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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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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).

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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)

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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.

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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’

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Direct Shear Test Results v’v’ Direct Shear Box (DSB) v’v’ Direct Simple Shear (DSS) ss

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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 '

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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 ee Volume Change is Contractive: vol = e/(1+e 0 ) < 0 Effective stress v ' c’=0

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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 uu max c u =s u Effective stress v '

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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

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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

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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 uu

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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 '

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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…..

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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 ’) ee at e ’ s uOC = s uNC

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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]

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Undrained Shear Strength from CSSM

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Porewater Pressure Response from CSSM

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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

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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

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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")

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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.

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

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(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° "

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

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

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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"

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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

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Port of Anchorage, Alaska

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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

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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

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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

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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

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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)

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

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

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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

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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 '

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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

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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

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MIT E-3 Constitutive Model Whittle (2005)

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

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

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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

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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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