Hardening Soil model with small strain stiffness - Zace Services Ltd.

Hardening Soil model with small strain stiffnessAndrzej TrutyZACE Servicesrevised 15.09.2008Andrzej Truty ZACE ServicesHardening Soil model with small strain stiffness

IntroductionHardening Soil (HS) and Hardening Soil-small (HS-small)models are designed to reproduce basic phenomena exhibitedby soils:densificationstiffness stress dependencyplastic yieldingdilatancystrong stiffness variation with growing shear strain amplitudein the regime of small strains (γ = 10 −6 to γ = 10 −3 )this phenomenon plays a crucial role for modeling deepexcavations and soil-structure interaction problemsNB. This model is limited to monotonic loadsAndrzej Truty ZACE ServicesHardening Soil model with small strain stiffness

IntroductionHS model was initially formulated by Schanz, Vermeer andBonnier (1998, 1999) and then enhanced by Benz (2006)Current implementation is slightly modified with respect tothe theory given by Benz:simplified treatment of dilatancy for the small strain version(HS-small)modified hardening law for preconsolidation pressureThis model seems to be one of the simplest in the class ofmodels designed to handle small strain stiffnessIt consists of the two plastic mechanisms, shear and volumetricSmall strain stiffness is incorporated by means of nonlinearelasticity which includes hysteretic effectsAndrzej Truty ZACE ServicesHardening Soil model with small strain stiffness

Notion of tangent and secant stiffness moduliInitial stiffness modulus E oUnloading-reloading modulus E urSecant stiffness modulus at 50 % of the ultimate deviatoricstress q f0250q [kpa]200150100501E o1 E 50q 50E ur1q f0.5 q fσ 3 =constq un0 0.05 0.1 0.15 0.2 0.25EPS-1 [-]Remark: All classical soil models require specification of E urmodulus (Cam-Clay, Cap etc..)Andrzej Truty ZACE ServicesHardening Soil model with small strain stiffness

Stiffness-strain relation for soils (G/G o (γ))G - current secant shear modulusG o - shear modulus for very small strainsAtkinson 1991Andrzej Truty ZACE ServicesHardening Soil model with small strain stiffness

Notion of treshold shear strain γ 07GTo describe the shape of (γ) curve an additionalG ocharacteristic point is neededIt is common to specify the shear strain γ 0.7 at which ratioGG o= 0.70.7γ 07Andrzej Truty ZACE ServicesHardening Soil model with small strain stiffness

Dynamic vs static modulusRelation between ”static” Young modulus E s , obtained fromstandard triaxial test at axial strain ε 1 ≈ 10 −3 , and ”dynamic”Young modulus (the one at very small strains) E d = E o isshown in diagram published by Alpan (1970) (after Benz)100EEdsRocks10cohesive soilsgranular soilsE s[kPa]11000 10000 100000 1000000Andrzej Truty ZACE ServicesHardening Soil model with small strain stiffness

HS model: general conceptDouble hardening elasto-plastic model (Schanz, Vermeer,Benz)Nonlinear elasticity for stress paths penetrating the interior ofthe elastic domain600q [kPa]500400300200Cap surface10000 100 200 300 400 500p [kPa]Graphical representation of shear mechanism and cap surfaceAndrzej Truty ZACE Services Hardening Soil model with small strain stiffness

HS model: shear mechanismDuncan-Chang model as the origin for shear mechanism250q fq [kPa]200150100500E 50½ q f11M-C limit0 0.01 0.02 0.03 0.04 0.05eps-1E urAndrzej Truty ZACE ServicesHardening Soil model with small strain stiffness

Stiffness stress dependencyRemarksE ur = E refurE 50 = E ref50( σ∗3 + c cotφσ ref + c cotφ( σ∗3 + c cotφσ ref + c cotφ) m) m1 Stiffness degrades with decreasing σ 3 up to σ 3 = σ L (bydefault we assume σ L =10 kPa)Andrzej Truty ZACE ServicesHardening Soil model with small strain stiffness

Extension to small strain: new ingredientsTo extend standard HS model to the range of small strain Benzintroduced few modifications:1 Strain dependency is added to the stress-strain relation, forstress paths penetrating the elastic domain2 The modified Hardin-Drnevich relationship is used to relatecurrent secant shear modulus G and equivalent monotonicshear strain γ hist3 Reversal points are detected with aid of deviatoric strainhistory second order tensor H ij ; in addition the currentequivalent shear strain γ hist is computed by using this tensorAndrzej Truty ZACE ServicesHardening Soil model with small strain stiffness

