1. bookVolumen 45 (2023): Edición 3 (September 2023)
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Studia Geotechnica et Mechanica
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An elastoplastic constitutive model for assessing ground settlements induced by deep excavations

Hiba El-Arja,
Hiba El-Arja,
Sébastien Burlon y
Sébastien Burlon y
Emmanuel Bourgeois
Emmanuel Bourgeois
Publicado en línea: 21 Sep 2023
Volumen & Edición: Volumen 45 (2023) - Edición 3 (September 2023)
Páginas: 231 - 245
Recibido: 16 Jan 2023
Aceptado: 13 May 2023
Studia Geotechnica et Mechanica's Cover Image
Studia Geotechnica et Mechanica
Detalles de la revista
License
Formato
Revista
eISSN
2083-831X
Primera edición
09 Nov 2012
Calendario de la edición
4 veces al año
Idiomas
Inglés
Introduction

The densification of urban areas implies the construction of deep excavations nearby sensitive constructions, for lack of available space at the surface. Most analyses are focused on wall displacements (Zhang et al., 2015a, Goh et al., 2017a), but the prediction of the displacements induced in the surrounding ground is a major challenge to avoid damages on existing buildings. Few calculation methods are able to assess not only the displacements of the retaining structure, but also the vertical displacements of the supported ground: the finite element method (FEM) makes it possible to evaluate such settlements, while taking into account a wide range of configurations and construction techniques.

However, FEM simulation results are not always consistent with observations: the removal of the weight of the excavated ground induces a heave of the bottom of the excavation and may lead to poor prediction of the settlement of the ground behind the retaining structure (Schweiger, 2002a, Coquillay et al., 2005, Elmi et al., 2006, Zhang et al., 2018). In terms of finite element simulations, constitutive models combining a linear elastic model and a shear strength failure criterion tend to predict ground heave behind the retaining structure instead of settlement (Jardine et al., 1986; Finno and Harahap, 1991; Simpson, 1992; Finno, 2008) and to overestimate the extent of the influence zone (Delattre, 2004). These two issues are fundamental in terms of design. In practice, designers take into account the limits of linear elasticity by assuming that the deformations are within a given range and then adjust deformation moduli to obtain acceptable horizontal displacements. Using nonlinear elastic models (Duncan and Chang, 1970; Jardine et al., 1986; Zdravkovic et al., 2005) gives very good results regarding the prediction of horizontal displacements of the retaining structures and improves the prediction of settlements behind the retaining structure. However, results for settlements can still be improved. Besides, the behavior of geomaterials is not only nonlinear, but also irreversible. It can be argued that, around deep excavations, plastic strains become prominent and must be precisely modeled to predict the settlements generated behind the retaining wall.

The choice of the yield function, the flow rule, and the hardening rule is, therefore, essential. A popular constitutive model for excavation problems is the “Hardening Soil Model” (Schanz et al., 1999) and its extension HSM-small strain (Benz, 2007). It is an elastoplastic model with a nonlinear elasticity and two plastic strain-hardening mechanisms: a deviatoric mechanism and a volumetric mechanism. It was first implemented in Plaxis (Plaxis, 2020), and other FE software reproduce it at least partially (for instance, Zsoil [Obrzud and Truty, 2018] or CESAR-LCPC [Itech, 2021]). This model has significantly improved the assessment of ground displacements induced by excavations (Obrzud, 2010; Obrzud and Truty, 2018), but its formulation is complex, so it is not easy to anticipate the influence of some of the parameters on the results of the simulation of an excavation.

In this paper, an elastoplastic constitutive model called “the Hardening Contractive Dilatant model,” or, in short, the HCD model is proposed. It aims to be easily used by engineers. First, the formulation is presented and a methodology is proposed for the determination of parameters. An application of the HCD model to the simulation of the construction of an experimental sheet pile wall in Germany is then presented.
Formulation of the HCD model

The HCD model is derived from a model developed for sands under monotonic and then under cyclic loading, called MODSOL (Chehade, 1991; Khoshnoudian, 2002; Khoshravan Azar, 1995; Shahrour and Ousta, 1998; Shahrour et al., 1995a; Zaher, 1995).

Elastic part of the model

The proposed approach is based on the idea that the plastic part of the strain plays a major role in the behavior of deep excavations, so that the elastic part of the model does not need to be described in detail. Only two parameters, Young's modulus and Poisson's ratio, are required. Thus, the formulation of the elastic part of the HCD model is not as flexible or versatile as that of the Hardening Soil Model or the HS-small model, but it is much simpler (it is worth recalling that the elastic part of these models requires five and six parameters, respectively, in addition to the strength parameters c and φ). In other words, we advocate the idea that, for deep excavations, the elastic part of the constitutive model can be kept simple; it may not be appropriate for other types of structures, such as shallow foundations or tunnels.

Note that the finite element code CESAR, in which the HCD model is programmed, makes it possible to combine the plastic part of the HCD model with a wide range of elastic models: linear models with moduli varying with depth, nonlinear models (the nonlinear elasticity of the modified Cam-Cay model, of HSM, or the Fahey–Carter model, etc.).
Plastic regime

Regarding the plastic regime, the model aims at reproducing the main phases defined by Fern and Soga (2018) as shear strain develops: initial contractive hardening phase, phase transition point (Tatsuoka et al., 1986), dilatant hardening phase, then evolution toward a regime where the volumetric plastic strain remains constant.

Yield function

The yield function is a function of the principal stresses and accounts for the influence of the Lode angle: fp, q, θ=q−Rθp+pc f\left( {p,\;q,\;\theta } \right) = q - R\left( \theta \right)\left( {p + {p_c}} \right) where p denotes the mean effective stress, q the deviatoric stress, and θ the Lode angle: p=13trσ¯¯ p = {1 \over 3}{\rm{tr}}\left( {\overline{\overline \sigma } } \right) q=32s¯¯:s¯¯ q = \sqrt {{3 \over 2}\overline{\overline s} :\overline{\overline s} } θ=13 Arcsin 33J32J23/2 \theta = {1 \over 3}{\rm{Arcsin\;}}\left( {{{3\sqrt 3 {J_3}} \over {2J_2^{3/2}}}} \right) s¯¯=σ¯¯−p1¯¯ ;J2=12s¯¯:s¯¯=12sijsij;J3=13s¯¯:s¯¯:s¯¯=13sijsjkski \matrix{ {\overline{\overline s} = \overline{\overline \sigma } - p\overline {\bar 1} \;;{J_2} = {1 \over 2}\overline{\overline s} :\overline{\overline s} = {1 \over 2}{s_{ij}}{s_{ij}};} \cr {{J_3} = {1 \over 3}\overline{\overline s} :\overline{\overline s} :\overline{\overline s} = {1 \over 3}{s_{ij}}{s_{jk}}{s_{ki}}} \cr } It follows from equation (1) that the yield surface is a cone in the principal stress state (with a noncircular trace in the deviatoric plane, because R(θ) depends on the the Lode angle, see Figures 1 and 2). In equation (1), care must be taken that the chosen yield surface is very simple: as a consequence, pc, which represents the distance from the apex of the yield surface to the origin of the effective principal stress space, should be interpreted as a parameter describing (together with R(θ)) the shear strength, rather than the material tensile strength. Besides, pc remains constant; by contrast, R(θ) is the local strain-hardening variable and varies as plastic strains evolve. Its initial value Ro(θ) defines the initial opening and shape of the elastic domain and is a function of the initial friction angle φo and the Lode angle θ according to the equation Roθ=6sinφo3−sinφosin3θ {R_o}\left( \theta \right) = {{6{\rm{\;sin\;}}{\varphi _0}} \over {3 - {\rm{\;sin\;}}{\varphi _o}{\rm{\;sin\;}}3\theta }}

Figure 1:

Yield surface in the principal stress space.

