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**Critical-State Soil Mechanics For Dummies**

Paul W. Mayne, PhD, P.E. Civil & Environmental Engineering Georgia Institute of Technology Atlanta, GA 2006

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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-logsv’ curves; and (2) shear stress-vs. normal stress (t-sv’) 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 (f', Cc, Cs) plus initial state (e0, svo', 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**

svo'=300 kPa sp'=900 kPa Cr = 0.04 Overconsolidation Ratio, OCR = 3 Cs = swelling index (= Cr) cv = coef. of consolidation D' = constrained modulus Cae = coef. secondary compression k ≈ hydraulic conductivity Cc = 0.38 sv’

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**sv’ sv’ t t t t Direct Shear Test Results gs d Direct Shear Box (DSB)**

Direct Simple Shear (DSS) t t t gs d Direct Shear Box (DSB)

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**CSSM for Dummies Void Ratio, e Void Ratio, e Effective stress sv'**

NC CSL CC CSL Void Ratio, e NC Log sv' CSL tanf' CSSM Premise: “All stress paths fail on the critical state line (CSL)” Shear stress t f c=0 Effective stress sv'

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**Volume Change is Contractive: evol = De/(1+e0) < 0**

CSSM for Dummies CC Void Ratio, e Void Ratio, e e0 NC ef De NC CSL CSL svo Log sv' Effective stress sv' CSL tmax = c + s tanf tanf' STRESS PATH No.1 NC Drained Soil Shear stress t Given: e0, svo’, NC (OCR=1) Drained Path: Du = 0 Volume Change is Contractive: evol = De/(1+e0) < 0 c’=0 svo Effective stress sv'

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**+Du = Positive Excess Porewater Pressures**

CSSM for Dummies CC Void Ratio, e Void Ratio, e e0 svo svo NC svf svf NC CSL CSL Effective stress sv' Log sv' CSL tanf' STRESS PATH No.2 NC Undrained Soil Du tmax = cu=su Shear stress t Given: e0, svo’, NC (OCR=1) Undrained Path: DV/V0 = 0 +Du = Positive Excess Porewater Pressures Effective stress sv'

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**CSSM for Dummies Void Ratio, e Void Ratio, e Effective stress sv'**

CC Void Ratio, e Void Ratio, e NC NC CSL CSL Effective stress sv' Log sv' CSL Note: All NC undrained stress paths are parallel to each other, thus: su/svo’ = constant tanf' Shear stress t DSS: su/svo’NC = ½sinf’ Effective stress sv'

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**CSSM for Dummies sp' sp' Void Ratio, e Log sv' CC Void Ratio, e CS OC**

NC NC CSL CSL sp' Effective stress sv' Log sv' CSL Overconsolidated States: e0, svo’, and OCR = sp’/svo’ where sp’ = svmax’ = Pc’ = preconsolidation stress; OCR = overconsolidation ratio tanf' Shear stress t sp' Effective stress sv'

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**CSSM for Dummies svo' svf' Log sv' svo' Void Ratio, e Void Ratio, e OC**

CC svo' e0 Void Ratio, e Void Ratio, e OC svf' CS NC NC CSL CSL Effective stress sv' Log sv' CSL Stress Path No. 3 Undrained OC Soil: e0, svo’, and OCR tanf' Du Shear stress t Stress Path: DV/V0 = 0 Negative Excess Du svo' Effective stress sv'

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**CSSM for Dummies svo' Log sv' Void Ratio, e Void Ratio, e OC NC NC CSL**

Effective stress sv' Log sv' CSL Stress Path No. 4 Drained OC Soil: e0, svo’, and OCR tanf' Shear stress t Stress Path: Du = 0 Dilatancy: DV/V0 > 0 Effective stress sv'

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**Critical state soil mechanics**

Initial state: e0, svo’, and OCR = sp’/svo’ Soil constants: f’, Cc, and Cs (L = 1-Cs/Cc) For NC soil (OCR =1): Undrained (evol = 0): +Du and tmax = su = cu Drained (Du = 0) and contractive (decrease evol) For OC soil: Undrained (evol = 0): -Du and tmax = su = cu Drained (Du = 0) and dilative (Increase evol) There’s more ! Semi-drained, Partly undrained, Cyclic…..

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**Equivalent Stress Concept**

CC NC svo' e0 Void Ratio, e svf' se' Void Ratio, e De OC NC CS sp' sp' ep CSL CSL Log sv' Effective stress sv' 1. OC State (eo, svo’, sp’) CSL 2. Project OC state to NC line for equivalent stress, se’ tanf' Shear stress t su at se’ suOC = suNC De = Cs log(sp’/svo’) De = Cc log(se’/sp’) 3. se’ = svo’ OCR[1-Cs/Cc] svo' se' Stress sv'

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**Critical state soil mechanics**

Previously: su/svo’ = constant for NC soil On the virgin compression line: svo’ = se’ Thus: su/se’ = constant for all soil (NC & OC) For simple shear: su/se’ = ½sin f’ Equivalent stress: se’ = svo’ OCR[1-Cs/Cc] Normalized Undrained Shear Strength: su/svo’ = ½ sinf’ OCRL where L = (1-Cs/Cc)

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**Undrained Shear Strength from CSSM**

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**Undrained Shear Strength from CSSM**

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**Porewater Pressure Response from CSSM**

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**Yield Surface Yield Surfaces Log sv' Void Ratio, e Void Ratio, e**

NC CSL NC OC OC Void Ratio, e Void Ratio, e sp' CSL Log sv' sp' Normal stress sv' CSL Yield Surface Yield surface represents 3-d preconsolidation Quasi-elastic behavior within the yield surface Shear stress t Normal stress sv'

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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 sinf’/(3-sinf’) and L = 1 – Cs/Cc 0.8 qc fs ub qT

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**Critical state soil mechanics**

Initial state: e0, svo’, and OCR = sp’/svo’ Soil constants: f’, Cc, and Cs (L = 1-Cs/Cc) 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) su/svo’ = ½ sinf’ OCRL where L = 1-Cs/Cc Yield surface represents 3-d preconsolidation

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