interface element

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Soil structure interaction analysis: modeling

the interface

Morched Zeghal and Tuncer B. Edil

Abstract: The sand–structure interface, developed under monotonic loading, was modeled based on physical observations.

The model takes into account the macroscopic conditions to yield a general constitutive law applicable to a wide range

of contact problems and the microstructural considerations constitute the specialization of the general equations to a

specific problem. The surface of slippage was idealized to be sinusoidal based on an intensive numerical simulation

program that made use of the discrete element technique. The model incorporates the effect of grain crushing found to

play a major role in the behavior of the interface. Analysis of laboratory data revealed a close relationship between

grain crushing and the work dissipated plastically during shear. The proposed elastoplastic model, requiring a limited

number of parameters, predicts the shear stresses for the modified direct shear test and reproduces the shaft resistance

of the shaft–sand interface pullout tests in a satisfactory manner.

Key words: sand–structure interface, microstructure, grain crushing, plastic work.

Résumé : L'interface sable–structure sous un chargement monotone a été modélisée sur la base d'observations physiques.Le modèle prend en compte les conditions macroscopiques pour donner une loi générale de comportement applicable à

une large plage de problèmes de contacts, et les considérations de microstructure constituent la spécialisation des équa-

tions générales pour un problème spécifique. La surface de glissement a été idéalisée sous une forme sinusoïdale dans

un programme élaboré de simulations numériques qui utilisait la technique d'éléments discrets. Le modèle incorpore

l'effet du broyage des grains qui, trouve-t-on, joue un rôle majeur dans le comportement de l'interface. L'analyse des

données de laboratoire permet d'identifier une relation serrée entre le broyage des grains et le travail dissipé en état

plastique durant le cisaillement. Le modèle élastoplastique proposé, requérant un nombre limité de paramètres, prédit

les contraintes au cisaillement pour l'essai de cisaillement direct modifié et reproduit de façon satisfaisante la résistance

du fût à l'interface sable–fût dans les essais d'arrachement.

Mots clés : interface sable–structure, microstructure, broyage de grains, travail en plasticité.

[Traduit par la Rédaction] Zeghal and Edil 628

Introduction

Over the last two decades a multitude of models wereused to predict the behavior of soils under both monotonicand cyclic loading (Chen and Saleeb 1982, 1986; Desai 1980;Desai and Faruque 1984; Nayak and Zienkiewicz 1972).Concurrent with soil modeling refinement has been the in-vestigation of another important aspect of behavior: the in-teraction between soils and structures. It is known that thisinteraction has a significant effect on the response. Manyconstitutive laws have been proposed to evaluate the behav-ior of the contact effectively. The first generation of inter-face models was elastic in nature, and was followed by

hyperbolic models. At a later stage, different models basedon plasticity were used to capture the interaction betweenvarious types of soils and structural surfaces under differentconditions (Boulon 1987; Desai et al. 1984; Heuze andBarbour 1982; Zamman and Desai 1983). However, they are

based on the macroscopic response usually seen during labo-ratory testing. Improved study of contact problems shouldincorporate not only modeling of the interface but also mi-croscopic features of the contact, i.e., degree of roughness of the structural surface and the interface zone characteristics.This paper introduces a model in which such critical ele-ments are identified. It is simple in concept and designed tohandle monotonic loading only. The model employs a non-associated flow rule and the parameters needed are few andeasy to determine from the modified direct shear test. Themodel constitutes a new way of looking at the interface anda different approach from previous models because it isbased on the physical characteristics of the interface.

Constitutive law

An interface constitutive model is presented below. Themodel is an elastoplastic Mohr-Coulomb isochoric model

Can. Geotech. J. 39: 620–628 (2002) DOI: 10.1139/T02-016 © 2002 NRC Canada

620

Received 10 March 2000. Accepted 11 December 2001. Published on the NRC Research Press Web site at http://cgj.nrc.ca on9 May 2002.