Figure 2:

Section of the yield surface in the deviatoric plane.
Hardening rule and failure surface – ultimate friction angle

As plastic strains develop, the hardening variable R(θ) evolves according to the equation Rθ=Roθ+Rfθ−Roθεdpb+εdp R\left( \theta \right) = {R_o}\left( \theta \right) + \left( {{R_f}\left( \theta \right) - {R_o}\left( \theta \right)} \right){{\varepsilon _d^p} \over {b + \varepsilon _d^p}} with Rfθ=6sinφult3−sinφultsin3θ {R_f}\left( \theta \right) = {{6{\rm{\;sin\;}}{\varphi _{ult}}} \over {3 - {\rm{\;sin\;}}{\varphi _{ult}}{\rm{\;sin\;}}3\theta }} where φult and b are material parameters. The Lode angle θ in equations (6)–(8) allows to distinguish compression stress paths (sin 3θ=1) and extension stress paths (sin 3θ=−1).

In equation (7), we use the plastic deviatoric strain εdp \varepsilon _d^p defined as (Shao and Desai, 2000): dεdp=23dε¯¯dp:dε¯¯dp d\varepsilon _d^p = \sqrt {{2 \over 3}d\overline{\overline \varepsilon } _d^p:d\overline{\overline \varepsilon } _d^p} Because εdp \varepsilon _d^p can only increase, according to (9), the model excludes any softening behavior.

For large values of εdp \varepsilon _d^p , the yield function tends to become ff=q−Rfθp+pc {f_f} = q - {R_f}\left( \theta \right)\left( {p + {p_c}} \right) φult is the ultimate friction angle, whereas parameter b describes the rate of strain hardening: for εdp=b \varepsilon _d^p = b , R takes the value (Ro+Rf)/2 at the center of the interval between its initial value Ro and its asymptotic value Rf.
Flow rule – characteristic friction angle

The originality of the proposed formulation lies in the fact that the flow rule is not defined by a plastic potential. Instead, the plastic strain increment is split into a deviatoric part dεdp¯¯ d\overline {\overline {\varepsilon _d^p} } and a volumetric part dενp¯¯ d\overline {\overline {\varepsilon _\nu ^p} } : dεp¯¯=dεvp¯¯+dεdp¯¯=dλGvp, q+Gdp, q d\overline {\overline {{\varepsilon ^p}} } = d\overline {\overline {\varepsilon _v^p} } + d\overline {\overline {\varepsilon _d^p} } = d\lambda \left[ {{G_v}\left( {p,\;q} \right) + {G_d}\left( {p,\;q} \right)} \right] where dλ is the plastic multiplier and the two functions Gv and Gd define the direction of the volumetric and deviatoric strain increments, respectively: Gvp, q=exp−αo εdpp+pc−qMcθI¯¯3 {G_v}\left( {p,\;q} \right) = {\rm{\;exp\;}}\left( { - {\alpha _o}\,\varepsilon _d^p} \right)\left[ {\left( {p + {p_c}} \right) - {q \over {{M_c}\left( \theta \right)}}} \right]{{\overline{\overline I} } \over 3} Gdp,q=p+pcMcθ3 s¯¯2 q {G_d}\left( {p,\;q} \right) = {{\left( {p + {p_c}} \right)} \over {{M_c}\left( \theta \right)}}{{3\;\overline{\overline s} } \over {2\;q}} where the parameter Mc(θ) depends on the Lode angle according to the equation Mcθ=6sinφc3−sinφcsin3θ {M_c}\left( \theta \right) = {{6{\rm{\;sin\;}}{\varphi _c}} \over {3 - {\rm{\;sin\;}}{\varphi _c}{\rm{\;sin\;}}3\theta }} which introduces an additional material parameter noted φc and called “characteristic angle.”

Three cases can be distinguished:

If q<Mc(θ)(p+pc), the behavior is contractive;

If q=Mc(θ)(p+pc), the soil reaches the characteristic state, which corresponds to the phase transition point from a contractive to a dilatant behavior; and

If q>Mc(θ)(p+pc), the behavior is dilatant.

 The proposed flow rule makes it possible to better represent the volumetric strains at the onset of the plastic regime because contractive plastic strain compensates for the dilatant volumetric elastic strains.

The transition between contractive and dilative regimes for the volumetric strain is similar to Rowe's Stress dilatancy equation (Rowe, 1962), which can be adapted to express the mobilized dilatancy angle as (Schanz and Vermeer, 1996): sin ψm=sin φm−sin φcv1−sin φm sin φcv sin\;{\psi _m} = {{sin\;{\varphi _m} - sin\;{\varphi _{cv}}} \over {1 - sin\;{\varphi _m}\;sin\;{\varphi _{cv}}}} But in the formulation of the HCD model, the plastic strain is not described by a plastic potential and a dilatancy angle: instead, equations (12) and (13) describe independently the deviatoric and the volumetric part of the plastic strain increment. Equation (12) shows that, for large deviatoric plastic strains, the volumetric plastic strain increment tends toward zero. Besides, the parameter αo makes it possible to control the rate at which plastic volumetric strain rate tends toward zero as the deviatoric plastic strain εdp \varepsilon _d^p increases. Note that the volumetric plastic strain increment is not related to a critical state defined by the stress ratio, for instance, in contrast to the approach proposed by Roscoe et al. (1958) or Schofield and Worth (1968).
Example of calibration of the HCD model on triaxial tests

A clear methodology for the determination of the model parameters based on data available for a real project is a crucial issue in practice. The formulation of the HCD model includes eight parameters:

the elastic parameters: Young's modulus E and Poisson's ratio ν;

the initial friction angle φo, the apex parameter pc and the ultimate friction angle φult; and

the characteristic angle φo and the hardening parameters αo and b.

 In this paragraph, the calibration of the model parameters is discussed using triaxial tests carried out for a metro station of line 15 south of the “Grand Paris Express” project. The data were provided by Soletanche-Bachy (a French deep foundations contractor). Tests with different confining stresses have been performed on samples from three layers (Table 1).

Summary of the triaxial tests considered in this study.

TestDepthType of soilType of test
B6.5–7Very fine silty sandConsolidated drained CD
C8.5–9.5Slightly sandy pebbles and gravelConsolidated drained CD
E13.7–14Silty clayConsolidated drained CD

In a first step, the apex parameter pc and the ultimate friction angle φult can be deduced from the stresses at failure in the drained triaxial tests.

As a first approximation, it is assumed that the elastic limit corresponds to an axial strain of 0.1%, which makes it possible to calculate Young's modulus E and Poisson's ratio ν. Denoting the mean effective stress and the deviatoric stress for this value of the axial strain as p0 and q0, parameter Ro is obtained from the equation (1): Ro=qopo+pc {R_o} = {{{q_0}} \over {{p_0} + {p_c}}} Angle φo can then be estimated using equation (6) with 3θ=1: φo=Arcsin3Ro6+Ro {\varphi _0} = {\rm{Arcsin\;}}{{3{R_o}} \over {6 + {R_o}}} The characteristic angle φc corresponds to the transition between the contractive and the dilatant behavior and its value can be determined by adjusting the axial strain εa–volumetric strain εv curve. The value of parameter b is determined by fitting the simulations on the axial strain εa–deviatoric stress q curve.

