Benz Small Strain Stifness in Geotechnical Analyses

Nonlinear soil behaviour at small strains is often neglected in geo -technical analyses. Doing so often leads to an overestimation offoundation settlements and retaining wall deflections. Settlementtroughs behind retaining walls or above tunnels, on the other hand,may be analysed as too flat and extended. Comparing measureddisplacements of piles or anchors within the working load range,to those calculated without considering small-strain stiffness,shows a considerably too soft response. This paper is concernedwith a qualitative and quantitative discussion of the small-strainstiffness phenomenon. In combination with the commerciallyavailable small-strain stiffness model introduced, it provides thebasics for incorporating small-strain stiffness into routine design.

1 Introduction

The maximum strain, at which soils exhibit almost fullyrecoverable behaviour, is found to be very small. The verysmall-strain stiffness associated with this strain range, i.e.shear strains lower than 1 × 10–5, is believed to be a funda-mental property of all types of geotechnical materials.With increasing strain, soil stiffness decays non-linearly.On a logarithmic scale, stiffness reduction curves exhibit acharacteristic S-shape similar to the one shown in Fig. 1.The borderline between small strains and larger strains,outlined in Fig. 1, is commonly drawn at the limit of classi-cal laboratory testing, that is triaxial or oedometer testingwithout special instrumentation as e.g. local strain gauges.Strains, at which soils exhibit almost fully recoverable oralmost elastic behaviour, are termed very small strains.

In German literature, dynamic stiffness is often used asa synonym for small-strain stiffness. This is presumably dueto the small strain amplitudes commonly used in dynamictesting, as static tests with local strain gauges reveal thatinertia effects only slightly increase a soil’s very-small strainstiffness, if it is increased at all (e.g. Stokoe et al. [30]).

Small-strain stiffness is mostly found to be a manifoldof the stiffness obtained in classical laboratory testing.Therefore, not accounting for it in geotechnical analyses maypotentially result in overestimating foundation settlementsand retaining wall deflections. The gradient of settlementtroughs behind retaining walls or above tunnels may beunderestimated. Piles or anchors within the working loadrange may show a considerably too soft response. The im-portance of small-strain stiffness to the engineering prac-tice has been highlighted, for example, in the Rankine lec-ture by Simpson [29], or the Bjerrum Memorial lecture byBurland [7]. As yet however, small-strain stiffness has notbeen widely implemented in engineering practice.

Settlement analyses, which introduces a limit depth,represents an exception to this statement. The limit depthis defined as a depth, where the soil is only slightly subjectedto additional stress due to external loading. These analysesindirectly consider small-strain stiffness in introducing anactive compression zone and an incompressible zone be-low the limit depth. The incompressible zone remains inthe range of small-strains and small-strain stiffness andhence, responds relatively stiffly (see Section 4.3).

If in numerical settlement analyses, a constitutivemodel is used which is not capable of simulating small-strain stiffness, a similar approach as the one describedabove has to be chosen: Either the lower mesh boundary ischosen at an assumed limit depth or the stiffness of deepersoil layers is manually increased. However, both methodsare dissatisfactory. A lower mesh boundary that is chosenin a distance, where it will influence calculation results, isgenerally to be avoided and more important the limitdepth is not known a priori. Manually increasing the stiff-ness of deeper soil layers requires knowledge about thestrain level in these layers after loading, which is also notknown a priori. Therefore the only satisfactory means ofaccounting for small-strain stiffness in numerical analysisis to adapt the constitutive model. To date, such constitu-tive models are rarely found in commercially availablecodes.

Small-strain stiffness in geotechnical analyses

Thomas BenzRadu SchwabPieter Vermeer

DOI: 10.1002/bate.200910038

Fig. 1. Characteristic stiffness-strain behaviour in logarith-mic scale according to [3].

