irection of lastic strain increment eenent on the stress state, ineenent of stress increment. The same for soil

55 Shear stress q t A t Mean effectie stress ' Flow rule ineenent of stress increment For a gien stress sate = r (, q )

56 Plastic otential Plastic otential g(,q) efine as: Mean effectie stress ' Shear stress q ( ) q r, = ( ) q r q g g, = =

57 Yiel surface & lastic otential for original Cam Clay moel Energy issiation equation Very imortance concet, a funamental law in natural science, a law in a higher orer than constitutie relations. Energy issiation of soil roose Soil is a frictional material an the issiation of the lastic energy is roortional to istortional strain increment an the mean effectie stress acting on it.

58 Yiel surface & lastic otential for original Cam Clay moel The issiation of the lastic energy is roortional to istortional strain increment an the mean effectie stress acting on it. Plastic work inut to an element of soil: Stress arameter an q an strain an work- conjugate. W = + q + q = Μ

59 Yiel surface & lastic otential for original Cam Clay moel The issiation of the lastic energy is roortional to istortional strain increment an the mean effectie stress acting on it. + q = Μ a thermoynamic material constant efining the rate of energy issiation

60 Drucker s stability criterion (952) The work one by the external agency on the change in islacement it rouces must be ositie or zero An element of soil consiere here. The element, an original equilibrium system, the stress (, q); an the islacement (, ). An aitional force, external agency, (, q) alie, an the islacement it rouces is (, ). ( + q ) 0

61 Drucker s stability criterion (952) The work one by the external agency on the change in islacement it rouces must be ositie or zero For an enclose stress ath, we obtain by integration ( + q ) 0 The eformation mae u of elastic art an lastic art e = + Elastic eformation comletely recoerable, no energy issiation

62 Drucker s stability criterion an normality Accoring to Drucker s stability criterion ( ) + q 0

63 Drucker s stability criterion an normality Shear stress q t A t Mean effectie stress ' ( ) + q 0 For all stress aths the stability criterion satisfie tt: the tangential line All the aths outsie tt line cause lastic eformation

64 Drucker s stability criterion an normality Shear stress q t A t Mean effectie stress ' ( ) + q 0 Plastic strain increment ector MUST E normal to the yiel surface If not, negatie work will be rouce. The angle between stress increment along tt line an >90. Normality

65 Drucker s stability criterion an normality Plastic strain increment normal to the yiel surface Associate flow rule: Normality Non-associate flow rule: lastic strain increment NOT normal to the yiel surface. For associate flow rule: g (, q, ) = (,, ) f q o o lastic otential yiel surface

66 For soil with associate flow rule With energy issiation equation ( ) ( ) o o q f q g =,,,, q f f = q = Μ +

67 Yiel surface & lastic otential y soling ifferential equations q = Μ + ( ) ( ) 0 ln,,,, = Μ = = q q g q f o c c efining the rate of energy issiation: a funamental material roerty.

68 Yiel surface f q Μ ( ) o, q, = ln = 0 c q ' o o : size of the yiel surface For isotroic loaing, σ = σ 2 = σ 3, then = o.

69 Flow rule Shear stress ratio η = = M q g g ( ) η Μ = = q g g q = η

70 Harening of soil If lastic olumetric eformation known, lastic shear eformation etermine Harening of soil: enlargement of yiel surface with lastic eformation Seek a relationshi between the enlargement of yiel surface an the lastic eformation ( ) η Μ = = q g g

71 Harening of the yiel surface A funamental assumtion for eriation The harening of yiel surface is eenent on the lastic olumetric eformation only. Consequently, () Plastic olumetric eformation etermine by change in size of yiel surface irresectie of loaing stress aths. = Λ (2) one to one relationshi between size of yiel surface an lastic olumetric eformation. o

72 Harening of the yiel surface = Λ at stress A: Three stress increment the same o the same o q All the loaing, on yiel surface f, resulting the same change in the yiel surface f the same lastic olumetric strain Soil eformation uring isotroic loaing well stuie. We try to work out relationshi between o an obtaine isotroic comression behaiour of soils. 2 Yiel surface f k A 3 2

73 Harening of the yiel surface For isotroic comression Stress state in the -q sace: o axis. Soil state in the e-ln sace ICL. At stress state A (= a, q=0) With an stress increment to ( a + ). For ois ratio at A: e A, At : e A + e e q k ln' ICL λ A o A o + o o + o o

74 Harening of the yiel surface For Isotroic Comression Line, Virgin comression: λ Elastic comression: κ q For comression from A an : Virgin comression The reuction in ois ratio λ e = λ Δ(ln ) = e k ln' ICL λ A o A o o + o +

75 Soil comression along ICL q The total olumetric strain increment e λ = = + e ( + e) ICL A o o + o Elastic olumetric strain increment y Hook s law κ e = + e ( ) e κ ln' λ A o + o o

76 ICL λ The total olumetric strain increment λ = ( + e) Elastic olumetric strain increment e κ A κ e = ( + e) Plastic olumetric strain increment ( λ κ ) ( + e) e = = o + o o ln'

77 Harening of the yiel surface Soil comresses along ICL, q The total olumetric strain increment λ = ( + e) Elastic olumetric strain increment κ e = ( + e) ICL A o o + o Plastic olumetric strain increment ( λ κ ) ( + e) e = = e κ ln' λ A o + o o

78 Harening of the yiel surface obtaine for yieling uring isotroic comression ( λ ) ( + e) κ = This equation is erie for ICL an ali for ICL only. Plastic olumetric eformation exresse in term of the mean effectie stress an the increment in mean effectie stress. ecause for IC, q 0.

