3.1. Strategy of Analyses
A comparison study of different constitutive soil models (e.g., the MCC model, the MC model, and the HSS model) is first presented in
Section 3.2. Additionally, three groups of parametric studies, which discuss the methods of considering soil–wall contact, the impact of the value of
μ, and the influence of the value of
τmax, respectively, are conducted. The results and analyses of these parametric studies are presented in
Section 3.3. Finally, a series of parametric studies discussing the impact of different soil features based on the MCC model are carried out, and the results are illustrated in
Section 3.4. To be noted, for models without special notes in this work, the applied input parameters are the same as the basic analysis.
3.2. Comparison of Different Soil Models
The MC model (i.e., the Mohr–Coulomb model), the MCC model (i.e., the Modified Cam-Clay model), and the HSS model (i.e., the Hardening Soil–small model) are three commonly applied soil models in deep excavation analyses. However, straight comparisons of them are lacking from previous studies. Based on the hypothetical excavation project, the performances of the three models are compared in this section. To be noted, the comparison is made based on a hypothetical project, which means there is no field data to calibrate the results of these models. Therefore, the study in this section only aims to find the difference rather than provide an evaluation of different models. Despite the lack of field data, a previous study by Dong [
18] has calculated a model sharing the same geometry as the current research, the results of which can be used as references. The input parameters of the MC model are shown in
Table 4, and the input parameters of the HSS model are presented in
Table 5.
The calculated diaphragm wall deflections, ground surface settlements, and ground surface horizontal movements at the middle section are presented and compared in
Figure 2. In this figure, the detailed description of the legend is presented in
Table 6.
In general, all three models yield displacement patterns similar to Dong’s results [
18]. The calculated maximum wall deflections of the MCC model, MC model, and HSS model are 13.3 mm, 14.1 mm, and 11.8 mm, respectively. In general, the differences in the wall deflections calculated from different models are insignificant; all of them (i.e., the MCC model, MC model, and the HSS model) seem to be applicable if the wall deflection is the main concern of a deep excavation.
By further examination, the comparison suggests that the HSS model yields the smallest wall deflection. However, it is not safe to say the reason for the small wall deflections of the HSS model is its consideration of the small strain stiffness of soil. The reason is that the determination of the parameters of the HSS model relies very much on experience correlations and an inverse analysis, which makes its parameters more flexible and favorable. In contrast, methods for determining the MCC parameters are relatively unified, and they all can be obtained from conventional laboratory tests. This advantage makes the MCC model a more appropriate choice for design purposes.
In terms of the ground settlements behind the wall, it is apparent that the MC model underestimates the settlement significantly. This is because the MC model produces unrealistically large upward movements of deep soils [
20], which has offset the settlement at the ground surface. The MCC model and the HSS model yield similar maximum ground settlements, and both of them are close to Dong’s results [
18]. However, considering the larger maximum wall deflection of the MCC model and the close relationship between the maximum wall deflection and the maximum ground settlement, it is safe to say that the MCC model would yield smaller maximum ground surface settlements if the wall deflection were the same as the HSS model. Meanwhile, the ground settlements far from the wall calculated from the MCC model are more evident than the HSS model and Dong’s result. This tendency should arise from the MCC model’s inability to account for the small strain behavior of the soil. In fact, a similar tendency was also reported from previous studies; for example, Kung [
21] concluded that soil models without considering small strain stiffness tend to underestimate surface settlements near the walls while overestimating settlements far from the walls. The horizontal ground movements show the same tendency with the settlements: the MC model underestimates the deformation significantly, and the MCC model and the HSS model yield a similar displacement profile.
Because of the lack of field data, the current study can only provide a preliminary comparison between different soil models. Assuming that Dong’s results [
18] are reliable, it can be said that both the MCC model (also denoted as the BA model subsequently) and the HSS model yield acceptable displacement fields, while the MC model underestimates the ground movements significantly. Considering that it is more convenient to determine the input parameters of the MCC model, the subsequent parametric studies in this work will be carried out based on the BA model.
3.3. Soil–Wall Interface Properties
3.3.1. Methods to Consider Soil–Wall Interaction
Several methods for considering soil–wall interactions (e.g., tie constraints, embedded approach, and surface-to-surface contact with different tangential behaviors) have been applied in previous studies. However, most of these studies generally assumed the methods used were reliable without detailed descriptions. To improve the understanding of this issue, the first parametric study in this section investigates the difference between different methods in considering the soil–wall contact. The analysis strategy and the corresponding descriptions of different cases are described in
Table 7.