How does it work ?N-1NN+1plot from paper by Ishihara 1986At step N : γ histN−1 = 8 × 10 −5 γ histN = 10 −4At step N + 1 : γ histN = 0 γ histN+1 = 2 × 10 −5Primary loading: γ histN+1 > γhistmaxUnloading/reloading: γ histN+1 ≤ γ maxHardin-Drnevich law: G =histG o1 + a γ histγ 0.7(secant modulus)Andrzej Truty ZACE ServicesHardening Soil model with small strain stiffness

Shear tangent modulus cut-offGG urγ cγ c = γ 0.7aAndrzej Truty ZACE Services(√ )Go− 1G urHardening Soil model with small strain stiffnessγ

Setting initial state variables: γ PSoGiven: σ o , OCRFind: γ PSand p coo and p co0600q [kPa]500400300200100Shear mechanismCap surfaceσ οσ SR0 100 200 300 400 500p [kPa]Procedure:Set effective stress state at the SR pointσySR = σ yo OCRσxSR = σzSR = σ y KoSRAndrzej Truty ZACE ServicesHardening Soil model with small strain stiffness

Setting initial state variables: γ PSoand p co600q [kPa]500400300200100Shear mechanismCap surfaceσ οσ SR00 100 200 300 400 500p [kPa]Procedure:For given σ SR state compute γoPSf 1 = 0from plastic conditionFor given σ SR state compute p co from plastic condition f 2 = 0Andrzej Truty ZACE ServicesHardening Soil model with small strain stiffness

Setting initial state variables: γ PSoand p coRemarks= KoNC(approximate Jaky’s formula)1 K SRo2 K SRo≈ 1 − sin(φ) in the standard applications= 1 for case of isotropic consolidation (used in triaxialtesting for instance)3 For sands notion of preconsolidation pressure is not asmeaningful as for cohesive soils hence one may assumeOCR=1 and effect of density will be embedded in H and MparametersAndrzej Truty ZACE ServicesHardening Soil model with small strain stiffness

Setting M and H parameters based on oedometric test600500σq [kPa]400q*300200100p*00 100 200 300 400 500p [kPa]σ ref 1oedE oedεAndrzej Truty ZACE ServicesHardening Soil model with small strain stiffness

Material propertiesParameter Unit HS-standard HS-smallEur ref [kPa] yes yesE50 ref [kPa] yes yesσ ref [kPa] yes yesm [—] yes yesν ur [—] yes yesR f [—] yes yesc [kPa] yes yesφ [ o ] yes yesψ [ o ] yes yese max [—] yes yesf t [kPa] yes yesD [—] yes yesM [—] yes yesH [kPa] yes yesOCR/q POP [—/kPa] yes yesEo ref [kPa] no yesγ 0.7 [—] no yesAndrzej Truty ZACE ServicesHardening Soil model with small strain stiffness

Converting MC to HS model: indentation problemAssumption: q = 0.5 q ultGiven: E for MC model and E urE 50= ...,E 50E oed= ...1mAq = 0.5 q ult10m10mFind: E refur , M and H for standard HS modelAndrzej Truty ZACE ServicesHardening Soil model with small strain stiffness

Example: triaxial test on dense Hostun sand65.55120000SIG-1 / SIG-3 [kPa]4.543.532.5HS-stdHS-smallG [kPa]100000800006000040000HS-stdHS-small21.520000100 0.02 0.04 0.06 0.08 0.10.00001 0.0001 0.001 0.01 0.1 1-EPS-Y [-]EPS-X - EPS-Y [-](a) σ 1σ 3(ε 1 ) (Z Soil)(b) G(γ) (Z Soil)43.50 0.02 0.04 0.06 0.08 0.10.08SIG-1 / SIG-3 [kPa]32.52HS-stdHS-small-EPS-V [-]0.070.060.050.040.030.02HS-stdHS-small1.50.0101-0.010 0.002 0.004 0.006 0.008 0.01-EPS-Y [-]-0.02-EPS-Y [-](c) σ 1σ 3(ε 1 ) (zoom) (Z Soil)(d) ε v (ε 1 ) (Z Soil)Andrzej Truty ZACE ServicesHardening Soil model with small strain stiffness