M. Zeghal.1 National Research Council of Canada, Institute for Research in Construction, Urban Infrastructure RehabilitationProgram, Building M-20, Ottawa, ON K1A 0R6, Canada.T.B. Edil. Department of Civil and Environmental Engineering, University of Wisconsin-Madison, Madison, WI 53706, U.S.A.

1Corresponding author (e-mail: [email protected]).



based on such work-hardening, nonassociative plasticitymodels as those proposed for granular soils. The dilatancy ismodeled outside of the constitutive model, replacing the realstructural surface by an inclined one. Hardening–softeningdepends on the plastic work. An additional term is incorpo-rated to account for the significant grain crushing that occursin the interface zone.

Incremental stress displacement relationsConsider two bodies (1 and 2) in contact. Each point on

the contact surface has a displacement vector of u1 or u2 if the point is considered a part of body 1 or 2, respectively.The relative tangential and normal displacements (gt and gn,respectively) can be defined by

[1a] g u u t t = −( ) 2 1

and

[1b] g u u nn = −( ) 2 1

where t and n are unit vectors in the tangential and normaldirections to the surface of contact.

For the purpose of obtaining the constitutive law, attentionshould be focused on the slip phase. The slip is character-ized by

[2] F = 0

where F is the yield (slip) function. If it is also assumed thatthe coefficient of friction between the grains is a function of the plastic work W p (used as an index of the amount of graincrushing) and that any hardening or softening is due to theplastic tangential work W t

p, the consistency equation

[3] dF = 0

can be rewritten as

[4] d d d dtp t

p

p

pF F F

W W

F

W W

i

i= + + =∂∂σ

σ ∂∂

∂∂

0

where σ i is the stress.The total incremental relative displacement is decomposed

into two parts: an elastic (recoverable) part and a plastic re-sidual part.

[5] d d de pg g gi i i= +

where dgi is the total incremental relative displacement; d egi

is the elastic part; and d pgi is the plastic incremental relativedisplacement.

If the change in stress is assumed to be due only to elastic

relative displacements then

[6] d d eσ i ij j E g=

The permanent relative displacements due to sliding andvolume change are related to the plastic potential functionthrough what is known as the plastic potential flow rule

[7] d

if or d

if and dpg

F F

GF F i

i

=< <

= =

0 0 0

0 0λ ∂∂σ

where G is the potential function and λ is a positive scalar.

Combining eqs. [4]–[7] and eliminating the λ term, wecan write the incremental stress – relative displacement rela-tion as

[8] d d

dp

pq q

p pq q

σ

∂∂σ

∂

∂σ

∂

∂σ

∂∂σi ij j

j

E g

F E g

F E

G H M

G= −− −

i j t n

p q t n

, ,

, ,

==

where dgi is the increment of relative displacement, E ij is theelastic stiffness, F and G are the yield and potential functions,respectively, H is the hardening term, and

M = ∂∂µ

∂µ∂

σ ∂∂σ

σ ∂∂σ

F

W

G Gp t

t

n

n

+

is a term incorporated to account for the changing coeffi-cient of friction. H will be zero if no hardening is present inthe model and M will also be zero if the coefficient of frictionis taken as a constant. At this stage, the model does not in-clude any hardening ( H = 0).

Yield and potential functionsDiscrete element simulations revealed that the slippage

surface showed some curvature and undulations that are afunction of the roughness of the structure present (Zeghal1993). The microstructural shear band, which is a thin layer,has a thickness. It is idealized by a sinusoidal wave. Itslength, height, and slope are denoted by Lk , h, and α, respec-tively, as shown in Fig. 1. Once the geometrical characteris-tics of the sinusoidal surface are defined, the slope at the

tangent point of contact can be expressed as a function of the sinusoidal length, the height, and the tangential displace-ment as (Plesha 1987)

[9] α π πk

k

t

k

= +

h

L

g

L2 21sin

where gt is the tangential displacement. If we assume thatfriction on the sinusoidal surface obeys

