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Simplified Critical-State Soil Mechanics Paul W. Mayne Georgia Institute of Technology Paul W. Mayne Georgia Institute of Technology.

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Presentation on theme: "Simplified Critical-State Soil Mechanics Paul W. Mayne Georgia Institute of Technology Paul W. Mayne Georgia Institute of Technology."— Presentation transcript:

1 Simplified Critical-State Soil Mechanics Paul W. Mayne Georgia Institute of Technology Paul W. Mayne Georgia Institute of Technology

2 PROLOGUE  Critical-state soil mechanics is an effective stress framework describing mechanical soil response  In its simple form here, we consider only shear loading and compression-swelling.  We merely tie together two well-known concepts: (1) one-dimensional consolidation behavior (via e-log  v ’ curves); and (2) shear stress-vs. normal stress (  v ’) plots from direct shear (alias Mohr’s circles).

3 Critical State Soil Mechanics (CSSM)  Experimental evidence  1936 by Hvorslev (1960, ASCE)  Henkel (1960, ASCE Boulder)  Parry (1961)  Kulhawy & Mayne (1990): Summary of 200+ soils  Mathematics presented elsewhere  Schofield & Wroth (1968)  Roscoe & Burland (1968)  Wood (1990)  Jefferies & Been (2006)  Basic form: 3 material constants (  ', C c, C s ) plus initial state parameter (e 0,  vo ', OCR)

4 Critical State Soil Mechanics (CSSM)  Constitutive Models in FEM packages:  Original Cam-Clay (1968)  Modified Cam Clay (1969)  NorSand (Jefferies 1993)  Bounding Surface (Dafalias)  MIT-E3 (Whittle, 1993)  MIT-S1 (Pestana, 1999; 2001)  Cap Model  “Ber-Klay” (Univ. California)  others (Adachi, Oka, Ohta, Dafalias, Nova, Wood, Huerkel)  "Undrained" is just one specific stress path  Yet !!! CSSM is missing from most textbooks and undergrad & grad curricula.

5 One-Dimensional Consolidation C c = 0.38 C r = 0.04  vo ' =300 kPa  p ' =900 kPa Overconsolidation Ratio, OCR = 3 C s = swelling index (= C r ) c v = coef. of consolidation D' = constrained modulus C  e = coef. secondary compression k ≈ hydraulic conductivity v’v’

6 Direct Shear Test Results v’v’  Direct Shear Box (DSB) v’v’  Direct Simple Shear (DSS)    ss

7 CSSM for Dummies  CSL ’  NC ’ Effective stress  v ' Shear stress  Void Ratio, e NC C tan  ' CSL Effective stress  v ' Void Ratio, e NC CSL CSSM Premise: “All stress paths fail on the critical state line (CSL)” CSL  c =0 e0e0  CSL ’  ½  NC ’ Log  v '

8 CSSM for Dummies Log  v ' Effective stress  v ' Shear stress  Void Ratio, e NC C tan  ' CSL STRESS PATH No.1 NC Drained Soil Given: e 0,  vo ’, NC (OCR=1) e0e0  vo Drained Path:  u = 0  max = c +  tan  efef ee Volume Change is Contractive:  vol =  e/(1+e 0 ) < 0 Effective stress  v ' c’=0

9 CSSM for Dummies Log  v ' Effective stress  v ' Shear stress  Void Ratio, e NC C tan  ' CSL STRESS PATH No.2 NC Undrained Soil Given: e 0,  vo ’, NC (OCR=1) e0e0  vo Undrained Path:  V/V 0 = 0 +  u = Positive Excess Porewater Pressures  vf uu  max  c u =s u Effective stress  v '

10 CSSM for Dummies Log  v ' Effective stress  v ' Shear stress  Void Ratio, e NC C tan  ' CSL Note: All NC undrained stress paths are parallel to each other, thus: s u /  vo ’ = constant Effective stress  v ' DSS: s u /  vo ’ NC = ½sin  ’ Void Ratio, e

11 CSSM for Dummies Log  v ' Effective stress  v ' Shear stress  Void Ratio, e NC C tan  ' CSL CSCS p'p' p'p' OC Overconsolidated States: e 0,  vo ’, and OCR =  p ’/  vo ’ where  p ’ =  vmax ’ = P c ’ = preconsolidation stress; OCR = overconsolidation ratio

12 CSSM for Dummies Log  v ' Effective stress  v ' Shear stress  Void Ratio, e NC C tan  ' CSL CSCS OC Stress Path No. 3 Undrained OC Soil: e 0,  vo ’, and OCR  vo ' e0e0 Stress Path:  V/V 0 = 0 Negative Excess  u Effective stress  v '  vf ' Void Ratio, e uu