Notice that there are two “Embedded” scenarios (i.e., the “Embedded” and the “Embedded (modified)”) in
Table 7. This is because when applying the “Embedded region” approach, the soil at the location of the diaphragm wall cannot be removed. Therefore, both the weight and stiffness of the soil accumulate in the diaphragm wall’s properties during the subsequent simulation processes, which may impact the calculated results. To quantify this impact, the “Embedded (modified)” case, in which the diaphragm wall applies a modified unit weight (i.e., by reducing the unit weight of the soil from the nominal concrete value), is further included in this study.
The calculated results of different models described in
Table 7 are presented in
Figure 3. According to the figure, the excavation-induced displacement field is sensitive to different modeling methods of soil–wall contact. Typically, the “Frictionless” model yields particularly large displacements from all aspects, indicating the significant impact of the friction at the soil–wall contact. On the other hand, all the models except the “Embedded” case yield the same displacement patterns, while the magnitudes vary with different modeling methods. The discrepancy of the “Embedded” model arises from the great accumulated weight of the diaphragm wall, which causes dramatic settlements. When the weight is modified (i.e., “Embedded (modified)”), the calculated ground displacement profiles become nearly identical with the “Rough” and “Tie” models. At the same time, despite the unrealistic ground movements of the “Embedded” case, the lateral wall deflection is similar to the “Rough” and “Tie” models (only with a difference of around 1 mm). This tendency suggests that the weight of the diaphragm wall has a significant impact on the surrounding soils while influencing the lateral deformations of the retaining structures negligibly.
The similar vertical ground movements near the wall (i.e., where the x coordinate is 0) between the “BC” model and the “Tie”, “Rough”, and “Embedded (modified)” models suggest that the relative sliding between the wall and the soil outside the pit of the “BC” model is minimal. However, the lateral wall deflection of the “BC” scenario is larger than the “Tie”, “Rough”, and “Embedded (modified)” cases significantly. This discrepancy might be because of different normal behaviors of the contact. In the “Tie”, “Rough”, and “Embedded (modified)” scenarios, the separation of the surfaces once they are closed is prohibited, while the “Hard contact” applied to represent the normal behaviors in the “BA” and “BC” models allows separations after contact. However, the contact problems are so complicated that more possibilities explaining the differences are likely. Overall, even if the relative sliding between the soil and the wall does not occur, the approaches of Tie constraints, Embedded region, and Rough tangential property should be applied with caution because they generally yield conservative lateral wall deflections.
3.3.2. Influence of the Shear Stress Limit, τmax
The difference between the “BC” and “BA” models, although very limited, is observed in
Figure 3, illustrating the influence of the shear stress limit,
τmax, set in the “BA” model. In order to make clear this influence, a further parametric study concerning the soil–wall contact was conducted. According to a previous study [
14], the value range of ultimate side friction of cast-in-place piles is 15 kPa~100 kPa. In the current parametric study, a domain of 5 kPa~100 kPa for the values of
τmax is considered. To be noted, the coefficient of friction,
μ, is kept to be 0.3 for all the models in this study. The results of this study, as well as the results of the “Frictionless” and “Rough” models described in
Table 7, are compared in
Figure 4.
As expected, the results from models with different values of
τmax are bounded by the “Frictionless” model and the “Rough” model (in which no shear stress limit is defined), and the displacements (in all aspects) generally increase with a decreasing
τmax value. At the same time,
Figure 4 shows that the “
τmax = 50 kPa”, “
τmax = 100 kPa”, and “BC” (i.e.,
τmax = ∞) models produce identical results, indicating that the vertical shear stress between the diaphragm wall and the soil in this study is smaller than 50 kPa. Furthermore, the difference between the “
τmax = 15 kPa” and “
τmax = 50 kPa” (i.e., the possible range for Shanghai soft soils) models is limited (e.g., about 2 mm of the lateral wall deflections, around 3 mm of horizontal ground movements, and approximately 3 mm of ground settlements), which indicates that the excavation-induced displacements are insensitive to
τmax values.
3.3.3. Influence of the Coefficient of Friction, μ
No specific scope of the
μ values of soil–wall contact has been reported previously; practitioners in Shanghai generally apply a scope of 0.25~0.75 by referring to the friction coefficient between the cushion cap bottom and foundation soils regulated by the Technical Code for Building Pile Foundations [
22]. This parametric study, fixing the shear stress limit at
τmax = 25 kPa, considers a
μ scope of 0.005~1.0. In addition, the results from the “Frictionless” and “Rough” cases are also included for comparison. The calculated displacements are presented in
Figure 5.
In general, the displacements decrease with the increasing μ values, while all the results are bounded by the “Frictionless” and “Rough” models. The identical results between the “μ = 0.6” and “μ = 1.0” models demonstrate that when μ is larger than 0.6, the interface property will be dominated by the shear stress limit. Considering the close results between the “μ = 0.3” and “μ = 0.6” models, the maximum value of μ should be somewhere slightly larger than 0.3. Similar to the shear stress limit, the difference between different models is insignificant, which suggests that calculated displacements are not sensitive to the coefficient ratio of friction, either.