[10] | |σ µσt n≤ −

where the coefficient of mobilized friction, µ, is a functionof the plastic work done, the corresponding slip (yield) func-tion in the case of smooth surface (flat slip surface) can be

written as

[11] F = +| |σ µσt n

Slip occurs when F = 0, and the behavior is elastic if F isnegative. The condition of F > 0 is not possible. The poten-tial function is taken as (Plesha 1987)

[12] G = | |σ t

For a sinusoidal surface, the slip and potential functionscan be rewritten as (by transformation; rotation of αk )

[13] F = + + −| ( sin cos )| ( cos sin )σ α σ α µ σ α σ αn k t k n k t k

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Zeghal and Edil 621



[14] G = +| ( sin cos )|σ α σ αn k t k

where αk and µ are as defined above and σ t and σn are thetangential and normal stresses, respectively. The coefficientof mobilized friction, µ, is defined as a function of the plas-tic work W p, which in turn is used as an index of graincrushing, as in

[15] µ =+W

a bW

pp

where a and b are two soil parameters. The following sec-tion presents evidence supporting the form of eq. [15].

Grain crushing

ExistenceAfter it was proposed that grain crushing was an impor-

tant factor in soil–structure interaction problems (Jensen etal. 2001), the results of modified direct shear tests providedby Hoteit (1990) were analyzed to confirm the existence of crushing and to evaluate the extent of its effect. Hoteit con-ducted an extensive experimental program on both loose anddense Quiou sand from Bretagne, France, at two initial con-fining pressures, 100 and 350 kPa, at the Institut NationalPolytechnique de Grenoble. Quiou sand is a uniform (unifor-mity coefficient, C u = 9.4) and calcareous sand with a meangrain size of 0.40 mm and an effective grain size of 0.05 mm. Furthermore, the test was performed for constantstress as well as for constant volume stress paths. Hoteitmeasured the grain size distribution prior to the test and alsomonitored its change during the shear test. The data wereanalyzed using the breakage factor defined by Lee andFarhoomand (1967). They evaluated the amount of crushingby defining the ratio D15i / D15f where D15i is the diametercorresponding to 15% finer before shearing, and D

15f

is thediameter still corresponding to 15% finer but after shearingoccurred. The analysis done using both D10 and D50 suggeststhat the D50i / D50f ratio is more suitable to describe crushing.Figure 2 shows the variation of the breakage factor as afunction of the plastic work (at the end of shear tests) using D10 and D50. The data is more scattered in the case of D10.This can be explained by the fact that this quantity is small,making it very sensitive to change. Figure 3, showing thepercentage retained by each sieve, clearly supports that thebigger grains are crushing first (by looking at the curve cor-responding to the beginning of shear) and crushing the most(curve at the end of shear).

Figure 4 shows that the crushing occurs, for loose anddense sand, at both high (350 kPa) and low (100 kPa) con-fining pressures and for both stress paths (constant stressand constant volume). This suggests that the amount of crushing can be assumed to be independent of the stress pathand initial confining pressure but that it is clearly a functionof the plastic work. Of course, plastic work is not solely dueto grain crushing, and there are other significant mechanismsthat contribute to plastic work; however, it seems that empir-ical data show that the degree of grain crushing is remark-ably well correlated to the amount of plastic work as shownin Fig. 4. More significantly, the relationship appears to beindependent of the direct shear stress path taken, i.e., con-stant normal stress or constant volume providing a basis fora unified formulation.

Coefficient of mobilized friction as a function of plasticwork

The same data were used to investigate the coefficient of mobilized friction. The ratio of the interface tangential andnormal stresses σ t / σn , referred to as the normalized shearstress, was used as a measure of the coefficient of mobilized

© 2002 NRC Canada

622 C an. G eotech. J. Vol. 39, 2002

Fig. 1. Shear band idealization. Lk , sinusoidal length; LL, left

half wavelength of the sinusoidal; LR, right half wavelength of

the sinusoidal; h, sinusoidal height; α, sinusoidal slope.