13 CSSM for Dummies Log  v ' Effective stress  v ' Shear stress  Void Ratio, e NC C tan  ' CSL CSCS OC Stress Path No. 4 Drained OC Soil: e 0,  vo ’, and OCR Stress Path:  u = 0 Dilatancy:  V/V 0 > 0  vo ' e0e0 Effective stress  v '

14 Critical state soil mechanics Initial state: e 0,  vo ’, and OCR =  p ’/  vo ’ Soil constants:  ’, C c, and C s (  = 1-C s /C c ) For NC soil (OCR =1):  Undrained (  vol = 0): +  u and  max = s u = c u  Drained (  u = 0) and contractive ( decrease  vol ) For OC soil:  Undrained (  vol = 0): -  u and  max = s u = c u  Drained (  u = 0) and dilative ( Increase  vol ) There’s more ! Semi-drained, Partly undrained, Cyclic response…..

15 Equivalent Stress Concept Log  v ' Stress  v ' Shear stress  Void Ratio, e NC C tan  ' CSL CSCS OC 1. OC State (e o,  vo ’,  p ’)  vo ' 2. Project OC state to NC line for equivalent stress,  e ’ 3.  e ’ =  vo ’ OCR [1-Cs/Cc]  vo ' e0e0 Effective stress  v ' Void Ratio, e p'p' p'p' e'e' e'e' susu  vf ' epep  e = C s log(  p ’/  vo ’)  e = C c log(  e ’/  p ’) ee at  e ’ s uOC = s uNC

16 Critical state soil mechanics Previously: s u /  vo ’ = constant for NC soil On the virgin compression line:  vo ’ =  e ’ Thus: s u /  e ’ = constant for all soil (NC & OC) For simple shear: s u /  e ’ = ½sin  ’ Equivalent stress: Normalized Undrained Shear Strength: s u /  vo ’ = ½ sin  ’ OCR  where  = (1-C s /C c )  e ’ =  vo ’ OCR [1-Cs/Cc]

17 Undrained Shear Strength from CSSM

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19 Porewater Pressure Response from CSSM

20 Yield Surfaces Log  v ' Normal stress  v ' Shear stress  Void Ratio, e NC CSL OC Normal stress  v ' Void Ratio, e p'p' p'p' OC  Yield surface represents 3-d preconsolidation  Quasi-elastic behavior within the yield surface

21 Critical state soil mechanics This powerpoint: geosystems.ce.gatech.edu Classic book: Critical -State Soil Mechanics by Schofield & Wroth (1968): Schofield (2005) Disturbed Soil Properties and Geotechnical Design Thomas Telford Wood (1990): Soil Behaviour and CSSM Jefferies & Been (2006): Soil liquefaction: a critical-state approach

22 ESA versus TSA Effective stress analysis (ESA) rules:  c' = effective cohesion intercept (c' = 0 for OCR < 2 and c' ≈ 0.02  p ' for short term loading)   ' = effective stress friction angle   = c' +  ' tan  ' = Mohr-Coulomb strength criterion   v ' =  v - u 0 -  u = effective stress Total stress analysis (TSA) is (overly) simplistic for clay with strength represented by a single parameter, i.e. "  = 0" and  max = c = c u = s u = undrained shear strength (implying "  u = 0")

23 Explaining the myth that "  = 0" The effective friction angle (  ') is usually between 20 to 45 degrees for most soils. However, for clays, we here of "  = 0" analysis which applies to total stress analysis (TSA). In TSA, there is no knowledge of porewater pressures (PWP). Thus, by ignoring PWP (i.e.,  u = 0), there is an illusional effect that can be explained by CSSM. See the following slides.

24 (Undrained) Total Stress Analysis - Consolidated Undrained Triaxial Tests Three specimens initially consolidated to  vc ' = 100, 200, and 400 kPa

25 (Undrained) Total Stress Analysis s u100 s u200 s u400 In TSA, however,  u not known, so plot stress paths for "  u = 0" Obtains the illusion that "  ≈ 0° "

26 Another set of undrained Total Stress Analyses (TSA) for UU tests on clays: UU = Unconsolidated Undrained

27 (Undrained) Total Stress Analysis susu Again,  u not known in TSA, so plot for stress paths for "  u = 0" Obtains the illusion that "  = 0° "

28 Explaining the myth that "  = 0"  Effective Stress Analyses (ESA) Drained Loading (  u = 0) Undrained Loading (  V/V 0 = 0)  Total Stress Analyses (TSA)  Drained Loading (  u = 0)  Undrained Loading with "  = 0" analysis:  V/V 0 = 0 and "  u = 0"

29 Cambridge University q-p' space P' = (  1 ' +  2 ' +  3 ')/3 q = (  1 -  3 ) Triaxial Compression CSL  2 ' =  3 ' 1'1' Undrained NC Stress Path Undrained OC Stress Path  vo ' = P 0 ' Drained Stress Path 3V : 1H