Eventually, the value of parameter αo is obtained by fitting the end of the simulated axial strain–volumetric strain curve on the results of the drained triaxial tests.

Table 2 summarizes the parameters obtained for each test. Numerical results are shown in Figures 3–5 and compared with test results. Note that, for tests E1, E2 and E3, the measured deviatoric stress presents a peak value and then a progressive decrease: as explained before, such a behavior cannot be reproduced using the HCD model.

Figure 3:

Calibration of the HCD model on tests B1, B2, and B3.

Figure 4:

Calibration of the HCD model on tests C2 and C3.

Figure 5:

Calibration of the HCD model on tests E1, E2, and E3.

Parameters of the HCD model for tests B, C, and E (σc denotes the confining pressure).

σcEvφopcφultαoφcb
kPaMPa-°kPa°-°-
Test B1150450.2710351310.005
2250700.2710351320.008
Test C34001070.2710351330.009
22001330.11515411380.002
33502330.1271515411390.005
Test E1200160.121.260231200.004
2350450.122.260231200.005
35501120.144.260231210.015

Besides, the results presented in Table 2 show the fitting of the parameters for each test: this leads to different values of Young's modulus for the different confining pressures. To overcome this difficulty, one can choose an average value or use a modulus value varying with the depth. This is the choice adopted in Section 3.

Two additional drained triaxial tests are performed on an alluvial soil: a monotonic test and a test with two loading–unloading cycles. In the latter case, the shear stress increases until the axial strain reaches 0.5%, then it returns to zero, then it increases up to 1.5% axial strain; a second unloading occurs followed by a final loading up to 15% axial strain.

The slope of the axial strain–deviatoric stress curves at the beginning of the monotonic test and on the unloading cycles is chosen to define the value of Young's modulus E. Poisson's ratio ν is determined using the slopes of the volumetric strain–axial strain curves at the beginning of the monotonic test and on the unloading cycles. Ultimate shear strength parameters are determined in the p-q plane for the highest levels of strain. Parameter φo is determined by assuming that the initial stress state of the triaxial test is on the boundary of the elastic domain. Parameters of the flow rule αo and φc are obtained by fitting the volumetric strain–axial strain curve on the experimental results. Parameter b, which governs the strain-hardening mechanism, is fitted to reproduce the deviatoric stress– axial strain curves.

Table 3 and Figures 6 and 7 show the results of the calibration.

Figure 6:

Calibration of the HCD model on the monotonic drained triaxial test on alluvial material.

Figure 7:

Calibration of the HCD model on the drained triaxial test with unloading cycles on alluvial material.

HCD model parameters.

Evφopcφultαoφcb
MPa °kPadeg deg
Monotonic test (confining pressure 175 kPa)570.4φmin0378270.001
Cyclic test (confining pressure 157 kPa)570.4φmin03720310.003

This example of calibration shows that the HCD model has the ability (or the flexibility) to reproduce accurately the results of triaxial tests and proposes to determine the values of the parameters in a given order. It remains to discuss the ability of the model to reproduce the behavior of a full-scale structure.
Case history: Hochstetten sheet pile wall

This section presents a numerical analysis of an experiment carried out in Hochstetten, Germany, by the University of Karlsruhe (von Wolfferdorff 1994a, 1994b). A sheet pile wall was monitored during several construction and loading stages. The experiment was the basis for a prediction exercise: 18 contributions were submitted using finite element models and 23 using the subgrade reaction coefficient method. Shahrour et al. (1995b) and Coquillay (2005) noted that the overall behavior and the horizontal displacements obtained by numerical simulations were reasonably close to the measures, but reproducing the settlements behind the wall remained a challenge.

In the following sections, we present finite element calculations for this sheet pile wall performed using the finite element software CESAR-LCPC in which the HCD model has been implemented.

Description of the experiment

The Hochstetten sheet pile wall was made of sheet piles vibrodriven down to a depth of 6 m in a sandy ground (see Figure 8). The water table was located 5.5 m below the natural ground level. The experiment includes the following stages:

0) Installation of the sheet piles

1) Excavation down to a depth of 1.75 m

2) Installation of three struts at a depth of 1.25 m with a spacing of 2.4 m and a prestressing force of 10 kN per strut

3) 4) 5) Excavations down to 3, 4, 5 m

6) Application of a 10 kPa pressure at the ground surface between 1 and 5 m from the wall (by means of a basin filled with water)

Figure 8:

Experimental setup (a) front view, (b) top view.

After the application of the surface load, the struts were progressively shortened. This phase is not taken into account hereafter.

The monitoring devices measured bending moments in the wall, forces in the struts, horizontal displacements of the wall, vertical displacements of the ground surface behind the wall, and earth pressure on both sides of the wall. Results of the experiment have been made public by von Wolffersdorff (1994b) and reproduced by Shahrour et al. (1995) and Coquillay (2005).

Case studies of monitored full-scale structures are always extremely valuable for the validation of numerical simulations, but this experiment is especially interesting because the structure, the hydraulic regime, and the lithology of the site are simple. Notably, the fact that the sand is relatively homogeneous avoids the addition of uncertainties attached to the determination of the parameters for several layers. Moreover, the experiment was carried out in well-controlled conditions, which limits uncertainties that are frequent in actual construction sites, such as undocumented transient phases (temporary storage of excavated material, for instance). Besides, the monitoring devices provide rich information, unlike many references in the literature for which it is difficult to precisely reconstruct both the settlement measurements behind the retaining structures and the geotechnical context.
Geometry, mesh, and modeling assumptions

The main assumptions are the following:

The wall is modeled in plane strain conditions. The excavation is 4 m wide and not symmetric: the struts are connected to another stiffer wall. The lateral extension of the calculation model is equal to 50 m and the vertical extension is equal to 15 m; the mesh, shown in Figure 9, comprises 1200 quadratic elements and 3700 nodes.

The walls are modeled using surface elements (as opposed to beam elements) and the row of struts is modeled using a simple “bar” element acting as a horizontal spring; in other words, the bending stiffness of the struts is neglected.

The installation stage is not modeled; the so-called “wished-in-place” method is used.

Interface elements are introduced between the wall and the ground.

The analysis is carried out in drained condition.

Figure 9:

Mesh used for the simulations.

The calculation comprises a sequence of six stages that match the description of the experiment given in Section 3.1.
Boundary conditions

The horizontal and vertical displacements are set to zero on the mesh lower boundary, and the horizontal displacement is zero on the left and right boundaries.
Initial stresses

The initial stresses are geostatic: σv′=γz,σh′=Ko γz \sigma _{_v}^\prime = \gamma z,\sigma _h^\prime = {K_o}\;\gamma z where the earth pressure coefficient Ko is chosen equal to 0.318 (which corresponds to Ko=1−sinφ with a friction angle φ of 43°, see Section 3.7) the value of the (dry) sand volumetric weight is taken equal to 17 kN/m3.
Properties of the walls and struts

The axial stiffness EA and the flexural stiffness EI (per unit length of wall) are known for the experimental sheet pile wall and for the reaction wall. Following Coquillay (2005), in the plane strain simulations, an “equivalent” thickness teq and an equivalent Young's modulus Eeq are used, so that the following conditions are fulfilled: Eeq teq=EA;Eeq teq3/12=EI {E_{eq}}\;{t_{eq}} = EA;{E_{eq}}\,t_{eq}^3/12 = EI The parameters are summarized in Table 4 (thicknesses being rounded to whole numbers in cm).