16 © Ernst & Sohn Verlag für Architektur und technische Wissenschaften GmbH & Co. KG, Berlin · Bautechnik Special issue 2009 – Geotechnical Engineering

09_016-027 Benz (1237).qxd:000-000 Benz (1237).qxd 30.07.2009 13:25 Uhr Seite 16

17Bautechnik Special issue 2009 – Geotechnical Engineering

Th. Benz/R. Schwab/P. Vermeer · Small-strain stiffness in geotechnical analyses

Constitutive models, that can be applied to engineeringproblems, should adequately simulate soil behaviour notonly at small strains, but also at larger strains in the senseof Fig. 1. The Hardening-Soil model [26], for example, is amodel that is often applied to geotechnical problems. TheHypoplastic law [15] with intergranular strain [20], includ-ing – in contrast to the Hardening-Soil model – a small-strain stiffness formulation, is less used in routine design. Infact, the use of a constitutive model in engineering prac-tice demands user-friendly models with input parameters,which are easy to understand in their physical meaning,and easy to quantify, based on test data or experience. Thispaper presents a possibility to enhance such user-friendlyelastoplastic formulations for non-linear stiffness variationat small-strains.

Although the paper’s main concern is with the devel-opment of a small-strain stiffness model, it also providesinformation which might be found helpful during its appli-ance. Firstly, small-strain stiffness is discussed from an ex-perimental point of view, with a focus on empirical corre-lations. Secondly, the new model is introduced, before itfinally is applied to various boundary value problems witha particular emphasis on the impact of small-strain stiff-ness on calculation results. Such an impact is to be ex-pected in most problems, including those expected tocause larger strains (Fig. 1); during construction, small-strain stiffness is always to be considered.

2 Soil stiffness at small strains

Small-strain stiffness is believed to be a fundamental propertyof geotechnical materials including clays, silts, sands, gravels,and rocks (Tatsuoka [33]) under static and dynamic loading(Burland [7]) and for drained and undrained loading con-ditions (Lo Presti et al. [16]). With increasing strain, soilstiffness decays non-linearly. On a logarithmic scale, stiffnessreduction curves exhibit a characteristic S-shape (Fig. 1).Experimental evidence shows that after a load reversal thevery small-strain stiffness is recovered.

These experimental observations can easily be ex-plained with a simple Coulomb-type frictional law, assumedto exist in-between the particles of granular material. Ini-tially, all inter-particle contacts are in a sticking mode. Dueto contact-elasticity, most contacts remain in the stickingmode for a finite strain amplitude. This finite strain ampli-tude can be considered to be equivalent to the quasi-elas-tic very small-strain regime. A further increase of the shearstrain will then cause an increasing number of particles toslip. The global material stiffness, a function of the stiffnessat all local contacts, thus decreases. Strain reversals allowthe soil skeleton’s inter-particle contacts to revert to stick-ing mode so that the maximum small-strain stiffness canbe recovered.

Assuming all particles of the granular matter to berigid (psammic behaviour [8]), volumetric strains cannotdecrease stiffness as it does not change the ratio of normaland tagential contact forces. Clearly, psammic behaviouris, like the frictional contact model, an abstraction of real-ity but shear strains actually influence small-strain stiff-ness degradation to a larger extent than volumetric strains.The reason, why stiffness reduction curves are commonlypresented in literature as function of shear strains, is prob-

ably different though: Small-strain stiffness was first dis-covered in soil dynamics. Shear strain γ s is defined as:

(1)

and therefore simplifies in triaxial conditions (ε2 = ε3) toγ s = |ε1 – ε3|.

In the following, secant moduli are used to describeexperimental results. Initial small-strain shear and Young’smoduli are denoted as G0 and E0 respectively. Assumingthat soils at very-small strains are elastic, the Young’smodulus can be related to the shear modulus as follows:

E0 = G0(1 + 2ν) (2)

where ν is Poison’s ratio. Next, the shear modulus G0 isdiscussed in more detail. After that, the focus will be onthe relationship between stiffness reduction and appliedshear strain.

2.1 Small-strain shear modulus G0

The small-strain shear modulus G0 in all types of soils ismainly affected by void ratio e and mean stress p. Cementa-tion, which is a very important influencing factor, too, willbe discussed later. In literature, many experimental resultsare presented in form of the modified Hardin [11] equation:

(3)

where A, k, and m are material constants, f gives the func-tional dependency of G0 on void ratio, OCR is the over-consolidation ratio, and pref is a reference pressure. Fig. 2depicts the results of Equation 3 for more than 30 parame-ter sets (A, k, m) for different soils published in literature.All references to the test data shown in Fig. 2 are providedin [4]. It turns out, that m for cohesionless soils equals ap-proximately 0.5 and slightly increases in cohesive soils. Incohesive soils, G0 increases with OCR. Viggiani & Atkinson[35], for example, found 0,20 < k < 0,25 for clays with plas-

G f e pp

k

ref

m

0 =⎛

⎝⎜⎜

⎞

⎠⎟⎟A OCR( ) ,

γ ε ε ε ε ε εs = − + − + −⎡⎣

⎤⎦

12 1 2

22 3

23 1

2( ) ( ) ( )

Fig. 2. Correlations between G0 and void ratio e


ticity Index 10 < IP < 40, which seems a relatively smallvariation for practical applications.