30 Port of Anchorage, Alaska

31 Cavity Expansion – Critical State Model for Evaluating OCR in Clays from Piezocone Tests where M = 6 sin  ’/(3-sin  ’) and  = 1 – C s /C c  0.8 qcqc fsfs ubub  qT qT qT qT

32 Cambridge University q-p' space P' = (  1 ' +  2 ' +  3 ')/3 q = (  1 -  3 ) CSL Yield Surface Original Cam Clay Modified Cam Clay Pc'Pc' Bounding Surface Cap Model

33 Anisotropic Yield Surface P’ = (  1 ’ +  2 ’ +  3 ’)/3 q = (  1 -  3 ) M c = 6sin  ’/(3-sin  ’) fctn(K 0NC ) Y 3 = Limit State Yield Surface e 0  vo ’ K 0 G 0 Y2Y2 CSL p’p’ Y1Y1 OC NC

34 Cambridge University q-p' space P' = (  1 ' +  2 ' +  3 ')/3 q = (  1 -  3 ) fctn(K 0NC ) Y 3 = Limit State Yield Surface CSL p’p’ Apparent M c

35 MIT q-p' space P' = ½(  1 ' +  3 ') q = ½(  1 -  3 ) fctn(K 0NC ) Yield Surface p’p’ OC Diaz-Rodriguez, Leroueil, and Aleman (1992, JGE)

36 Diaz-Rodriguez, Leroueil, & Aleman (ASCE Journal Geotechnical Engineering July 1992) Yield Surfaces of Natural Clays

37 Friction Angle of Clean Quartz Sands (Bolton, 1986 Geotechnique)

38 State Parameter for Sands,  (Been & Jefferies, 1985; Jefferies & Been 2006) log p' void ratio e p' = ⅓ (  1 '+  2 '+  3 ') VCL CSL 10 p0'p0'  = e 0 - e csl Dry of Critical (Dilative) Wet of Critical (Contractive) e0e0 e csl

39 State Parameter for Sands,  (Simplified Critical State Soil Mechanics) log p' void ratio e p' = ⅓ (  1 '+  2 '+  3 ') VCL CSL 10 p0'p0'  = e 0 - e csl  = (C s - C c )∙log[ ½ cos  ' OCR ]  u  = (1 - ½ cos  ' OCR  ]∙  vo ' e0e0 e csl then CSL = OCR = 2/cos  '

40 Georgia Tech State Parameter for Sands,  (Been, Crooks, & Jefferies, 1988) log OCR p = log2  +  where OCR p = R = overconsolidation ratio in Cambridge q-p' space, , = C c /ln(10) = compression index, and   C s /ln(10) = swelling index log OCR p = log2  +  where OCR p = R = overconsolidation ratio in Cambridge q-p' space, , = C c /ln(10) = compression index, and   C s /ln(10) = swelling index

41 MIT Constitutive Models  Whittle et al. 1994: JGE Vol. 120 (1) "Model prediction of anisotropic behavior of Boston Blue Clay"  MIT-E3: 15 parameters for clay  Pestana & Whittle (1999) "Formulation of unified constitutive model for clays and sands" Intl. J. for Analytical & Numerical Methods in Geomechanics, Vol. 23  MIT S1: 13 parameters for clay  MIT S1: 14 parameters for sand

42 MIT E-3 Constitutive Model Whittle (2005)

43 MIT S-1 Constitutive Model Pestana and Whittle (1999)

44 MIT S-1 Constitutive Model Predictions for Berlin Sands (Whittle, 2005)

45 Critical state soil mechanics Initial state: e 0,  vo ’, and OCR =  p ’/  vo ’ Soil constants:  ’, C c, and C s Link between Consolidation and Shear Tests CSSM addresses:  NC and OC behavior  Undrained vs. Drained (and other paths)  Positive vs. negative porewater pressures  Volume changes (contractive vs. dilative)  s u /  vo ’ = ½ sin  ’ OCR  where  = 1-C s /C c Yield surface represents 3-d preconsolidation State parameter:  = e 0 - e csl Yield surface represents 3-d preconsolidation State parameter:  = e 0 - e csl

46 Simplified Critical State Soil Mechanics Log  v ' Effective stress  v ' Shear stress  Void Ratio, e NC C CSL CSCS p'p' p'p' OC Four Basic Stress Paths: 1. Drained NC (decrease  V/V o ) 2. Undrained NC (positive  u) 3. Undrained OC (negative  u) 4. Drained OC (increase  V/V o ) '' e NC consolidation swelling e OC Yield Surface dilative contractive +u+u -u-u  CS  ½  p  max = s u NC  max =  tan  s u OC  max = c+  tan  c'

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