3.3.4. General Evaluation
In general, the parametric studies in this section suggest that the soil–wall interface properties can impact the excavation-induced displacement fields. Typically, the modeling methods to consider the contact affect the calculated results dramatically. Even under situations without relative sliding between the wall and the soils, Tie constraints, Embedded region, and Rough tangential properties are not recommended because they generally produce over-conservative lateral wall deflections. When the soil–wall contact is simulated by surface-to-surface contact with a tangential behavior represented by the Coulomb friction model with a shear stress limit, both the coefficient ratio of friction and the shear stress limit, if not values that are apparently unreliable, influence the displacements negligibly. Therefore, the μ and τmax values can generally be determined according to experiences.
3.4. Soil Parameters
Studies in this work generally involve the MCC model and the HSS model. Numerous studies concerning the input parameters of the HSS model have been reported [
23,
24,
25]. In contrast, the influence of the input parameters of the MCC model on the excavation-induced displacement has never been investigated. In order to fill this gap in knowledge, as well as to improve the understanding of the application of the MCC model in deep excavation problems, this section presents a series of parametric studies based on the MCC model.
The discussed items include Poisson’s ratio, ν, a void ratio, e0, the coefficient of lateral earth pressure at rest, K0, a frictional constant, M, an isotropic logarithmic compression index, λ, and the swelling index, κ. To be noted, there will be only one variable, which is the corresponding discussed parameter, in each parametric study, while the possible correlation between the discussed parameter and other unchanged ones (e.g., λ and κ; ν, M, and K0) is not considered. The parameter bound of each group of the input parameters can generally cover most of the possible values of Shanghai soft soils, while specific values within the bounds applied in the parametric studies are only to describe an increasing or decreasing order but do not necessarily correspond to particular soils.
3.4.1. Influence of the Poisson’s Ratio, ν
Figure 6 presents the calculated results of models with different Poisson’s ratios,
ν, which range from 0.2 to 0.4. According to the figure, the changing Poisson’s ratio impacts the excavation performance significantly. Both deformations and the influence weight of
ν increase with an increasing value in
ν. For example, the maximum wall deflection increased by 19%, 23%, 28%, and 37%, respectively, for each 0.5 increment of the
ν value from 0.2 to 0.4. When the value of
ν changed from 0.2 to 0.4, the maximum wall deflection, the maximum ground surface settlement, and the maximum horizontal movement of the ground surface increased by 98%, 98%, and 151%, respectively.
The influence of the Poisson’s ratio can be explained by the expression of Young’s modulus in the MCC model:
where
K is the bulk modulus, which can be expressed by
where
e0 is the void ratio,
p′ is the mean effective stress, and
κ is the unloading–reloading line slope. Substituting Equation (1) into Equation (2), we can obtain
Therefore, when other variables in the expression remain unchanged, a larger Poisson’s ratio reflects a weaker stiffness of the soil, and thus larger deformations occur under the excavation-induced unloading.
Unfortunately, despite the great impact of Poisson’s ratio on the calculated performance of deep excavations, corresponding values for in situ soils can only be determined empirically or computed through back analyses.
3.4.2. Influence of the Void Ratio, e0
According to the Technical Code for Building File Foundations [
22], the void ratio of Shanghai soft soils is generally between 0.6 and 1.6. To cover this range, five
e0 values (i.e., 0.5, 0.9, 1.2, 1.6, and 2.0) are compared in
Figure 7.
In general, all aspects of deformations decreased with the increasing void ratio. When the input value of e0 reduced from 2.0 to 0.5, the maximum wall deflections, the maximum ground surface settlements, and the maximum horizontal ground surface movements increased by 43%, 31%, and 33%, respectively. According to Equation (2), the bulk modulus, K, of soils is proportional to the void ratio, e0. Therefore, soils with a larger void ratio accordingly have a larger bulk modulus and, thus, a smaller rebound during unloading. In addition, the results also indicate that, compared with the ground surface movements, the lateral wall deflection was more sensitive to the value of e0.
3.4.3. Influence of the Coefficient of Lateral Earth Pressure at Rest, K0
The impact of the coefficient of lateral earth pressure at rest on deep performance reflects the influence of the magnitude of the initial horizontal stresses in the ground. The sensitivity of the excavation-induced deformations to different values of
K0 is investigated in
Figure 8.
The figure shows that the differences of the wall deflections, the vertical and the horizontal ground movements between the “K0 = 0.5” model and the “K0 = 1” model were around 12%, 50%, and 23%, respectively. The diaphragm wall deflections increased negligibly during K0 value’s increase from 0.5 to 1. Theoretically, larger values of K0 mean larger lateral earth pressure on the back of the diaphragm wall and, thus, a larger lateral stress relief when the soil inside the excavation is removed. However, the influence of the K0 is insignificant. Furthermore, the limited differences of the lateral wall deflections with different K0 values are generally above the bottom of the excavation, while the lower part of the wall inserted into the soil remains stable.