As a rule of thumb, either the relationship by Hardin &Black [9]:

(4)

or the more simple expression by Biarez & Hicher [6]:

(5)

can be used to estimate small-strain stiffness moduli E0and G0 in various soils. Here again, e is void ratio, p is themean effective stress in kPa, and pref = 100 kPa is the ref-erence pressure equal to atmospheric pressure. The rela-tionship by Biarez & Hicher was proposed for soils with wl< 50%, whereas the one by Hardin & Black was derivedfor undisturbed clayey soils and crushed sands.

The scatter band highlighted in Fig. 2 suggests, thatthese correlations should be used only if no direct tests areavailable. In this case, small-strain shear moduli can bederived from indirect test results, too. Fig. 3 gives an ex-

E MPa ppref

0

0 540[ ]

,

=⎛

⎝⎜⎜

⎞

⎠⎟⎟

1e

G MPa ppref

0

0 5

[ ],

=( )

+

⎛

⎝⎜⎜

⎞

⎠⎟⎟33

2,97–e

1 e

2

ample of a correlation between G0 and cone penetrationtip resistance.

The chart published by Alpan [1] shown in Fig. 4which – as he stated – relates static to dynamic soil stiff-ness (see axis labels Es, Ed), can provide estimates for thesmall-strain modulus E0, too. Interpreting the dynamicmodulus Ed as the initial or very small-strain modulus E0,and the static modulus Es as the apparent elastic modulusin conventional soil testing, e.g. the secant modulus Eur inlarger triaxial unloading- reloading loops, E0 can be esti-mated from triaxial test results. To the authors’ experience,Alpan’s chart provides reasonable estimates for the stiff-ness of soils, when interpreted in the way described above.

2.2 Modulus reduction – the reference shear strain γ 0,7

Fig. 5 shows exemplary stiffness reduction curves for sandand cohesive soils, which were originally published bySeed & Idris [28] and Vucetic & Dobry [36], respectively.According to these sources, stiffness reduction curves ofcohesive soils with IP < 15 % are comparable to those ofsands. However, with increasing soil plasticity, soil stiff-ness decays more gradually with applied strain.

In soil dynamics, experimentally acquired stiffness re-duction curves were first approximated with mathematicalexpressions. The best known models of soil dynamics arethe Hardin-Drnevich model [10], the Ramberg-Osgoodmodel [23], and the bilinear model. Fig. 6 gives a compari-son of these models in a loading-unloading-reloading loop.The bilinear model gives only a very rough approximationof real soil behaviour and is therefore not discussed in anymore detail. Excellent approximations, on the other hand,are obtained with hyperbolic laws as used in the Ramberg-Osgood model and in the Hardin-Drnevich model. The ba-sic difference between these models is, that they are formu-lated in stress space and strain space, respectively:

Ramberg-Osgood (6)

Hardin-Drnevich (7)GG

s

r

0

1

1

=

+γ

γ

GG

y

0

1

1

=

+ α ττ

κ

18 Bautechnik Special issue 2009 – Geotechnical Engineering


Fig. 3. Correlation between G0 and cone penetration tipresistance after Lunne et al. [17]

Fig. 4. Alpan chart: Original publication [1] (left) and its interpretation after [5] with test results by [37] (right)




where G0, α, κ, τy are the material parameters of the Ram-berg-Osgood model and G0, γ r are the material parametersof the Hardin-Drnevich model. Hence, the Hardin-Drnevichmodel is well suited for practical application not only be-cause of its data fitting qualities, which are often pointedout in literature, but also because of its simple structurerequiring two input parameters only.

Next, the hyperbolic Hardin-Drnevich model is adoptedfor quantifying stiffness reduction curves. For this, the ref-erence shear strain γ r, introduced above, is replaced by thereference shear strain γ 0,7 = 3γ r/7:

. (8)

The reference shear strain γ 0,7 is the shear strain at whichthe shear modulus G has been reduced to 70 % of its in-itial value G0. In the following, this reference shear strainis utilized to quantify the importance of several influen-cing factors on the decay of small-strain stiffness.

