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

Hardening Soil model with small strain stiffnessAndrzej TrutyZACE Services25.08.2009Andrzej 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 pressuremodified form of the cap yield surface (2009)This 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

Influence of void ratio and confining stress p ′onG/G o (γ))Cohesionless soilsWichtmann and Triantafyllidis (after Benz)Andrzej Truty ZACE ServicesHardening Soil model with small strain stiffness

Influence of plasticity index PI on G/G o (γ))Cohesive soilsRemarks(Vucetic and Dobry (after Benz (PhD thesis))1 Results for PI < 30 are confirmed by other researchers whilethese for PI > 30 should be used with a special care (Benz)2 Stokoe proposed linear interpolation for γ 0.7γ 0.7 = 10 −4 for PI = 0 to γ 0.7 = 6 × 10 −4 for PI = 100Andrzej 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

Poisson coefficient for very small strainsPoisson ratio varies in the range ν = 0.1..0.3 in small straindomainIts value in further derivations will be kept constant (bydefault ν = 0.25)Andrzej 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 and cap yield surfacesGraphical representation of shear mechanism and cap surfaceAndrzej Truty ZACE ServicesHardening 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

HS model: shear mechanism700600500γ=0.1=const.M-Cq [kPa]400300γ=0.01=const.2001000γ=0.001=const.γ=0.0001=const.0 100 200 300 400 500 600p [kPa]f 1 = q a qE 50 q a − q − 2 q − γ PSE urq f =2 sin(φ)1 − sin(φ) (σ 3 + c ccotφ)q a = q fR fAndrzej Truty ZACE ServicesHardening Soil model with small strain stiffness

Flow rule for shear mechanism, dilatancy and hardeningg 1 = σ 1 − σ 3− σ 1 + σ 3sin ψ m2 2sin ψ m =sin φ m − sin φ cs1 − sin φ m sin φ csσsin φ m = 1 − σ 3σ 1 + σ 3 + 2c cotφsin psi_m0.60.50.40.30.20.10-0.1-0.2-0.3-0.4-0.5Contractancy cut-offDomain ofcontractancy0 10 20 30 40 50 60phi_m [deg]Domain ofdilatancy(dγ PS ∂g1= dλ 1 − ∂g 1− ∂g )1= dλ 1∂σ 1 ∂σ 2 ∂σ 3Rowe’s dilatancyAndrzej Truty ZACE ServicesHardening Soil model with small strain stiffness

Cap mechanismq 2Yield condition: f 2 =M 2 r 2 (θ) + p2 − pc2r(θ) is defined via van Ekelen’s formula (like in Cam-ClaymodelPlastic potential: g 2 = q2M 2 + p2( )pc + c cotφ mHardening law: d p c = dλ 2 2Hσ ref p+ c cotφRemarks:1 M and H parameters can be estimated for assumed KoNCtangent E oed modulus set up at a given vertical stressandAndrzej Truty ZACE ServicesHardening Soil model with small strain stiffness

Additional strength criteriaMohr-Coulomb yield conditionf1 ∗ = σ 1 − 1 − sin φ1 + sin φ σ 2c cos φ3 −1 + sin φ = 0Mohr-Coulomb plastic flow ruleg ∗ 1 = g 1NB. Here same plastic flow rule is used as for the shearmechanism f 1Rankine yield condition (tensile cut-off)f 3 = σ 1 − f t = 0where: f t is the assumed tensile strength (default is f t = 0)Rankine plastic flow rule(associated flow rule is used)g 3 = f 3Andrzej 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 tensor4 Hardening laws for γ PS and p c are modified by introducing h ifactor; this factor for very small strains is much larger than1.0 and decreases to 1.0 once the shear strain γ hist exceedscertains strain amplitude γ c5 Certain constractancy is allowed in the plastic flow rule forshear mechanismAndrzej 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

Modifications: DilatancyPHI = 40, PSI=10PHI = 30, PSI=5PSI_m [deg]10Scaled Rowe’s dilatancy5D = 0.2500-510 20 30 40-10-15-20-25-30-35Dafalias,Li(after Benz)Rowe’s dilatancyPHI_m [deg]PSI_m [deg]20151050-50 10 20 30 40-10-15-20-25-30Scaled Rowe’s dilatancyD = 0.25Dafalias,Li(after Benz)Rowe’s dilatancyPHI_m [deg]Andrzej Truty ZACE ServicesHardening Soil model with small strain stiffness

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

Setting M and H parameters based on oedometric testAssumptions:1 At a given σoed ref vertical stress both shear and volumetricmechanisms are active2 p ∗ 1 + 2KNCo= σoed ref3while q∗ = (1 − KoNC )σoedref3 A strain driven program is applied with vertical strainamplitude ∆ε = 10 −5 and resulting tangent oedometricmodulus is computed as E oed = ∆σ∆ε4 The two conditions must be fulfiled: K o coefficient generatedby the model must be equal to the one set by the user (usingJaky’s formula for instance K o = 1 − sin φ) and tangentoedometric modulus generated by the model must be equal tothe value given by the user5 If we take the data from the experiment we must be sure thatthe given oedometric modulus corresponds to the primaryloading branch of σ − ε curveAndrzej 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

User interfaceRemark1 HS/HS-small model can be actived only in the ⊠ AdvancedmodeAndrzej Truty ZACE Services Hardening Soil model with small strain stiffness

User interface: Elastic propertiesAndrzej Truty ZACE ServicesHardening Soil model with small strain stiffness