Example: triaxial test on dense Hostun sandSIG-1 / SIG-3 [kPa]65.554.54HS-std3.5HS-small32.521.510 0.02 0.04 0.06 0.08 0.1EPS-1 [-]G [kPa]2000001800001600001400001200001000008000060000400002000000.00001 0.0001 0.001 0.01 0.1 1EPS-1 - EPS-3 [-]HS-stdHS-small(a) σ 1σ 3(ε 1 ) (Z Soil)(b) G(γ) (Z Soil)43.50 0.02 0.04 0.06 0.08 0.10.08SIG-1 / SIG-3 [kPa]32.52HS-stdHS-smallEPS-V [-]0.070.060.050.040.030.02HS-stdHS-small1.50.0101-0.010 0.002 0.004 0.006 0.008 0.01-0.02EPS-1 [-]EPS-1 [-](c) σ 1σ 3(ε 1 ) (zoom) (Z Soil)(d) ε v (ε 1 ) (Z Soil)Andrzej Truty ZACE ServicesHardening Soil model with small strain stiffness

Example: triaxial test on dense Hostun sand65.53000005250000SIG-1 / SIG-3 [kPa]4.543.532.5HS-stdHS-smallG [kPa]200000150000100000HS-stdHS-small21.550000100 0.02 0.04 0.06 0.08 0.10.00001 0.0001 0.001 0.01 0.1 1EPS-1 [-]EPS-1-EPS-3 [-](a) σ 1σ 3(ε 1 ) (Z Soil)(b) G(γ) (Z Soil)40 0.02 0.04 0.06 0.08 0.13.50.08SIG-1 / SIG-3 [kPa]32.52HS-stdHS-smallEPS-V [-]0.070.060.050.040.030.02HS-stdHS-small1.50.0101-0.010 0.002 0.004 0.006 0.008 0.01-0.02EPS-1 [-]EPS-1 [-](c) σ 1σ 3(ε 1 ) (zoom) (Z Soil)(d) ε v (ε 1 ) (Z Soil)Andrzej Truty ZACE ServicesHardening Soil model with small strain stiffness

Estimation of material properties: input dataGiven 3 drained triaxial test results for 3 confining pressures:σ 3 = 100 kPaσ 3 = 300 kPaσ 3 = 600 kPaShear characteristics q − ε 1Dilatancy characteristics ε v − ε 1Stress paths in p − q planeMeasurements of small strain stiffness moduli E o (σ 3 ) for theassumed confining pressures (through direct measurement ofshear wave velocity in the sample)Andrzej Truty ZACE ServicesHardening Soil model with small strain stiffness

Estimation of material properties: stress paths in p-q planeEstimation of friction angle φ = φ cs and cohesion cqResidual M-C envelope16sinφM*=3 − sinφ6 cosφc*= c3 − sinφpIf we know M ∗ and c ∗ then we can compute φ and c:φ = arcsin 3 M∗6 + M ∗ c = c ∗ 3 − sin φ6 cos φAndrzej Truty ZACE ServicesHardening Soil model with small strain stiffness

Estimation of material properties: stress paths in p-q planeEstimation of friction angle φ = φ cs and cohesion c3000q [kPa]25002000150010002358 12358/1386=1.7500138600 300 600 900 1200 1500 1800p [kPa]Here: φ = arcsin 3 ∗ 1.76 + 1.7 ≈ 42o c = 0Andrzej Truty ZACE ServicesHardening Soil model with small strain stiffness

Estimation of material properties: dilatancy angle0.060.05EPS-V [-]0.040.030.020.01d1Dilatancy cut-off0-0.01-0.020 0.02 0.04 0.06 0.08 0.1ψ = arcsinEPS-1 = - EPS-3 [-]( d2 + d)Andrzej Truty ZACE ServicesHardening Soil model with small strain stiffness

Estimation of material properties: dilatancy angle0.060.050.04ε 0.03 V0.020.010-0.01d=0.75-0.020 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1ε 1( ) 0.75ψ = arcsin≈ 16 o2 + 0.751Andrzej Truty ZACE ServicesHardening Soil model with small strain stiffness

Estimation of material properties: E refoand mAnalytical formula: E o = E refo( σ∗3 + c cotφσ ref + c cotφ) mMeasured: shear wave velocity v s at ε 1 = 10 −6 and at givenconfining stress σ 3Compute : shear modulus G o = ρv 2 sCompute : Young modulus E o = 2 (1 + ν) G oσ 3 [kPa] E o [kPa]100 250000300 460000600 675000Andrzej Truty ZACE ServicesHardening Soil model with small strain stiffness

Estimation of material properties: E refoand mAnalytical formula: E o = E refo( σ∗3 + c cotφσ ref + c cotφ) mMeasured: shear wave velocity v s at ε 1 = 10 −6 and at givenconfining stress σ 3Compute : shear modulus G o = ρv 2 sCompute : Young modulus E o = 2 (1 + ν) G oσ 3 [kPa] E o [kPa]100 250000300 460000600 675000Andrzej Truty ZACE ServicesHardening Soil model with small strain stiffness