The influence of void ratio on the reference shear strainin sand is very limited. At a reference pressure of 100 kPa,it is tested within the limits of 6 × 10–5 < γ 0,7 < 3 × 10–4 (seeFig. 5). Soil plasticity, on the other, hand has a significant

GG s0

0 7

1

1 37

=

+γ

γ ,

impact on γ 0,7. Vucetic & Dobry [36] proposed the modu-lus reduction chart shown in Fig. 5. Stokoe et al. [31] pro-posed a linear dependency of γ 0,7 on the plasticity index(IP) of the form:

γ0,7 = (γ0,7)ref + 5 × 10–6 IP (OCR)0,3, (9)

where (γ 0,7)ref is the reference shear strain for IP = 0, whichis about 1 × 10–4. Compared to the chart published byVucetic & Dobry, the correlation by Stokoe et al. suggestslower reference shear strains for higher plasticity indices.

In all soils, the reference shear strain γ 0,7 is addition-ally influenced by confining stress: The reference shearstress increases with confining stress.

2.3 In-situ tests

Comparisons of laboratory and in-situ small-strain stiff-ness measurements for a given material mostly yieldsmaller stiffness in the laboratory. Fig. 7 shows experimen-tal results from studies conducted in Japan and in theUSA. Correlations derived from laboratory tests generallyindicate the lower limit of in-situ stiffness. When adoptingmodulus reduction curves derived in the laboratory to in-situ measured shear moduli G0, they are usually scaled in

Fig. 5. Stiffness reduction curves after Seed & Idris [28] (left) and Vucetic & Dobry [36] (right)

�

�

G0

1

�y

�y

�

�

G0

1

�r

�f

�

�

G0

1

�y

�y

G0

1

a b c

Fig. 6. One-dimensional models known from soil dynamics: (a) Bilinear model, (b) Ramberg-Osgood model, (c) Hardin-Drnevich model


stiffness only [32]. Thus, the reference shear strain γ 0,7 isnot believed to be much affected by diagenesis.

3 Small-strain stiffness models

Small-strain stiffness was not applied to static problemsuntil the 1980’s. Among the first small-strain stiffness modelswere models that used one or more kinematic yield surfacese.g. Mroz et al. [19]. The idea of a very small inner yieldsurface (bubble) was invented by Al-Tabbaa in 1987. It wasafterwards that Al-Tabbaa & Wood [2] published their bubbleextension of the CamClay model. In his 1992 Rankine lec-ture, Simpson [29] made the analogy between soil behaviourand a man pulling bricks behind him. Simpson’s analogy,together with his interpretation in strain space, becamewell known by the term Simpson brick model. Within theHypoplastic framework, the intergranular strain concept[20] was invented. The previously discussed models fromsoil dynamics were applied to static problems, too. In multi-axial constitutive models, these one-dimensional modelsneed to be extended. An example for a multi-axial formu-lation is given by the MIT S1 model [21].

As yet, however, small-strain stiffness has not widelybeen implemented in routine design. Considering numericalanalyses, this may be due to the complexity of the availablemodeling concepts summarized above and/or the fact thatthey lack some substantial features of real soil behaviour.Models used in engineering practice today, rarely have morethan one elastic domain. The Cam-Clay or the HardeningSoil models are two well known examples of this modelclass. A possibility to incorporate small-strain stiffness inthem is discussed in this section. The proposed model ex-tension is based on a multi-axial formulation of theHardin-Drnevich model.

3.1 A small-strain stiffness extension for elastoplastic models

It is impossible for small-strain stiffness to be associated withthe entire elastic domain of isotropically hardened elasto-plastic models as e.g. the Cam-Clay, or the Hardening Soilmodel, as they would respond too stiff in larger unloading-

reloading cycles as a consequence. Therefore, their elasticstiffness is generally taken as a secant stiffness of largerunloading-reloading loops (see Fig. 8). Small-strain stiffnessis neglected. Unlike the stress-strain path sketched in Fig. 8,unloading-reloading loops, which are based on a constantsecant stiffness, show no hysteretic behaviour.