User interface: Elastic properties (HS)Remarks1 Standard HS model is activated if ⊠ Advanced checkbox isset OFFis the unloading/reloading Young modulus given at thereference stress σ ref2 E refur3 ν ur is the unloading/reloading Poisson coefficient; it variesfrom 0.15 to 0.3, hence for sands it is recommended toassume ν ur = 0.2..0.25 and for clays ν ur = 0.25..0.34 m is the exponent in stress dependency power law; it variesfrom m = 0.4 to m = 0..6; it is smaller for dense sands andlarger for clays5 σ L is the minimum allowed reference stress value used forevaluation of stiffness moduliAndrzej Truty ZACE ServicesHardening Soil model with small strain stiffness

User interface: Elastic properties (HS-small)Remarks1 HS-small model is activated if ⊠ Advanced checkbox is setON2 The HS-small model requires two additional parameters:Young modulus at very low strains Eoref at the reference stressσ ref and threshold shear strain γ 0.73 In case of lack of information on Eoref one may try to estimatebased on Alpan’s diagram assuming E s = E urE refo4 In the current implementation γ 0.7 is assumed to be constant5 In case of lack of information on γ 0.7 the diagram by Vuceticand Dobry can be used for cohesive soils and diagram byWichtmann and Triantafyllidis for cohesionless onesAndrzej Truty ZACE ServicesHardening Soil model with small strain stiffness

User interface: Plastic properties (HS/HS-small)Andrzej Truty ZACE ServicesHardening Soil model with small strain stiffness

User interface: Plastic properties (HS/HS-small)Remarks1 All material properties collected in group Nonlinear arecommon for HS and HS-small models2 In the advanced mode one may activate tensile and dilatancycut-off conditions, set up the multiplier D for Rowe’sdilatancy law in the contractant domain (for HS model thedefault value is D = 0.0 and for HS-small D = 0.25),is the secant Young modulus at 50 % of failure deviatoricstress q f derived from the q − ε 1 curve in drained triaxial test4 φ is the friction angle3 E50 ref5 ψ is the dilatancy angle6 c ′ is the effective cohesion7 R f is the failure ratio (default R f = 0.9)8 f t is the tensile strength (default f t = 0)9 e max is the maximum allowed void ratio; if current void ratioexceeds the e max dilatancy angle is switched to ψ = 0Andrzej Truty ZACE ServicesHardening Soil model with small strain stiffness

User interface: Plastic properties (HS/HS-small)Remarks1 Cap surface parameter M and hardening parameter H arederived by using a simple calculator which simulates anoedometric test; for given tangent oedometric modulus E oedat a given reference vertical stress σoed ref and for assumed KNCoparameter (here Jaky’s formula can be used) values of H andM are evaluated (press button Evaluate M,H ); one mayassume E oed = E ref50( σrefoed + c cotφσ ref + c cotφ) mas a default value2 Setting the initial state variables γoPS and p co can be carriedout by means of assumed OCR or preoverburden pressureq POP3 To compute KoNC from Jaky formula press buttonUse Jaky’s formula for KoNCAndrzej Truty ZACE ServicesHardening Soil model with small strain stiffness

User interface: Plastic properties (HS/HS-small)Remarks1 Pairs KoSR and OCR (OCR ≥ 1.0) or KoSR and q POP areneeded to setup the initial position of the cap surface and theinitial value of the hardening parameter γ PSis the minimum allowed value for the initialpreconsolidation stress2 p mincoAndrzej Truty ZACE ServicesHardening Soil model with small strain stiffness

Converting MC to HS model: general ideaQuestion: Having calibrated standard MC can we convert it to HSmodel ?Stiffness modulus Eurref and cap surface parameters H and Mcan be estimated by running an inverse analysis of a planestrain problem of a soil layer loaded by a strip loading qq = 0.5 q ult with q ult being the approximate ultimate limitload densityThe template data files for MC and HS model can be found inthe CFG directory under names: template-foot-MC andtemplate-foot-HSAndrzej Truty ZACE ServicesHardening Soil model with small strain stiffness

Converting MC to HS model: indentation problem1mAq = 0.5 q ult10m10mAndrzej Truty ZACE ServicesHardening Soil model with small strain stiffness

User interface: Converting MC to HS modelGiven: γ dry , K insituoEurrefE ref50= .... and E 50refEoedref,ν ur , σ ref , σ L , m, φ, ψ, c ′ , OCR, K SRo ,= .... and Young modulus that userwould assume in the simulation with a standard MC modelFind: E refurAndrzej Truty ZACE ServicesHardening Soil model with small strain stiffness

Convert MC to HS model: algorithmThe estimation idea is as follows:1 We know parameters to be used in the simulation with aid ofa standard MC model: E, γ dry , K insituo ,ν ur , φ, ψ, c ′2 Now we want to use HS/HS-small model but we do not knowon how to estimate E refur , H and M parameters3 We select a plane-strain problem of a strip loading q appliedto a uniform layer of soil as a template problem4 We assume the additional parameters for HS model: σ L , m,OCR, KoSR and the two coefficients E urrefE50ref = .... (default is 3)and E 50refEoedref= .... (default is 1.0)5 We run the optimization procedure which yields the E refur , Mand H such that the settlement at point A obtained from MCand standard (!!!) HS model are the sameAndrzej 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]200000180000160000140000120000HS-std100000HS-small8000060000400002000000.00001 0.0001 0.001 0.01 0.1 1EPS-1 - EPS-3 [-](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-qplaneEstimation 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-qplaneEstimation 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 mPlot E o vs σ 3E o[kPa]80000070000060000050000040000030000020000010000000 100 200 300 400 500 600 700σ3[kPa]Andrzej 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 ratioNote: this is very rough estimationBest solution: Perform triaxial testing and project on CPTprofile to adjust empirical coefficient (49.4) for a given siteAndrzej 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

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