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2006 Critical-State Soil Mechanics For Dummies Paul W. Mayne, PhD, P.E. Civil & Environmental Engineering Georgia Institute of Technology Atlanta, GA

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PROLOGUE Critical-state soil mechanics is an effective stress framework describing mechanical soil response In its simplest form here, we consider only shear- induced loading. We merely tie together two well-known concepts: (1) one-dimensional consolidation behavior, represented by e-log v ’ curves; and (2) shear stress-vs. normal stress ( v ’) from direct shear box or simple shearing. Herein, only the bare essence of CSSM concepts are presented, sufficient to describe strength & compressibility response.

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Critical State Soil Mechanics (CSSM) Experimental evidence provided by Hvorslev (1936; 1960, ASCE); Henkel (1960, ASCE Boulder) Henkel & Sowa (1961, ASTM STP 361) Mathematics presented elsewhere, including: Schofield & Wroth (1968); Burland (1968); Wood (1990). In basic form: 3 material constants ( ', C c, C s ) plus initial state (e 0, vo ', OCR) Constitutive Models, include: Original Cam-Clay, Modified Cam Clay, NorSand, Bounding Surface, MIT-E3 (Whittle, 1993) & MIT-S1 (Pestana) and others (Adachi, Oka, Ohta, Dafalias) "Undrained" is just one specific stress path Yet !!! CSSM is missing from most textbooks and undergrad & grad curricula in the USA.

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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 Log v ' 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

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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…..

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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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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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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 ) Using effective stresses, 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

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