Parameters of the reaction wall and the sheet pile wall.

WallE(MPa)I(cm4/m)A(cm2/m)Eeq (MPa)teq (cm)
Reaction wall210,00096810624,40010
Sheet pile wall210,00011,610116680035

The row of struts is modeled by a two-node linear element withstanding only tension/compression forces, with a linear elastic behavior; the elastic parameters are deduced from the axial stiffness of the struts and the spacing. In the simulations presented hereafter, we adopt the values reported in Coquillay (2005): E = 210,000 MPa and A = 0.02 m2/m. At the installation (stage 2), a prestressing force of 4.2 kN/m is taken into account, corresponding to the force in each strut (10 kN) divided by the spacing between struts (2.4 m).
Interface model and parameters

For the simulations with both models (HSM and HCD), the interface between the walls and the sand is modeled by zero-thickness elements called “joint elements,” based on the approach proposed by Goodman et al. (1968), characterized by their cohesion cint and friction angle φint. It is often suggested that these parameters can be deduced from the cohesion and friction angle csand and φsand of the sand by cint = Rinter tan csand and tan φint = Rinter tan φsand, where Rinter is a coefficient that remains to be discussed. By contrast, in the following, the interface strength properties are set independently from those of the sand. Since no specific data was available, after preliminary parametric studies, the following values are selected: cint=1 kPa and φint=5o {c_{int}} = 1\;{\rm{kPa}}\;{\rm{and}}\;{\varphi _{{\mathop{ int}} }} = {5^{\rm{o}}} A short discussion regarding the influence of this choice is presented in Section 4.
Sand parameters

The files of the prediction exercise contained the results of triaxial tests carried out on the sand of the site. The results of three monotonic drained tests are reproduced in Coquillay (2005), under the names S5DK15, S5DK21, and S5DK31 for confining pressures of 100, 200, and 300 kPa, respectively.

We have performed a calibration of the parameters of two models: the Hardening Soil Model (Schanz et al., 1999), because it is very widely used by engineering companies or firms and has become a standard, and the HCD model.

The calibration aimed at reproducing as much as possible the three tests considered with the same parameters. The calibration led to the parameters given in Table 5. The result of the calibration is illustrated in Figure 10 for the Hardening Soil Model and in Figure 11 for the HCD model.

Figure 10:

Results of the calibration of the Hardening Soil Model parameters.

Figure 11:

Results of the calibration of the HCD model parameters.

Reference parameters (calibrated on the triaxial tests) for the Hochstetten sand.

HSMEurref=63 MPa, E50ref=21 MPa, ν=0.1, m=0.5, pref=100 kPa, c=5 kPa, φ=43°, ψ=14°, Rf=0.9
HCD modelE=75 MPa, ν=0.3, Ro=0, pc=2 kPa, φult=43°, φc=29°, αo=3, b=0.0025

For the HSM parameters, we have adopted the choices commonly adopted in engineering practice: Poisson's ratio equal to 0.1; reference stress equal to 100 kPa; Eur=3E50, Rf=0.9. It can be kept in mind for the following discussion that, with the proposed parameters, the simulation slightly overestimates the stiffness for the test with 100 kPa confining pressure; but, in a general way, both models reproduce the test results reasonably well, with the volumetric strain being relatively less well approximated.
Results
Results obtained with HSM

Table 6 shows the values of the maximum horizontal displacement of the sheet pile wall for the last three stages of the simulation: excavation down to 4 m, then 5 m, and application of a 10 kPa surcharge behind the wall, and of the settlement of the ground at 1 m behind the wall for the last two stages.

Results obtained with HSM (displacements in mm).

MeasuresHSM (ref.)HSM (adj.)
Max horizontal displacement – 4 m1.831.34 (−27%)0.91 (−50%)
Max horizontal displacement – 5 m2.954.49 (+52%)2.56 (−13%)
Max horizontal displacement –surcharge3.365.80 (+73%)3.28 (−2%)
Settlement 1 m behind the wall – 5 m2.171.21 (−44%)0.80 (−63%)
Settlement 1 m behind the wall – surcharge2.804.18 (+49%)2.26 (−19%)

With the values of the reference parameters resulting from the calibration, the horizontal displacement of the wall is significantly overestimated for the last two stages, and the settlement behind the wall is also overestimated (Table 6 shows the values of the computed displacements in mm and the relative error with the measured values), in spite of the fact that the calibration of the parameters tended to overestimate the sand initial stiffness, as mentioned in Section 3.7.

With the initial estimate of the maximum horizontal displacement and of the maximum settlement in the final state being much too large, increasing the sand stiffness is a straightforward way to improve the results, without modification of other parameters. No systematic calibration procedure has been used for the model parameters, such as the ones proposed by Zhang et al. (2015b) or Moussaoui et al. (2022). The difference between simulations and measurements is simply reduced by increasing the values of the elastic moduli. The values Eurref=135 MPa and E50ref=45 MPa are selected (instead of 63 MPa and 21 MPa, respectively; with no modification of the ratio Eurref/E50ref). Table 6 shows that the adjusted moduli improve the estimated maximal horizontal displacement of the sheet pile wall at the final stage (relative error of 4%) and the settlement 1 m behind the wall at the same stage (relative error of 17%), but, of course, the simulation of the triaxial tests gives results that are less close to the laboratory data.
Results obtained with the HCD model

Table 7 shows the same comparisons for the HCD model. Again, we present the results with the parameters obtained by the calibration presented in Section 3.7. The results are clearly closer to the measures than those obtained with HSM for the reference parameters.

Results obtained with the HCD model (displacements in mm).

MeasuresHCD (ref.)HCD (adj.)
Max. horizontal displacement – 4 m1.830.86 (−53%)1.37 (−25%)
Max. horizontal displacement – 5 m2.952.53 (−14%)2.99 (+1%)
Max. horizontal displacement – surcharge3.362.86 (−14%)3.51 (+4%)
Settlement 1 m behind the wall – 5 m2.171.81 (−17%)1.81 (−17%)
Settlement 1 m behind the wall – surcharge2.802.65 (−5%)3.04 (+8%)

We have also adjusted the model parameters to reduce the difference with the measured values. The proposed adjustment consists in letting Young's modulus vary with the depth z according to (z)=27+5z, where E is in MPa and z is in m. With this adjustment, the model reproduces reasonably well the experimental results.
Comparisons HSM/HCD

This section presents a more direct comparison of the results obtained with the two models and the experimental values. Results shown below correspond to the adjusted parameters of both models.

Figure 12 compares the horizontal displacements of the wall: the simulations with the HCD model give values of the maximum displacement and of the toe displacement closer to the measures than the simulations with HSM. On the other hand, the horizontal displacement in the upper part of the wall is better reproduced using HSM.

Figure 12:

Comparison of the horizontal displacements (last three stages).