A straight forward method to incorporate small-strainstiffness in elastoplastic frameworks is to define their elasticstiffness as a function of the actual strain amplitude. At smallstrain amplitudes, the large strain secant unloading-reload-ing stiffness is increased. The MIT S1 model [21] is an ex-ample, where strain dependent elastic moduli are imple-mented. In contrast to the MIT S1 model, the approachproposed here is based on a generalization of the Hardin-Drnevich model:

(10)

where e.is the devaitroric strain rate and H* is a tensor which

memorizes deviatoric strain history [4]. In proportionalloading, H* equals the integral of deviatoric strain rates, so

GG

mitHist

Hist0

0 7

1

1 37

3=

+

=γ

γ

γ

,

*H e

e

�

�



Holocene sand

(thin-wall sampling)

Holocene clay Pleistocene clay

Soft rock

(rotary core)

In-situ frezzing

method

GravelSoft rock

(block)

Range from

ROSRINE

study

Remolded

cemented

sandy soils

Range often found

with rock cores

150

Velocity ratio v /vS,Lab S,Field

0.0 0.5 1.0 1.5

300

450

600

750

900

1050

0

General trend

In-s

itu

shea

rw

ave

vel

oci

tyv

[m/s

]S

,Fie

ld

2.001.501.000.500.250.10 0.80

Modulus ratio G /G0,Lab 0,Field

Pleistocene sand

(tube sampling)

100010010

G0,Field [MPa]

2

Mo

du

lus

rati

oG

/G0,

Lab

0,F

ield

1

0

Fig. 7. Differences in the ratio of laboratory-to-field stiffness. Left: Data from Japanese case studies after Toki et al. [34].Right: Results from the US American ROSRINE study after Stokoe & Santamarina [32]

�axial

q

�0

�0

�0

��ur

��

0,5 qf

qf

Fig. 8. The elastic secant modulus Eur




that Equation 10 simplifies to Equation 7. In non-propor-tional loading, however, H* can be partially or fully resetaccording to the following transformation rule:

(11)

where u(x) is the Heaviside function and λ(i) are theEigenvalues of the deviatoric strain rate. Underlined ten-sorial quantities are transformed in the strain rate’s Eigen-system. Poison’s ratio is assumed to be constant.

The resulting stress-strain law is not continuous in strain(load reversals), so that by definition it does not qualify asan elastic stress-strain law. Nevertheless, strain on a closedstress cycle starting from a stress reversal point is typicallyrecovered. In the following, the new stress-strain law istherefore denominated paraelastic (Hueckel & Nova [13]).However, the terminology “elastic stiffness” is here main-tained for the use of the above defined stiffness in an elasto-plastic framework as a reminder that it applies to all stressstates, within as well as on the yield surface, similar to theelastic stiffness known from traditional elastoplastic models.

3.2 Primary loading, unloading, and reloading – the Masing rules

To systematically describe the behaviour of brass under cyclicloading, Masing [18] proposed the following two rules: 1. The shear modulus in unloading is equal to the initialtangent modulus for the initial loading curve. 2. The shapeof the unloading and reloading curve is equal to the initialloading curve, except that its scale is enlarged by a factorof two.

H T HT T*

= =

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

−111

22

33

1

0 00 00 0

mitT

TT

und

T 1111

111 11

222

1

11 1 1

1

=+

+ + −( )⎡⎣⎢

⎤⎦⎥

=

Hu H H

TH

( )( )λ

22

122 22

3333

11 1 1

1

11

++ + −( )⎡

⎣⎢⎤⎦⎥

=+

+

u H H

TH

( )( )λ

uu H H( )( )λ 133 33 1 1+ −( )⎡

⎣⎢⎤⎦⎥

Given that a functional form exists which can describethe initial loading curve, the above rules can likewise beapplied to construct the hysteresis loops of soils for symmet-rical or periodic loadings. However, if the loading is irreg-ular, i.e. not symmetrical or periodic, the current loading orunloading curve may intersect a previous one. Fig. 9 showsthree possible extensions of the original Masing rules, thatwere proposed in literature: Path A: Rosenblueth & Her-rera [25], path B: Jennings [14], path C: Richart [24]).

All extended Masing rules require an internal mem-ory of previous changes in the stress-strain path, i.e. a lociin stress space. Such an internal memory in stress space isnot a practicable constitutive ingredient for the extendedHardin-Drnevich model: In combination with an elasto-plastic model, the memory would be highly influenced byplastic yielding, so that the paraelastic stiffness could onlybe determined in an iterative scheme. The yield loci of acombined isotropically hardened elastoplastic model canenforce the memory in stress space instead. The resultingstress-strain path is equivalent to that proposed by Richart(Fig. 9, path C).