However, the obtained results might only represent the current idealized excavation because the tendency is not observed universally. In fact, the general influence of the value of
K0 is much more complicated. Several researchers have previously investigated the impact of the coefficient of lateral earth pressure at rest on excavation performances, but no consistent conclusion has been drawn. For example, Potts [
26] carried out a series of 2D numerical analyses and found that the wall displacements were very much dependent on the values of
K0, in which a larger value of
K0 led to much larger wall deflections. A series of 3D numerical analyses conducted by Dong [
18] indicated that changes in the values of the coefficient of lateral earth pressure at rest had an insignificant influence on diaphragm wall movements. At the same time, Xu [
27], through his parametric studies, summarized that the diaphragm wall deflections decrease along with the increasing
K0 values. Although the influence tendency of
K0 in this study is minimal, it is opposite to Xu’s results [
27]. These contradictory conclusions might arise from various factors (such as different modeling methods, excavation scales, and constitutive models), which need more targeted investigations. Unmistakably, the impact of
K0 on excavation-induced displacements is still ambiguous, and further research is desired.
Unlike the lateral wall deflections, the ground surface movements seem to be less complicated. The lateral movements generally increased, while the vertical movements generally decreased with an increasing value of K0. Nevertheless, considering that the excavation-induced ground movements are closely related to lateral wall deflections, the influence of K0 on ground movements should be further investigated.
3.4.4. Influence of the Slope, λ, of the Normal Consolidation Line in the Plane
The values of
λ reported from the several realistic projects in Shanghai range from 0.07 to 0.17, reflecting a general scope of the parameter for Shanghai soft soils [
13,
14]. This parametric study expands this scope to 0.03~0.4, and the calculated results are compared in
Figure 9.
The figure shows that λ values have a very limited impact on the lateral wall deflections (i.e., the maximum wall deflections only increased by 6.5% when the λ value increased from 0.03 to 0.4). This is because λ mainly reflects soil characteristics under loading conditions, but deep excavations are basically unloading processes.
Different from the lateral wall deflections, the ground outside the pit is very sensitive to the variation in the value of λ; i.e., when the λ value differs from 0.03 to 0.4, the maximum settlements and the maximum horizontal movements of the ground surface increased by 445% and 42%, respectively. This tendency indicates the existence of loading conditions outside the pit during the excavation. In fact, the loading conditions arise from the upward movement of the deep soil caused by the excavation process.
3.4.5. Influence of the Slope, κ, of the Unloading–Reloading Line in the Plane
The influence of the values of
κ is described in
Figure 10. The parameter is the slope of the unloading–reloading lines in the
plane and mainly reflects the unloading behavior of the soils. During the excavation, the soil in front of the wall was in the unloading condition. Consequently, the wall deflections were directly related to the swelling index. Moreover, according to Equation (2), the bulk modulus is in inverse proportion to the value of
κ. Therefore, as shown in the figure, the maximum wall deflections increased by as much as 115% when the
κ value increased from 0.005 to 0.025.
Theoretically, the great increase in the lateral wall deflections brings great volume loss to the ground behind the wall and hence should increase the ground settlements. Yet, as shown in
Figure 10, the ground settlements decrease with the increasing value in
κ. This indicates that the deep soil is more sensitive to the value of
κ. The larger
κ causes more significant upward movements of the deep soil, which outweighs the additional volume loss caused by the increased lateral wall deflections, and thus, decreasing settlements are produced.
3.4.6. Influence of the Slope, M, of the Critical State Line in the Plane
The MCC model does not involve a cohesion parameter, and
M is derived from the effective friction angle through Equation (4):
In addition,
M is the only parameter reflecting the shear strength of MCC soils. The results of models with different
M values are presented in
Figure 11. In general, the diaphragm wall deflections decrease with the increase in the
M value, and the deformations are very sensitive to
M with small values. This tendency is because
M affects the shape of the plastic surface, and a small value of
M means the soil can enter the plastic zone easily. When the value of
M was larger than one, the diaphragm wall was no longer sensitive to the change of
M, indicating the soil was in the elastic behavior.
The ground settlements increased slightly when the value of M differed from 0.3 to 0.5. When the value of M was larger than 0.5, however, the settlements decreased with the increasing value in M. Different from the diaphragm wall, a further increase in the value of M when it is larger than one can still reduce the ground settlements, which should arise from the additional upward movement of the deep soil. Meanwhile, the lateral ground movement seems to be insensitive to the change in the value of M.