Figure 13 compares the vertical displacements of the ground surface behind the wall for distances between 1 and 5 m: both models give similar results and close to the measured values. It can be estimated that the HCD model gives values rather closer to the measurements for the excavation down to 5 m.

Figure 13:

Comparison of the settlements behind the wall (last two stages).

The results of the experiment also presented the bending moments in the sheet pile wall. The comparison between models and measures is not shown here for brevity, but confirms a strong consistency between models and experiment.

In the last place, we can compare the values of the forces in the struts for the different stages of the experiment (Figure 14). Both models give very similar results and tend to overestimate the normal forces in the struts for the last two stages.

Figure 14:

Comparison of the forces in the struts (for four stages: excavation down to 3, 4, and 5 m, and surcharge).
Discussion

Several factors may be responsible for the differences observed in the horizontal displacement of the upper part of the wall, which are as follows:

The sheet piles were vibrodriven; for that type of installation, one can question the assumption that the stress state in the sand remains geostatic.

Regarding the calibration on the triaxial tests, the determination of the cohesion c for the HSM model and of pc for the HCD model is not very precise, which is an issue because a sensitivity analysis (not presented here) shows that their influence on the results is not negligible.

The parameters of the sand–sheet pile wall interface have not been discussed in detail, yet they can also have a strong influence on the wall behavior. To illustrate this point, we have rerun the simulations with φint = 20° instead of 5° (the value of 20° is found in the analyses of the same sheet pile wall presented by Mestat and Arafati (1998), Coquillay et al. (2005), and Elmi et al. (2006), focusing on the horizontal displacements and giving poor results in terms of vertical displacements behind the wall). For both the HSM and HCD models, the maximum horizontal displacement is reduced by approximately 10% for all stages. The forces in the struts are reduced, very similar for both models, and slightly underestimated at the initial stages and close to the measurements at the surcharge step. Vertical displacements behind the wall are presented in Figure 15. The agreement with the measured settlements is less satisfactory: notably, for the excavation down to 5 m, the maximum settlement is 0.5 mm for the HSM model and 1 mm for the HCD model, while the measures give 2.2 mm. This led us to retain the value of 5° in Section 3.

 Nevertheless, from a qualitative point of view, the HCD model gives good results for the simulation of the Hochstetten pile wall and, for the special case discussed here, gives better results with the parameters resulting directly from the calibration on triaxial tests as HSM. HSM performs very well too, but it was necessary to adjust the parameters derived from the calibration on the triaxial tests. It must also be kept in mind that the absolute displacements measured on the sheet pile wall were small, so a small deviation between the computed and measured values results in apparently large relative errors.

Figure 15:

Comparison of the settlements behind the wall (last two stages) with a friction angle of 20° for the sand–wall interface.

In the last place, it can be mentioned that the HCD model has also been used to perform simulations for a 30-m deep excavation in Berlin sand, with horizontal displacements in the order of 30 mm. This case is well documented (Schweiger, 2002b) and often used to validate or calibrate numerical models; the results obtained with the HCD model have been presented in detail in El Arja (2020) and summarized in El Arja et al. (2019). The conclusions regarding the proximity between the HSM and HCD models are similar, but the comparison is less complete, since vertical displacements behind the wall had not been measured. Also, it can be noted that no attempt has been made yet to model deep excavation in clays with the HCD model or to analyze the effect of groundwater drawdown (Zhang et al., 2018a, 2018b).
Conclusion

This paper presents a simple elastoplastic constitutive model developed to improve the prediction of the vertical displacements behind retaining walls.

One of the advantages of the HCD model is its simplicity: it involves only one plastic mechanism, and the elastic part of the model is linear and elastic. The yield surface is relatively classic and the hardening is isotropic. The originality of the model lies in the formulation of the flow rule that allows accounting for a contractive–dilatant behavior in the plastic regime.

Among the eight parameters, four are familiar (Young's modulus, Poisson's ratio, tensile strength, and friction angle); one defines the initial position of the yield surface and the last three are clearly associated with the characteristic state, the gradual decrease in plastic volumetric strain, and the rate at which the yield surface evolves toward the failure surface. Identification of the parameters based on triaxial tests is explained, and an example of calibration is presented.

For the numerical analysis of the experimental sheet pile wall of Hochstetten, the HCD model gives results that are rather closer to the measured values than HSM. It can be argued that the nonassociated flow rule of the HCD model allows to better control the volumetric plastic strains; at least, the influence of the model parameters is easy to anticipate, which facilitates the recalibration of a numerical simulation on experimental results.

Figure 1:

Yield surface in the principal stress space.
Yield surface in the principal stress space.

Figure 2:

Section of the yield surface in the deviatoric plane.
Section of the yield surface in the deviatoric plane.

Figure 3:

Calibration of the HCD model on tests B1, B2, and B3.
Calibration of the HCD model on tests B1, B2, and B3.

Figure 4:

Calibration of the HCD model on tests C2 and C3.
Calibration of the HCD model on tests C2 and C3.

Figure 5:

Calibration of the HCD model on tests E1, E2, and E3.
Calibration of the HCD model on tests E1, E2, and E3.

Figure 6:

Calibration of the HCD model on the monotonic drained triaxial test on alluvial material.
Calibration of the HCD model on the monotonic drained triaxial test on alluvial material.

Figure 7:

Calibration of the HCD model on the drained triaxial test with unloading cycles on alluvial material.
Calibration of the HCD model on the drained triaxial test with unloading cycles on alluvial material.

Figure 8:

Experimental setup (a) front view, (b) top view.
Experimental setup (a) front view, (b) top view.

Figure 9:

Mesh used for the simulations.
Mesh used for the simulations.

Figure 10:

Results of the calibration of the Hardening Soil Model parameters.
Results of the calibration of the Hardening Soil Model parameters.

Figure 11:

Results of the calibration of the HCD model parameters.
Results of the calibration of the HCD model parameters.

Figure 12:

Comparison of the horizontal displacements (last three stages).
Comparison of the horizontal displacements (last three stages).

Figure 13:

Comparison of the settlements behind the wall (last two stages).
Comparison of the settlements behind the wall (last two stages).

Figure 14:

Comparison of the forces in the struts (for four stages: excavation down to 3, 4, and 5 m, and surcharge).
Comparison of the forces in the struts (for four stages: excavation down to 3, 4, and 5 m, and surcharge).

Figure 15:

Comparison of the settlements behind the wall (last two stages) with a friction angle of 20° for the sand–wall interface.
Comparison of the settlements behind the wall (last two stages) with a friction angle of 20° for the sand–wall interface.

Reference parameters (calibrated on the triaxial tests) for the Hochstetten sand.

HSM Eurref=63 MPa, E50ref=21 MPa, ν=0.1, m=0.5, pref=100 kPa, c=5 kPa, φ=43°, ψ=14°, Rf=0.9
HCD model E=75 MPa, ν=0.3, Ro=0, pc=2 kPa, φult=43°, φc=29°, αo=3, b=0.0025

HCD model parameters.

E v φo pc φult αo φc b
MPa ° kPa deg deg
Monotonic test (confining pressure 175 kPa) 57 0.4 φmin 0 37 8 27 0.001
Cyclic test (confining pressure 157 kPa) 57 0.4 φmin 0 37 20 31 0.003

Parameters of the reaction wall and the sheet pile wall.

Wall E(MPa) I(cm4/m) A(cm2/m) Eeq (MPa) teq (cm)
Reaction wall 210,000 968 106 24,400 10
Sheet pile wall 210,000 11,610 116 6800 35

Results obtained with HSM (displacements in mm).