Fig. 2. Breakage factor for calcareous sand (a) based on D10 and

(b) D50 versus plastic work at the end of test.



friction. Figure 5a shows that the normalized shear stress isnot constant but changes with the tangential displacementfor both constant stress and constant volume tests based onHoteit’s tests on calcareous sand. To confirm that this be-havior is not unique to this type of calcareous sand, the sameanalysis was extended to a silica sand (Portage sand) testedby Sengara (1991) for both constant volume and constantstress tests under three different initial normal stresses (35,69, and 138 kPa) in a modified direct shear box. Figure 5bshows essentially the same behavior for Sengara’s silicasand. However, the small peak behavior evident in Hoteit’stests on the calcareous sand is absent, perhaps due to themore resistant mineralogical composition of the silica sand.It is also noted that the higher the initial confining pressure,the lower the normalized shear stress before all the curvesconverge asymptotically to the same value.

The experimental evidence from two different sourcessupports the idea of a changing coefficient of mobilized fric-tion during the interface shear. Figure 6 shows the normal-

ized shear stress as a function of total plastic work (used asan index of grain crushing and other plastic mechanisms) forthe experimental data provided by both Hoteit and Sengara.The results show that the curves for constant stress and con-stant volume for the same initial confining pressure coincideeven though the constant volume curve continues as a con-stant function of plastic work beyond the termination of theconstant stress curve.

The idea of changing the coefficient of mobilized frictionas a function of the plastic work appears to be reasonableand is backed by the experimental data. The curves showingthe change in the coefficient of friction as a function of plas-tic work appear very similar to the widely used hyperbolicmodel used in soil mechanics i.e.,

[16] µ =+W

a bW

p

p

where a and b are two parameters that are related to the ini-tial slope and the asymptotic limit of the above curves.These parameters can be obtained as the intercept and the

© 2002 NRC Canada

Zeghal and Edil 623

Fig. 4. Breakage factor versus plastic work for calcareous sand.CS, constant stress test; CV, constant volume test.

Fig. 5. Normalized shear stress versus tangential displacement

for (a) calcareous sand and (b) silica sand. CS, constant stress

test; CV, constant volume test.

Fig. 3. Breakage analysis for calcareous sand.



slope of a plot of the ratio of plastic work to normalizedshear stress versus plastic work. Indeed, the data plotted inthis fashion resulted in a linear plot, allowing for the deter-mination of a and b (Zeghal 1993). Figure 7 shows a com-parison between the experimental data of the constantvolume test run with an initial confining pressure of 138 kPaand the numerical data obtained using eq. [16]. The numeri-cal and experimental curves show a good fit.

Parameters needed

The model requires two sets of parameters. The first set,referred to as physical parameters, are related to the mechan-ical properties of the soil. The second set are referred to asthe geometrical parameters, and they are related to themicrostructural feature of the shear band at the interface i.e.,sinusoidal surface.

Physical parametersFour parameters are needed and related to the physical

properties of the interface.

Shear and normal stiffnesses

These two parameters are related to the elastic response.The shear stiffness E t is the slope of the unloading part of the shear stress (σ t ) versus relative tangential displacement(gt) curve. It can be taken as the initial tangent of the un-loading curve or as the average slope of the unloading–re-loading phase. The normal stiffness E n denotes the slope of the unloading part of the normal stress (σn) versus relativenormal displacement (gn) curve. These parameters can bedetermined from a variety of tests, such as the modified di-rect shear test.

Grain crushing parameters

Two parameters are needed to describe the process of grain crushing as a function of plastic work. They can be de-termined from the modified direct shear test. At any giventime during shear, both the incremental plastic work d d dp

t tp

n npW g g= +σ σ and the ratio of the tangential stress

to the normal stress σ t / σn (normalized shear stress) can beevaluated. The plot of the ratio of the accumulated plasticwork and normalized shear stress (W p nσ / σ t ) as a function of plastic work (W p) yields a straight line. The intercept of thisline is the parameter a and the slope is the parameter b of eq. [16].

© 2002 NRC Canada

624 C an. G eotech. J. Vol. 39, 2002

Fig. 7. A comparison between experimental data and hyperbolic

curve fit.