3.3 HS-Small – a small-strain extension of the Hardening Soil model

The combination of the multiaxial small-strain formula-tion, introduced above, with the Hardening Soil model isdiscussed next. In the finite element code Plaxis, this com-bination is referred to as HS-Small model. Its stress andstrain history dependent elastic stiffness is controlled bytwo additional material parameters. These are the initialshear modulus Gref,0 defined for the reference pressure prefand the shear strain γ 0,7, at which the shear modulus hasdecreased to 70 percent of its initial value. The shear mod-ulus G is calculated from:

(12)

where σ3 is the minor principal stress and c cot ϕ is the max-imum allowable isotropic tensile stress, which can be limitedwith an additional input parameter. The exponent m gives

G Gc

p crefHist ref

=+

+

+00 7

0 7

37

7 3,,

,

cotcot

γ

γ γ

σ ϕ

ϕ

⎛⎛

⎝⎜⎜

⎞

⎠⎟⎟

m

,

8.0

�/�y

� �/ y

6.04.02.0-2.0-4.0-6.0-8.0

0.2

0.4

0.6

0.8

1.0

-1.0

-0.8

-0.6

-0.4

-0.2

a b

8.06.04.02.0-2.0-4.0-6.0-8.0

� �/ y

0.2

0.4

0.6

0.8

1.0 �/�y

-1.0

-0.8

-0.6

-0.4

-0.2

02

3

1

4

C

BA

Fig. 9. Hysteresis loops in symmetric (left) and irregular (right) loading according to the extendedMasing rules after Pyke [22]


the stress dependent increase of stiffness. A single exponentm is used for scaling all stiffness moduli in the model. Poi-son’s ratio is assumed constant at all strain amplitudes.

A lower cut-off limit in the hyperbolic small-strainstiffness reduction curve is introduced at the shearstrain, where tangent stiffness is reduced to the unload-ing-reloading stiffness Gur in larger strain cycles. Fromthen on, the small-strain formulation is inactive. It is ac-tivated again whenever a significant change in strain rateis detected (Equation 11). The resulting small-strain stiff-ness behaviour of the HS-Small model is illustrated inFig. 10.

Stress relaxation erases a soil’s memory of previousapplied stress. Soil ageing in the form of particle (or as-sembly) reorganization during stress relaxation and for-mation of bonds between them can erase a soil’s strainhistory. Considering that the second process in a naturallydeposited soil develops relatively fast, the strain historyshould start from zero in most boundary value problems.This is the default setting in the HS-Small model.

However, sometimes an initial strain history may bedesired. In this case, the strain history can be triggeredby applying an extra load step before starting the actualanalysis. Such an additional load step might also beused to model overconsolidation. Usually the overcon-solidation’s cause has vanished long before the start ofcalculation, so that the strain history should be reset af-terwards.

4 Model application

All problems discussed in this section are analyzed usingthe finite element method. A more detailed description ofthe various problems including material data sets, modelgeometry, loading history etc. is given in Benz [4]. In or-der to identify effects that can be attributed to small-strainstiffness, all calculations were run with and without small-strain stiffness formulation, i.e. with the HS-Small modeland the Hardening Soil model, respectively. In the spreadfooting example, presented in Section 4.3, the Hypoplasticmodel with intergranular strain is considered as an alter-native small-strain model.

4.1 Triaxial test

In triaxial testing, small-strain stiffness does not exessivelyaffect the overall stress-strain curve. Fig. 11 illustrates this,with a drained triaxial test on dense Hostun sand and itsnumerical back calculation. Only in up-scaling the veryfirst part of the stress-strain curve on the right hand sideof Fig. 11, the differences become visible.