Measures HSM (ref.) HSM (adj.)
Max horizontal displacement – 4 m 1.83 1.34 (−27%) 0.91 (−50%)
Max horizontal displacement – 5 m 2.95 4.49 (+52%) 2.56 (−13%)
Max horizontal displacement –surcharge 3.36 5.80 (+73%) 3.28 (−2%)
Settlement 1 m behind the wall – 5 m 2.17 1.21 (−44%) 0.80 (−63%)
Settlement 1 m behind the wall – surcharge 2.80 4.18 (+49%) 2.26 (−19%)

Summary of the triaxial tests considered in this study.

Test Depth Type of soil Type of test
B 6.5–7 Very fine silty sand Consolidated drained CD
C 8.5–9.5 Slightly sandy pebbles and gravel Consolidated drained CD
E 13.7–14 Silty clay Consolidated drained CD

Parameters of the HCD model for tests B, C, and E (σc denotes the confining pressure).

σc E v φo pc φult αo φc b
kPa MPa - ° kPa ° - ° -
Test B 1 150 45 0.2 7 10 35 1 31 0.005
2 250 70 0.2 7 10 35 1 32 0.008
Test C 3 400 107 0.2 7 10 35 1 33 0.009
2 200 133 0.1 15 15 41 1 38 0.002
3 350 233 0.127 15 15 41 1 39 0.005
Test E 1 200 16 0.12 1.2 60 23 1 20 0.004
2 350 45 0.12 2.2 60 23 1 20 0.005
3 550 112 0.14 4.2 60 23 1 21 0.015

Results obtained with the HCD model (displacements in mm).

Measures HCD (ref.) HCD (adj.)
Max. horizontal displacement – 4 m 1.83 0.86 (−53%) 1.37 (−25%)
Max. horizontal displacement – 5 m 2.95 2.53 (−14%) 2.99 (+1%)
Max. horizontal displacement – surcharge 3.36 2.86 (−14%) 3.51 (+4%)
Settlement 1 m behind the wall – 5 m 2.17 1.81 (−17%) 1.81 (−17%)
Settlement 1 m behind the wall – surcharge 2.80 2.65 (−5%) 3.04 (+8%)

Benz, T. (2007). Small strain stiffness of soils and its numerical consequences, PhD Thesis, Universität Stuttgart. BenzT. 2007 Small strain stiffness of soils and its numerical consequences PhD Thesis, Universität Stuttgart Search in Google Scholar

Chehade, W. (1991). Méthodologie pour la validation des modèles des géomatériaux-Application aux modèles élastoplastiques, PhD Thesis, Univ Sciences et Technologies de Lille (in French). ChehadeW. 1991 Méthodologie pour la validation des modèles des géomatériaux-Application aux modèles élastoplastiques PhD Thesis, Univ Sciences et Technologies de Lille (in French) Search in Google Scholar

Coquillay, S. (2005) Prise en compte de la non linéarité du comportement des sols soumis à de petites déformations pour le calcul des ouvrages géotechniques, PhD Thesis, Ecole Nationale des Ponts et Chaussées (in French). CoquillayS. 2005 Prise en compte de la non linéarité du comportement des sols soumis à de petites déformations pour le calcul des ouvrages géotechniques PhD Thesis, Ecole Nationale des Ponts et Chaussées (in French) Search in Google Scholar

Coquillay, S., Bourgeois, E., and Mestat, Ph. (2005) A non-linear elastic-perfectly plastic model for the simulation of the Hochstetten sheet pile wall. 11th International Conference of IACMAG, Turin, Italy, June 19–24, 2005. CoquillayS. BourgeoisE. MestatPh. 2005 A non-linear elastic-perfectly plastic model for the simulation of the Hochstetten sheet pile wall 11th International Conference of IACMAG Turin, Italy June 19–24, 2005 Search in Google Scholar

Delattre, L. (2004). A century of retaining wall computation methods III: Modeling of retaining walls by means of the finite element method, Bull Lab Ponts et Chaussées, n°252–253, 95–117, www.ifsttar.fr/collections/BLPCpdfs/blpc_252-253_95-117_en.pdf. DelattreL. 2004 A century of retaining wall computation methods III: Modeling of retaining walls by means of the finite element method Bull Lab Ponts et Chaussées 252–253 95 117 www.ifsttar.fr/collections/BLPCpdfs/blpc_252-253_95-117_en.pdf. Search in Google Scholar

Duncan, J.M., and Chang, C.-Y. (1970). Nonlinear Analysis of Stress and Strain in Soils. Journal of the Soil Mechanics and Foundations Division 96, 1629–1653. DuncanJ.M. ChangC.-Y. 1970 Nonlinear Analysis of Stress and Strain in Soils Journal of the Soil Mechanics and Foundations Division 96 1629 1653 Search in Google Scholar

El Arja, H. (2020). Contribution à la modélisation numérique des excavations profondes, PhD Thesis, Université Paris-Est. El ArjaH. 2020 Contribution à la modélisation numérique des excavations profondes PhD Thesis, Université Paris-Est Search in Google Scholar

El Arja, H., Bourgeois, E., and Burlon, S. (2019). Prise en compte du mécanisme des déformations plastiques dans les calculs des excavations profondes. Proc XVII ECSMGE-2019, Reykjavik. El ArjaH. BourgeoisE. BurlonS. 2019 Prise en compte du mécanisme des déformations plastiques dans les calculs des excavations profondes Proc XVII ECSMGE-2019 Reykjavik Search in Google Scholar

Elmi, F., Bourgeois, E., Pouya, A., and Rospars, C. (2006) Elastoplastic Joint Element for the Finite Element Analysis of the Hochstetten Sheet Pile Wall, in Numerical methods in geotechnical engineering NUMGE 2006, Schweiger (ed), Graz, Austria, 411–416 ElmiF. BourgeoisE. PouyaA. RosparsC. 2006 Elastoplastic Joint Element for the Finite Element Analysis of the Hochstetten Sheet Pile Wall, in Numerical methods in geotechnical engineering NUMGE 2006 Schweiger (ed), Graz, Austria 411 416 Search in Google Scholar

Fern, E.J., and Soga, K. (2018). The dilatancy conditions at critical state and its implications on constitutive modelling, Numerical Methods in Geotechnical Engineering NUMGE 2018, Porto, 8 p. FernE.J. SogaK. 2018 The dilatancy conditions at critical state and its implications on constitutive modelling Numerical Methods in Geotechnical Engineering NUMGE 2018 Porto 8 p. Search in Google Scholar

Finno, R.J. (2008). General Report: Analysis and Numerical Modeling of Deep Excavations, Proc 6th Int Symp on Geotechnical Aspects of Underground Construction in Soft Ground, Shanghai, China, Ng, Nuang and Liu, eds., Taylor & Francis, 87–97. FinnoR.J. 2008 General Report: Analysis and Numerical Modeling of Deep Excavations Proc 6th Int Symp on Geotechnical Aspects of Underground Construction in Soft Ground Shanghai, China Ng Nuang Liu eds., Taylor & Francis 87 97 Search in Google Scholar