Fig. 8. The modified direct shear interface test.

Fig. 6. Normalized shear stress versus plastic work for (a) cal-

careous sand and (b) silica sand. CS, constant stress test; CV,

constant volume test.



Geometrical parametersThe geometrical parameters are related to the shape of the

idealized shear band surface. A sinusoidal surface can be de-fined either by its half wavelength Lk and height (h) or by itswavelength and slope (α). The length of the sinusoidal sur-face is assumed to be the same for different stress paths. It isdeduced from the maximum tangential displacement needed

to reach the ultimate shear strength. Since the displacementnecessary to reach this point is higher for constant volumetests than for constant stress tests, the former is adopted tobe the length of the sinusoidal surface. The slope, definingthe second parameter, is determined from the modified directshear tests as the dilatancy angle characteristic of the volumechange behavior.

Model performance

To check the performance of the interface model, two lab-oratory tests that were developed and conducted at the Uni-versity of Wisconsin-Madison were modeled, i.e., the

modified direct shear test (Sengara 1991) and the shaft–sandinterface pullout test (El Sakhawy 1991; El Sakhawy andEdil 1996). In both tests, Portage sand was used at a relativedensity of 70%. It is a uniform (C u = 1.8) angular silica sandwith a mean grain size of 0.32 mm and an effective grainsize of 0.22 mm. The same method of preparing rough struc-tural surfaces was followed; consisting of gluing Portagesand grains to the surface of the structural element (steel).The newly developed interface model was implemented inPLASOIL, a general finite element code developed at theUniversity of Wisconsin-Madison (Abdel Rahman 1988;Edil and Abdel Rahman 1993). This code supports manysoil models, of which the elastic constitutive law was used,to model the mass of soil beyond the interface (Hoteit 1990).

Modified direct shear testsFigure 8 presents the modified direct shear test (Sengara

1991). It is similar to the interface friction tests used by

© 2002 NRC Canada

Zeghal and Edil 625

Fig. 10. A comparison of the experimental results with the

model prediction in a constant volume modified shear test on

silica sand for a rough surface at 69 kPa initial normal stress.

Fig. 9. A comparison of the experimental results with the model

prediction in a constant stress modified shear test on silica sand

for a rough surface at 69 kPa normal stress.



Plytas (1985), Hoteit (1990), and numerous other investiga-tors, except that the normal stress is applied using a pressur-ized bag instead of a rigid plate over the sand. In the lowerhalf of the 50 mm square direct box, sand is replaced withthe structural material (smooth steel plate or rough platewith sand grains glued on), and a flexible membrane bound-ary consisting of a rubber bag replaces the upper loadingplaten. The bag is filled with water and pressurized. This

setup allows a shear test to be conducted under a constantnormal stress path in which the change in volume of the sandduring shear is measured. Alternatively, the tests can be per-formed under a constant volume stress path in which thechange in normal stress can be measured during shearing.

Rough surfaceUsing the results of the modified direct shear test for three

different confining pressures, namely 35, 69, and 138 kPa,the interface model parameters are determined as (i) tangen-tial stiffness: E t = 181 055 kN/m3; (ii) normal stiffness: E n =542.9 × 106 kN/m3; (iii) grain crushing parameters (a and b):0.03 and 1.43, respectively; and (iv) geometrical parameters:

the half wavelength was taken to be 5 mm and the slope tobe 4.4° as an average of the results of the tests under threedifferent confining pressures.

Figure 9, presenting the comparisons between the back prediction and the experimental results of tangential stressand normal displacement versus tangential displacement be-havior in a constant normal stress test performed under aconfining pressure of 69 kPa, shows that they are in reason-

ably good agreement. Figure 10, giving the results of thesesimulations in terms of tangential and normal stresses versustangential displacement in a constant volume test, clearlyshows that the model is capable of reproducing the experi-mental data in a satisfactory manner

Smooth surfaceThe case of a smooth surface was also modeled. The pa-

rameters corresponding to this case are similar to those for therough case except for the following: (i) grain crushing param-eters (a and b): they were determined to be 0.01 and 2.5, re-spectively; the method of determining these parameters issimilar to the method used in the rough surface case; and

© 2002 NRC Canada

626 C an. G eotech. J. Vol. 39, 2002

Fig. 12. The axisymmetric sand–shaft interface test (small-scale

pile shaft load test).