4.2 Excavation problem 2D

Simpson introduced his brick model in a case study of theBritish Library deep excavation. The Houses of Parliamentunderground car park case study is another deep excava-



Fig. 10. Stiffness-reduction curve of the HS-Small model. Left: Elastic secant shear modulus. Right: Elastic tangentshear modulus

0.00 0.02 0.04 0.06 0.08 �1[-]0

2

4

�1/�3�3 = 300 kPa CD

0.0001 0.001 0.01 �1-�3[-]0

40000

80000

120000

160000

GSecant[kN/m2]

Experiment

Hardening-Soil

HS-Small

Fig. 11. Triaxial test on dense Hostun sand




tion study, that is often cited along with small-strain stiff-ness. The tilt of the Big Ben clock tower right next to thisexcavation nicely illustrates the effects of advanced (small-strain stiffness) models. These can predict the total exca-vation heave more correctly and thus do not suggest thatthe clock tower tilts away from the pit, which less refinedmodels may do. Beyond the constitutive model however,there are a lot more things to consider in the analysis ofdeep excavations, for example: the interface between soiland retaining structure, initial stress conditions, structuralelements, time effects, and pore water pressures. Some ofthese issues should always be thoroughly considered inexcavation analysis (interface, initial stresses, structural el-ements), others are less critical under less difficult soilconditions. An example with less critical soil conditions ischosen here: An excavation in Berlin sand.

The working group 1.6 „Numerical methods in Geo -technics“ of the German Geotechnical Society (DGGT)has organized several comparative finite element studies(benchmarks). One of these benchmark examples is theinstallation of a triple anchored deep excavation wall inBerlin sand. The reference solution by Schweiger [27] isused here as the starting point: Both, the mesh shown in

Fig. 12, and the soil parameters are taken from this refer-ence solution. However, the bottom soil layer defined bySchweiger could be omitted in the analysis when using theHS-Small model. In the reference solution, the only pur-pose of this layer is the simulation of small-strain stiffnessdue to a lack of small-strain stiffness constitutive modelsback then.

Figs. 13, 14, and 15 show results from the finite elementcalculations. The small-strain stiffness formulation accu-mulates more settlements right next to the wall, whereas thesettlement trough is smaller. The triple anchored retainingwall is deflected less when using the HS-Small model(Fig. 13). Excavation heave is considerably less (Fig. 14)and the elastic stiffness expressed as ratio G/Gur reduces alot around the excavation pit (Fig. 15). In the example G0equals 3 Gur.

4.3 Spread footing problem

The next boundary value problem considered is a stripfooting on dense Hostun sand, first published by Hintneret al. [12]. The strip footing has a width of 1 meter, a thick-ness of 1 meter and an embedded depth of 1 meter. The

Fig. 12. Geometry and anchor detail

Fig. 13. Surface settlement trough (left) and lateral wall deflection (right) at the end of excavation


vertical load of 150 kPa is distributed over the width of thefooting. The homogeneous soil layer’s material parametersare identical to those back calculated from the triaxial testdiscussed in Section 4.1. In the analysis, three different con-stitutive models are employed: The original HardeningSoil model, the HS-Small model, and the Hypoplastic modelwith intergranular strain. All models were calibrated usingtriaxial and oedometer test data. For details on the calibra-tion of the constitutive models the reader will refer to [12].

The results of the strip footing case study are illustratedin Fig. 16. Apparently, both models incorporating small-strain stiffness give similar results. They both predict lowersettlements than the Hardening Soil model. As the verticaldisplacements with depth are decaying much faster in thesmall-strain models, the assumption of a limit depth for theactive compression zone, at the point where the additionalvertical stress due to external loads is not more than twentypercent of the effective vertical pressure of the overburden(e.g. German code of practice), seems justified. The ratioG/Gur plotted in Fig. 16 indicates elastic stiffness at theend of loading. In the example G0 equals 3 Gur.

4.4 Excavation problem 3D

The HS-Small model is deployed last in a 3D finite ele-ment analysis of an interaction between the excavation pitfor the lock Sülfeld (Fig. 17) and two railway bridge abut-ments which are located in a distance of approximately 15 m

to the excavation pit. Fig. 18 shows the geometry, struc-tural features, and excavation stages modeled within thefinite element analysis. The structural features include twoabutments on pile foundations, back-anchored and strut-ted retaining walls, sheet pile walls, and a floor slab in thestrutted section. Embedded elements are used to modelthe floor slab, sheet pile walls and piles. All other struc-tural features and the subsoil are discretized by volume el-ements. Four excavation stages are introduced: One be-fore wall construction (pre-excavation), and three after



Fig. 14. Vertical displacements in section A-A at the end ofexcavation

3.0

2.0

1.0

G/G

ur[-

]

0 1004020 60 80-20 120

0

40

20

80

60

100

Fig. 15. Ratio of the elastic moduli at the end of excavation

Fig. 16. Calculated settlements (right) and ratio G/Gur (left) at p = 150 kPa/m

Fig. 17. Strutted and back-anchored Sülfeld excavation nearthe railway bridge




wall construction (excavation step 1 to 3). Total excava-tion depth is 14 m. Construction of the retaining walls isnot modeled accurately. Instead, they are placed as awhole. The phreatic level is adjusted several times in thecalculation, due to groundwater discharge in the bedrockor groundwater lowering in the fill. After the final excava-tion step, the groundwater table is lowered due to worksnear the pumping station. This change in phreatic level isthe final load step considered in the analysis.