Finno, R.J., and Harahap, I.S. (1991). Finite Element Analyses of HDR-4 Excavation. Journal of Geotechnical Engineering 117, 1590–1609. FinnoR.J. HarahapI.S. 1991 Finite Element Analyses of HDR-4 Excavation Journal of Geotechnical Engineering 117 1590 1609 Search in Google Scholar

Goh, A.T.C., Zhang, F., Zhang, W., Zhang, Y., and Liu, H. (2017a) A simple estimation model for 3D braced excavation wall deflection, Computers and Geotechnics 83, 106–113. http://dx.doi.org/10.1016/j.compgeo.2016.10.022 GohA.T.C. ZhangF. ZhangW. ZhangY. LiuH. 2017a A simple estimation model for 3D braced excavation wall deflection Computers and Geotechnics 83 106 113 http://dx.doi.org/10.1016/j.compgeo.2016.10.022 Search in Google Scholar

Goh, A.T.C., Zhang, F., Zhang, W., and Chew, O.Y.S. (2017b) Assessment of strut forces for braced excavation in clays from numerical analysis and field measurements, Computers and Geotechnics 86, 141–149. http://dx.doi.org/10.1016/j.compgeo.2017.01.012 GohA.T.C. ZhangF. ZhangW. ChewO.Y.S. 2017b Assessment of strut forces for braced excavation in clays from numerical analysis and field measurements Computers and Geotechnics 86 141 149 http://dx.doi.org/10.1016/j.compgeo.2017.01.012 Search in Google Scholar

Goodman, R. E., Taylor, R. L., and Brekke, T. L. (1968). A model for the mechanics of jointed rock. Journal of the Soil Mechanics and Foundations Division ASCE, Vol.94, No. SM3, 637–659. GoodmanR. E. TaylorR. L. BrekkeT. L. 1968 A model for the mechanics of jointed rock Journal of the Soil Mechanics and Foundations Division ASCE 94 SM3 637 659 Search in Google Scholar

Itech (2021) General overview - Geotechnical analysis - v2021 (https://www.cesar-lcpc.com/documents/CESAR-FC-2021-EN.pdf) Itech 2021 General overview - Geotechnical analysis - v2021 (https://www.cesar-lcpc.com/documents/CESAR-FC-2021-EN.pdf) Search in Google Scholar

Jardine, R.J., Potts, D.M., Fourie, A.B., and Burland, J.B. (1986). Studies of the influence of non-linear stress–strain characteristics in soil–structure interaction. Géotechnique 36(3), 377–396 (https://doi.org/10.1680/geot.1986.36.3.377). JardineR.J. PottsD.M. FourieA.B. BurlandJ.B. 1986 Studies of the influence of non-linear stress–strain characteristics in soil–structure interaction Géotechnique 36 3 377 396 (https://doi.org/10.1680/geot.1986.36.3.377). Search in Google Scholar

Khoshnoudian, F. (2002). Numerical Analysis of the Seismic Behavior of Tunnels Constructed in Liquefiable Soils, Soils and Foundations, 42(6), 1–8. KhoshnoudianF. 2002 Numerical Analysis of the Seismic Behavior of Tunnels Constructed in Liquefiable Soils Soils and Foundations 42 6 1 8 Search in Google Scholar

Khoshravan Azar, A. (1995). Problèmes de sols saturés sous chargement dynamique: modèle cyclique pour les sols et validation sur des essais en centrifugeuse, PhD Thesis, Univ Sciences et Technologies de Lille (in French). Khoshravan AzarA. 1995 Problèmes de sols saturés sous chargement dynamique: modèle cyclique pour les sols et validation sur des essais en centrifugeuse PhD Thesis, Univ Sciences et Technologies de Lille (in French) Search in Google Scholar

Mestat, Ph., and Arafati, N. (1998) Finite element modelling of the performance of the experimental sheet pile at Hochstetten, Bulletin de Liaison des Laboratoires des Ponts et Chaussées, 216, 19–39. MestatPh. ArafatiN. 1998 Finite element modelling of the performance of the experimental sheet pile at Hochstetten Bulletin de Liaison des Laboratoires des Ponts et Chaussées 216 19 39 Search in Google Scholar

Moussaoui, M., Rehab Bekkouche, S., Kamouche, H., Benayoun, F., and Goudjil, K. (2022) Identification of Soil Mechanical Parameters by Inverse Analysis Using Stochastic Methods, SSP-Journal of Civil Engineering, 17(1), DOI: 10.2478/sspjce-2022-0018 MoussaouiM. Rehab BekkoucheS. KamoucheH. BenayounF. GoudjilK. 2022 Identification of Soil Mechanical Parameters by Inverse Analysis Using Stochastic Methods SSP-Journal of Civil Engineering 17 1 10.2478/sspjce-2022-0018 Abierto DOISearch in Google Scholar

Obrzud, R.F. (2010). On the use of the Hardening Soil Small Strain model in geotechnical practice, Numerics in Geotechnics and Structures, 15–42. ObrzudR.F. 2010 On the use of the Hardening Soil Small Strain model in geotechnical practice Numerics in Geotechnics and Structures 15 42 Search in Google Scholar

Obrzud, R.F., and Truty, A. (2018). The Hardening soil model - A practical guidebook, Z Soil.PC 100701 report. ObrzudR.F. TrutyA. 2018 The Hardening soil model - A practical guidebook Z Soil.PC 100701 report. Search in Google Scholar

Plaxis (2020). Material Models Manual. Plaxis 2020 Material Models Manual Search in Google Scholar

Roscoe, K.H., Schofield, A.N., and Wroth, C.P. (1958). On the Yielding of Soils, Géotechnique 8(1), 22–53 (https://doi.org/10.1680/geot.1958.8.1.22). RoscoeK.H. SchofieldA.N. WrothC.P. 1958 On the Yielding of Soils Géotechnique 8 1 22 53 (https://doi.org/10.1680/geot.1958.8.1.22). Search in Google Scholar

Rowe, P.W. (1962) The stress-dilatancy relation for static equilibrium of an assembly of particles in contact. Proc. Royal Society of London, Mathematical and Physical Sciences, Vol. 269, Series A, 500–557. RoweP.W. 1962 The stress-dilatancy relation for static equilibrium of an assembly of particles in contact Proc. Royal Society of London, Mathematical and Physical Sciences 269 Series A, 500 557 Search in Google Scholar

Schanz, T., and Vermeer, P. (1996) Angles of friction and dilatancy of sand, Géotechnique 46 (1), 145–151. SchanzT. VermeerP. 1996 Angles of friction and dilatancy of sand Géotechnique 46 1 145 151 Search in Google Scholar

Schanz, T., Vermeer, P.A., and Bonnier, P.G. (1999). The hardening soil model: Formulation and verification, Beyond 2000 in Computational Geotechnics, R.B.J. Brinkgreve, ed., 281–296. SchanzT. VermeerP.A. BonnierP.G. 1999 The hardening soil model: Formulation and verification Beyond 2000 in Computational Geotechnics BrinkgreveR.B.J. ed. 281 296 Search in Google Scholar

Schofield, A., and Worth, P. (1968). Critical State Soil Mechanics, Mc Graw-Hill. SchofieldA. WorthP. 1968 Critical State Soil Mechanics Mc Graw-Hill Search in Google Scholar