79 Harening of the yiel surface To obtain lastic olumetric eformation equation for loaing along general stress ath The harening of yiel surface is eenent on the lastic olumetric eformation only. Equation for shoul be written in terms of size of yiel surface o, an change of yiel surface in size o.

80 Harening of the yiel surface We know, for Isotroic comression, o= an o= From to ( λ ) ( ) κ = + e ( λ ) ( ) o + e κ = o The equation ali for all stress increments that result in the same change in the yiel surface

81 Plastic eformation Yiel surface Flow rule Harening of the yiel surface Plastic eformation comletely efine From the yiel surface function ( ) ( ) o e + = κ λ ( ) 0 ln,, = Μ = q q f o c ( ) ( ) c c q g q f =,,,, ( ) η Μ = Μ + = q o o o ln

82 Plastic eformation Yiel surface Flow rule Harening of the yiel surface Plastic eformation obtaine ( ) ( ) o o e + = κ λ ( ) 0 ln,, = Μ = q q f o c ( ) η Μ = ( ) ( ) ( ) ( )( ) Μ + Μ + = Μ + + = q e q e o o ln ln η κ λ κ λ

83 Total increment stress an strain relationshi Plus elastic eformation, total increment stress an strain relationshi = = 9 Now that κ ( λ κ ) + ( + e) ( + e) 2κ ( + ν ) q + ( + e)( 2ν ) Elastic ln ( λ κ ) ( + e)( Μ η) o q + Μ o ln lastic efine. A moel is comlete. + q Μ The change in the strain state of an element of material to the change in the stress state acting on the element. The moel is the original Cam Clay moel.

84 Original Cam Clay moel Unify consistently the mechanical roerties of soil into a simle an elegant theoretical framework. Most imortant eeloment in moern soil mechanics Two istinguishe features of the original Cam Clay moel an moels of the Cam Clay family () Volumetric eenent of harening The harening of yiel surface is eenent on the lastic olumetric eformation only. Plastic olumetric eformation uniquely etermine by the change of yiel surface in size; an There is one to one relationshi between the size of the yiel surface an lastic olumetric eformation.

85 (2) Preiction of the existence of a critical state of eformation Let s examine flow rule an energy issiation function ( Μ η ) At a secial stress state, η=μ, from flow rule, = + q = Μ = ( Μ η ) = At this secial state, soil has no resistance to shear eformation. Like water or gas, it can not resist istortion.

86 (2) Preiction of the existence of a critical state of eformation At this secial state, η=m, from the energy issiation. For no change of, i.e., =0, then + q = Μ q = Μ any alue of can satisfy the equation. Soil can remain at the state η=μ with no change in its stress, o = 0, an no lastic olumetric eformation. Howeer, the shear lastic eformation can be infinitie. This is a Critical State of eformation

87 Original Cam Clay moel A Critical State of eformation: At a Critical State of eformation, a soil has no resistance to shear eformation an the soil can be istorte continuously with its stress state an ois ratio remain unchange. It is a final failure state. Cam Clay moel is the first Critical State moel. The theoretical framework, unifying consistently the mechanical roerties of soil into a simle an elegantly by original Cam Clay moel, the Critical State Soil Mechanics (CSSM). CSSM establishe soil mechanics as a cohesie science. The crown earl of soil mechanics

88 5: Soil lasticity From the introuction of the original Cam Clay moel basics of soil lasticity are introuce () Soil eformation iie into elastic an lastic arts e = + (2) Elastic eformation: Hook s law σ σ σ 2 3 = ν 2 ν 3 Deformation in irection E E2 E3

89 5: Soil lasticity (3) Moelling lastic Three comonants require: (i) Yiel surface, (ii) Flow rule an (iii) Harening function (i) Yiel surface in the stress sace f (, q, ) = 0 o Size of the yiel surface iiing soil behaiour into two regions: loaing insie the surface: elastic, loaing on the surface, lastic. During lastic eformation, the yiel surface changes with lastic eformation an the current stress state remains on the yiel surface.

90 5: Soil lasticity Moelling lastic eformation (ii) Flow rule: the irection of lastic strain increment Direction of lastic strain increment eenent on the stress only, i.e., ineenent of the stress increment. = r (, q) Direction of lastic strain increment normal to the yiel surface: associate flow rule. Direction of lastic strain increment NOT normal to yiel surface: non-associate flow rule.

91 5: Soil lasticity Moelling lastic eformation (iii) Harening of the yiel surface Variation of yiel surface with lastic eformation. For moels of the Cam Clay family, harening of yiel surface: lastic olumetric eformation eenent ( q) o = h, A more general harening function can be eenent on the lastic strain as o = h ( ), q,,

92 5: Soil lasticity Moelling lastic eformation f (, q, ) = 0 (i) Yiel surface: (ii) Flow rule: o (, q ) (iii) Harening of the yiel surface = r ( q) o = h, With these three arts efine, lastic eformation foun. From (i), the size change of the yiel surface obtaine as o = f + f o f q q

93 5: Soil lasticity Moelling lastic eformation (i) Yiel surface; (ii) Flow rule; (iii) Harening of the yiel surface Then lastic strain increment can be exresse as ( ) ( ) ( ) + = + = o o f q q f f q r q h f q q f f q h,,,

94 Thank you ery much!

95 6: Moels of the Cam Clay family Moifie Cam Clay moel Dafalias moel Volumetric eenent harening