A two-step model calibration procedure was employed:First, all laboratory test results were statistically averagedbefore secondly, selected tests were back analyzed in numer-ical element tests. Properties of the sand layer were esti-mated from in-situ heavy dynamic penetration test results.The obtained material parameters were finally calibratedin a 2D finite element model by comparing calculated dis-placements of the existing lock during operation to meas -ured ones. In the calibration procedure, the confidence inthe correlated small-strain stiffness parameters could bespecifically increased.

The Sülfeld analysis’ main focus is on displacementsof the railway bridge abutments and their backfill. The wellequipped site allows for various other comparisons, as well.From extensometer readings in the excavation pit, the finalheave during excavation step 3 can be quantified as 2–3 mm.In the analysis, heave < 4 mm is obtained when using theHS-Small model. The HS analysis over-predicts the excava-tion heave by more than 8 mm. The measured settlement(extensometer) of the abutment closest to the excavation pitis shown in Fig. 19. From geodetic measurements, it can be

additionally derived that the abutments are not tilting. Ex-actly the same behaviour is found in the HS-Small analysis:Almost no tilt of the abutments occurs with a total settle-ment of approximately 2 mm. The HS calculation on the

Fig. 18. Geometry of the Sülfeld excavation in the 3D FE model

u - Hardening-Soilz

u [mm]z

0.0

-5.0

-10.0

-15.0

u - HS-Smallz

Fig. 19. Vertical displacements of the bridge abutments (hereshown with backfill)


other hand shows settlements of up to 10 mm in combina-tion with tilt.

In summary, the small-strain stiffness formulation inthe HS-Small model could be successfully deployed in theSülfeld 3D finite element analysis. It significantly improvedall results obtained from the Hardening Soil model. As inthe previous calculations, the improvement is achieved withzero strain history at the onset of loading.

5 Summary

The hyperbolic Hardin-Drnevich model was generalizedto multi-axial strain space, so that it can easily be com-bined with many existing elastoplastic constitutive modelsthat are based on isotropic elasticity and hardening. Infact, these are the kind of models most commonly used inengineering practice today. One of these is the HardeningSoil model, as implemented in the finite element codePlaxis V8. The small-strain stiffness enhanced version ofthe Hardening Soil model, introduced in this paper, isnamed HS-Small model.

Application of the HS-Small model is straightforward.Its Small-Strain Overlay extension requires only two addi-tional parameters. These are the initial small-strain stiffnessG0 and the threshold shear strain γ 0.7 at which stiffness isreduced to 0,7 G0. In the case of no small-strain stiffnessin-situ or laboratory test data being available, these twoparameters can be correlated with various other soil or testparameters.

The HS-Small model was applied in various bound-ary value problems. In these, all effects that are commonlyattributed to small-strain stiffness in soil-structure interac-tion could be recognized: The width and shape of settlementtroughs, for example, are more precisely modeled and ex-cavation heave is reduced to a realistic value. The overallreliability of numerical displacement analysis is consider-ably increased. Analysis results are also less sensitive tothe choice of proper boundary conditions. Large meshesno longer cause extensive accumulation of displacements,because marginally strained mesh parts are very stiff.

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Authors:Prof. Dr.-Ing. Thomas Benz, Geotechnical Division, Dept. of Civil andTransport Engineering, NTNU, Høgskoleringen 7a, N-7034 Trondheim,Norway, [email protected]. Radu Schwab, Federal Waterways Engineering and Research Institute, Kußmaulstraße 17, D-76187 Karlsruhe, Germany,[email protected]. Dr.-Ing. Pieter A Vermeer, Institute of Geotechnical Engineering, University Stuttgart, Pfaffenwaldring 35, D-70569 Stuttgart, Germany,[email protected]


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