Estimation of material properties: E refoand mReanalyze E o vs σ 3 in logarithmic scales13.1 − 12.55Averaged slope yields m; here m =1.0Find(intersection of the line with axis ln E o atσ∗ )3 + c cotφlnσ ref = 0+ c cotφHere the intersection is at 12.43 hence= e 12.43 ≈ 2.718 12.43 = 250000 kPaE refo= 0.5513.6ln E o13.413.213m12.812.6⎛ σ3+ c cotφ⎞1ln⎜⎟12.4 12.43ref⎝ σ + c cotφ⎠12.20 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2Andrzej Truty ZACE ServicesHardening Soil model with small strain stiffness

Estimation of E refofrom CPT testingTo estimate small strain modulus G o at a certain depth onemay use empirical formula by Mayne and Rix:G o = 49.4 q0.695 te 1.13[MPa]q t is a corrected tip resistance expressed in MPae is the void ratioAndrzej Truty ZACE ServicesHardening Soil model with small strain stiffness

Estimation of material properties: E ref50Lets us find E 50 for each confining stressf50( σ3=q250020001500100)10001E50( σ3= 300kPa)1E50( σ3= 600kPa)E50( σ3= 100kPa)1qf50( σ3=qf50( σ3=100)500100)00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1Andrzej Truty ZACE ServicesHardening Soil model with small strain stiffness

Estimation of material properties: E ref50Reanalyze E 50 vs σ 3 in logarithmic scalesHere we can fix m to the one obtained for small strain moduliFind(intersection of the line with axis ln E 50 atσ∗ )3 + c cotφlnσ ref = 0+ c cotφHere the intersection is at ≈ 10.30 henceE50 ref ≈ e10.30 ≈ 2.718 10.30 ≈ 30000 kPaln E 5011.411.21110.810.6⎛ σ3+ c cotφ⎞ln⎜⎟σ + c cotφref10.410.30⎝⎠10.20 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2Andrzej Truty ZACE ServicesHardening Soil model with small strain stiffness

Estimation of material properties: E refurThe unloading reloading modulus as well as oedometricmoduli are usually not accessibleWe can use Alpans diagram to deduce E refurE refo(default is E urrefEorefthis value is largeronce we know= 3); for cohesive soils like tertiary claysFor oedometric modulus at the reference stress σ ref = 100kPa we can assume Eoed ref = E 50refγ 0.7 = 0.0001...0.0002 for sands and γ 0.7 = 0.00005...0.0001for claysSmaller γ 0.7 values yield softer soil behaviorAndrzej Truty ZACE ServicesHardening Soil model with small strain stiffness

Excavation in Berlin Sand: engineering draftAndrzej Truty ZACE ServicesHardening Soil model with small strain stiffness

Excavation in Berlin Sand: FE discretizationSeepageElementsFluid head BCyxInterface wall-soilSeepageElements +Fluid head BC90mFictitious interface to modelimpermeable barrier30m120mAndrzej Truty ZACE ServicesHardening Soil model with small strain stiffness

Excavation in Berlin Sand: Bending moments-600 -400 -200 0 200 4000-5-10Y [m[]-15-20-25-30HSHS-smallMCM [kNm/m]-35Andrzej Truty ZACE ServicesHardening Soil model with small strain stiffness

Excavation in Berlin Sand: Wall deflections-0.04 -0.03 -0.02 -0.01 00-5-10Y [m]-15-20-25-30HS-smallHSMCMeasurementUx [m]-35Andrzej Truty ZACE ServicesHardening Soil model with small strain stiffness

Excavation in Berlin Sand: Soil deformation in crosssection x =20mY [m]-10-20-30-40-50-60-70-80-90-1000 0.01 0.02 0.03 0.040Uy [m]Vertical heaving of subsoil at last stage of excavationHSHS-smallMCAndrzej Truty ZACE ServicesHardening Soil model with small strain stiffness

Excavation in Berlin Sand: Settlements behind the wall0 20 40 60 80 100 120 1400.010.0050UY [m]-0.005-0.01HSHS-smallMC-0.015-0.02-0.025X [m]Settlements behind the wall at last stage of excavationAndrzej Truty ZACE ServicesHardening Soil model with small strain stiffness

ConclusionsModel properly reproduces strong stiffness variation with shearstrainIt can be used in simulations of soil-structure interactionproblemsImplementation is ”rather heavy”It should properly predict deformations near the excavationsModel reduces excessive heavings at the bottom of theexcavationAndrzej Truty ZACE ServicesHardening Soil model with small strain stiffness