Schweiger, H.F. (2002a). Benchmarking in Geotechnics_Part I: Results of benchmarking. Technical report CGG IR006 2002, Institute for Soil Mechanics and Foundation Engineering, Graz University of Technology. SchweigerH.F. 2002a Benchmarking in Geotechnics_Part I: Results of benchmarking Technical report CGG IR006 2002, Institute for Soil Mechanics and Foundation Engineering, Graz University of Technology Search in Google Scholar

Schweiger, H.F. (2002b). Results from numerical benchmark exercises in geotechnics, 5th Conf on Num Meth in Geotechnical Engineering NUMGE 2002, Mestat (ed), Paris, volume 1, 305–314. SchweigerH.F. 2002b Results from numerical benchmark exercises in geotechnics 5th Conf on Num Meth in Geotechnical Engineering NUMGE 2002 Mestat (ed), Paris 1 305 314 Search in Google Scholar

Shahrour, I., and Ousta, R. (1998). Numerical analysis of the behavior of piles in saturated soils under seismic loading, 11th European Conf. on Earthquake Engineering (Rotterdam). ShahrourI. OustaR. 1998 Numerical analysis of the behavior of piles in saturated soils under seismic loading 11th European Conf. on Earthquake Engineering (Rotterdam) Search in Google Scholar

Shahrour, I., Benzenati, I., and Khoshravan Azar, A. (1995a). Validation of a nonlinear coupled dynamic model on centrifuge tests of VELACS project, Numerical Models in Geomechanics NUMOG V, (Rotterdam), pp. 269–274. ShahrourI. BenzenatiI. Khoshravan AzarA. 1995a Validation of a nonlinear coupled dynamic model on centrifuge tests of VELACS project Numerical Models in Geomechanics NUMOG V Rotterdam 269 274 Search in Google Scholar

Shahrour I., Ghorbanbeigi S., and Von Wolffersdorff P.A. (1995b) Comportement des rideaux de palplanche : expérimentation en vraie grandeur et prédictions numériques. Revue Française de Géotechnique, 71, 39–47 (in French). Shahrour I.Ghorbanbeigi S. Von WolffersdorffP.A. 1995b Comportement des rideaux de palplanche : expérimentation en vraie grandeur et prédictions numériques Revue Française de Géotechnique 71 39 47 (in French). Search in Google Scholar

Shao, C., and Desai, C.S. (2000). Implementation of DSC model and application for analysis of field pile tests under cyclic loading, Int J for Numerical and Analytical Methods in Geomechanics, 24(6), 601–624. ShaoC. DesaiC.S. 2000 Implementation of DSC model and application for analysis of field pile tests under cyclic loading Int J for Numerical and Analytical Methods in Geomechanics 24 6 601 624 Search in Google Scholar

Simpson, B. (1992). Retaining structures : displacement and design, Géotechnique 42(4), 541–576 (https://doi.org/10.1680/geot.1992.42.4.541). SimpsonB. 1992 Retaining structures : displacement and design Géotechnique 42 4 541 576 (https://doi.org/10.1680/geot.1992.42.4.541). Search in Google Scholar

Tatsuoka, F.M., Sakamoto, M., Kawamura, T., and Fakushiima, S. (1986). Strength and deformation characteristics of sand in plane strain compression at extremley low pressures, Soils and Foundations, 26(1), 65–84. TatsuokaF.M. SakamotoM. KawamuraT. FakushiimaS. 1986 Strength and deformation characteristics of sand in plane strain compression at extremley low pressures Soils and Foundations 26 1 65 84 Search in Google Scholar

Von Wolffersdorff P.-A. (1994a) Spundwand-Feldversuch in Hochstetten / Sheetpile wall field test in Hochstetten. Part A: Presentation of the problem. Part B: Test results and predictions. Workshop Sheet Pile Test Karlsruhe, Delft University, October 6 and 7, 1994 Von WolffersdorffP.-A. 1994a Spundwand-Feldversuch in Hochstetten / Sheetpile wall field test in Hochstetten Part A: Presentation of the problem. Part B: Test results and predictions. Workshop Sheet Pile Test Karlsruhe Delft University October 6 and 7, 1994 Search in Google Scholar

Von Wolffersdorff P.-A. (1994b) « Results of field test and evaluation of the predictions and subsequent calculations », Workshop Sheet Pile Test Karlsruhe, Delft University. Von WolffersdorffP.-A. 1994b « Results of field test and evaluation of the predictions and subsequent calculations » Workshop Sheet Pile Test Karlsruhe, Delft University Search in Google Scholar

Zaher, M. (1995). Validation des modèles de sols sur ouvrages types. PhD Thesis, Université des sciences et technologies de Lille. ZaherM. 1995 Validation des modèles de sols sur ouvrages types PhD Thesis, Université des sciences et technologies de Lille Search in Google Scholar

Zdravkovic, L., Potts, D.M., and St John, H.D. (2005). Modelling of 3D excavation in finite element analysis. Géotechnique 55(7), 497–513 (https://doi.org/10.1680/geot.2005.55.7.497). ZdravkovicL. PottsD.M. St JohnH.D. 2005 Modelling of 3D excavation in finite element analysis Géotechnique 55 7 497 513 (https://doi.org/10.1680/geot.2005.55.7.497). Search in Google Scholar

Zhang, W., Goh, A.T.C., and Xuan, F. (2015a) A simple prediction model for wall deflection caused by braced excavation in clays, Computers and Geotechnics 63, 67–72. http://dx.doi.org/10.1016/j.compgeo.2014.09.001 ZhangW. GohA.T.C. XuanF. 2015a A simple prediction model for wall deflection caused by braced excavation in clays Computers and Geotechnics 63 67 72 http://dx.doi.org/10.1016/j.compgeo.2014.09.001 Search in Google Scholar

Zhang, W., Goh, A.T.C., and Zhang, Y. (2015b) Updating soil parameters using spreadsheet method for predicting wall deflections in braced excavations. Geotechnical and Geological Engineering 33, 1489–1498, DOI 10.1007/s10706-015-9914-4 ZhangW. GohA.T.C. ZhangY. 2015b Updating soil parameters using spreadsheet method for predicting wall deflections in braced excavations Geotechnical and Geological Engineering 33 1489 1498 10.1007/s10706-015-9914-4 Abierto DOISearch in Google Scholar

Zhang, W.G., Goh, A.T.C., Chew, O.Y.S., Zhou, D., and Zhang, R. (2018a) Performance of braced excavation in residual soil with groundwater drawdown, Underground Space 3, 150–165. https://doi.org/10.1016/j.undsp.2018.03.002 ZhangW.G. GohA.T.C. ChewO.Y.S. ZhouD. ZhangR. 2018a Performance of braced excavation in residual soil with groundwater drawdown Underground Space 3 150 165 https://doi.org/10.1016/j.undsp.2018.03.002 Search in Google Scholar

Zhang, W., Wang, W., Zhou, D., Zhang, R., Goh, A.T.C., and Hou, Z. (2018b) Influence of groundwater drawdown on excavation responses - A case history in Bukit Timah granitic residual soils. Journal of Rock Mechanics and Geotechnical Engineering 10, 856–864. (https://doi.org/10.1016/j.undsp.2018.03.002) ZhangW. WangW. ZhouD. ZhangR. GohA.T.C. HouZ. 2018b Influence of groundwater drawdown on excavation responses - A case history in Bukit Timah granitic residual soils Journal of Rock Mechanics and Geotechnical Engineering 10 856 864 (https://doi.org/10.1016/j.undsp.2018.03.002) Search in Google Scholar

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