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

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

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

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.

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

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

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.).

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

The yield function is a function of the principal stresses and accounts for the influence of the Lode angle:
_{c}_{c}_{o}(θ)_{o}

As plastic strains develop, the hardening variable _{ult}

In equation (7), we use the plastic deviatoric strain

For large values of
_{ult}_{o}_{f}_{o}_{f}

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
_{v}_{d}_{c}(θ)_{c}

Three cases can be distinguished:

If _{c}_{c}

If _{c}_{c}

If _{c}_{c}

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):
_{o}

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

the initial friction angle _{o}_{c}_{ult}

the characteristic angle _{o}_{o}

Summary of the triaxial tests considered in this study.

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 |

In a first step, the apex parameter _{c}_{ult}

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 _{0} and q_{0}, parameter _{o}_{o}_{c}_{a}_{v}_{a}

Eventually, the value of parameter _{o}

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.

Parameters of the HCD model for tests B, C, and E (_{c}

_{c} | _{o} | _{c} | _{ult} | _{o} | _{c} | |||||
---|---|---|---|---|---|---|---|---|---|---|

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 |

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 _{o}_{o}_{c}

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

HCD model parameters.

_{o} | _{c} | _{ult} | _{o} | _{c} | ||||
---|---|---|---|---|---|---|---|---|

Monotonic test (confining pressure 175 kPa) | 57 | 0.4 | _{min} | 0 | 37 | 8 | 27 | 0.001 |

Cyclic test (confining pressure 157 kPa) | 57 | 0.4 | _{min} | 0 | 37 | 20 | 31 | 0.003 |

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.

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

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.

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)

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

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.

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.

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

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.

The initial stresses are geostatic:
_{o}_{o}^{3}.

The axial stiffness _{eq}_{eq}

Parameters of the reaction wall and the sheet pile wall.

^{4}/m) | ^{2}/m) | _{eq} | _{eq} | ||
---|---|---|---|---|---|

Reaction wall | 210,000 | 968 | 106 | 24,400 | 10 |

Sheet pile wall | 210,000 | 11,610 | 116 | 6800 | 35 |

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): ^{2}/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).

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 _{int}_{int}_{sand}_{sand}_{int}_{inter} tan _{sand}_{int}_{inter} tan _{sand}_{inter} 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:

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

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.

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

HSM | _{ur}^{ref}_{50}^{ref}=21 MPa, ^{ref}_{f} |

HCD model | _{o}_{c}_{ult}_{c}_{o} |

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; _{ur}_{50}, _{f}

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).

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%) |

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 _{ur}^{ref}_{50}^{ref}=45 MPa are selected (instead of 63 MPa and 21 MPa, respectively; with no modification of the ratio _{ur}^{ref}_{50}^{ref}). 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.

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).

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%) |

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

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 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.

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.

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}

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 _{int}

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

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.

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

HSM | _{ur}^{ref}_{50}^{ref}=21 MPa, ^{ref}_{f} |

HCD model | _{o}_{c}_{ult}_{c}_{o} |

#### HCD model parameters.

_{o} |
_{c} |
_{ult} |
_{o} |
_{c} |
||||
---|---|---|---|---|---|---|---|---|

Monotonic test (confining pressure 175 kPa) | 57 | 0.4 | _{min} |
0 | 37 | 8 | 27 | 0.001 |

Cyclic test (confining pressure 157 kPa) | 57 | 0.4 | _{min} |
0 | 37 | 20 | 31 | 0.003 |

#### Parameters of the reaction wall and the sheet pile wall.

^{4}/m) |
^{2}/m) |
_{eq} |
_{eq} |
||
---|---|---|---|---|---|

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).

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.

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).

_{c} |
_{o} |
_{c} |
_{ult} |
_{o} |
_{c} |
|||||
---|---|---|---|---|---|---|---|---|---|---|

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).

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%) |

^{th} Conf on Num Meth in Geotechnical Engineering NUMGE 2002, Mestat (ed), Paris, volume 1, 305–314.