Fig. 11. A comparison of the experimental results with the model

prediction in (a) a constant stress and (b) a constant volume

modified shear test on silica sand for a smooth surface at 69 kPa

normal stress.



(ii) geometrical parameters: the sinusoidal half wavelengthwas also taken to be 5 mm. The slope of the sinusoidal curvewas taken as zero. More precisely, the sliding surface is notsinusoidal any more, but it is now a flat surface.

Figure 11, giving the results of these simulations in a con -stant stress and a constant volume test, clearly shows thatthe model is capable of reproducing the experimental data in

a satisfactory manner for a smooth surface as well.

Shaft–sand testsThe second test used in evaluating the interface model

performance is the shaft–sand interface pullout test (ElSakhawy 1991; El Sakhawy and Edil 1996). It is an originalpiece of axisymmetrical interface shear testing equipmentthat uses a cylindrical soil specimen (117 mm in diameter,267 mm in height) in a test cell similar to the triaxial com-pression test with a shaft (27 mm in diameter) in the centerof the specimen. The shaft surface is made rough by gluingsand grains on it. The boundary stresses and displacementsof the soil specimen are independently controlled and thevertical and horizontal boundaries are decoupled using pres-

surized bags. Figure 12 shows the shaft–sand interface pull-out test. It was intended to simulate a soil elementsurrounding an axially loaded inclusion. Initial lateral nor-mal stresses of 46, 69, and 104 kPa were applied on thesame sand that was used in the modified direct shear tests.

The mass of the sand beyond the interface was also mod-eled as elastic. Since the same sand used in the modifiedshear test is used in this test too, the same parameters deter-mined from the former test and for the rough case wereused. Figure 13 shows that the model is capable of replicat-ing the mechanism occurring at the pile–sand interface atdifferent confining pressures.

ConclusionsA new constitutive law was developed to model sand–

structure interfaces. It requires relatively few parameters thatcan easily be determined from one laboratory test, namelythe modified direct shear test. The model captures signifi-cant aspects of both macro- and micro-structural features of the contact and incorporates the effect of grain crushingthrough a close correlation between the coefficient of fric-tion and plastic work. To validate the new interface model,the modified direct shear and the pile shaft resistance testswere modeled. The results showed that the model provides aunified formulation capable of replicating and predicting thebehavior at the sand–structure contact independent of the di-rect shear path taken (i.e., for both constant normal stressand constant volume tests) and independent of the test con-figuration (i.e., for both direct shear and axisymmetric inter-face shear tests) using the same model parameters. However,it requires further refinement to better define the geometricalparameters and to include features such as hardening andhandling of cyclic loading.

Acknowledgment

The test data used in this paper were generated by other in-vestigators. The authors acknowledge the valuable contribu-tion of these investigators, namely Dr. N. Hoteit under the

© 2002 NRC Canada

Zeghal and Edil 627

Fig. 13. A comparison of the experimental results with the

model prediction in an axisymmetric shaft–sand interface test on

silica sand for a rough shaft at (a) 46 kPa, (b) 69 kPa, and

(c) 104 kPa lateral confining stress.



© 2002 NRC Canada

628 C an. G eotech. J. Vol. 39, 2002

supervision of Dr. Marc Boulon at L’Institut NationalPolytechnique de Grenoble, Drs. W. Sengara and N. ElSakhawy at the University of Wisconsin-Madison. The sec-ond author, T. Edil, also acknowledges the direct contribu-tions of Dr. Marc Boulon to his understanding of the interfacebehavior. The authors also express their gratitude to Prof. Mi-chael E. Plesha of the University of Wisconsin-Madison for

his critique and discussions of the constitutive model.

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