Keywords

1 Introduction

Deep braced excavations are generally known to be associated with risks from various sources such as retaining systems, subsurface soil conditions, and construction techniques. The inherent uncertainty of the strength properties of soil is one of the primary factors that influence the deformation of the retaining wall and the ground settlement. It is of great importance to study the uncertainty of braced excavations in face with the heterogeneous soil strength properties.

Many scholars used the random field to represent the soil spatial variability and study the uncertainty of braced excavations by probabilistic methods. Luo et al. [1] presented a simplified approach to consider the effect of spatial variability in a two-dimensional random field for reliability analysis of basal heave in a braced excavation in clay. Wu et al. [2] proposed a novel method of updating the probability distribution of the maximum wall displacement based on the measurements at earlier stages. Qi and Zhou [3] presented an efficient Bayesian back-analysis procedure for braced excavations using wall deflection data at multiple points. Lo and Leung [4] introduced an approach using field measurements to update the parameters characterizing spatial variability of soil properties for subsequent construction stages. Gholampour and Johari [5] presented a practical approach for reliability analysis of braced excavation in spatially varied unsaturated soils. Most of the researchers used PLAXIS software with the Mohr-Coulomb failure criterion to build the models of braced excavations. They found that it is necessary to consider soil spatial variability during braced excavations. Seldom studies discussed the impact of struts on the uncertainty reduction of braced excavations.

In this study, the numerical model of a braced excavation with struts is firstly established by an elastic-plastic model with Drucker-Prager failure criterion in COMSOL Multiphysics. Random field theory is used to simulate the spatial variability of Young’s modulus. The uncertainty of ground settlement and deflection of retaining wall controlled by struts are studied by Monte Carlo simulation based on 500 random fields.

2 Methods

2.1 Numerical Model of Braced Excavation with Struts

The numerical model of braced excavation is modeled by COMSOL Multiphysics. Drucker-Prager criterion is used to simulate the elastic-plastic behavior of excavation. Drucker-Prager criterion is the linear combination of the first invariant of stress tensor I1 and the square root of the second invariant of the deviatoric stress tensor J2:

$$ F = \sqrt {J_{2} } + \alpha I_{1} - k $$
(1)

where α and k are two material constants and can be calculated by the following equations:

$$ \alpha = \frac{\tan \phi }{{\sqrt {9 + 12\tan^{2} \phi } }} $$
(2)
$$ k = \frac{3c}{{\sqrt {9 + 12\tan^{2} \phi } }} $$
(3)

where c and ϕ are the cohesion and the angle of internal friction, respectively. Note that the hardening or softening of soil behavior is not considered, so the dilatation angle is not included.

Fig. 1.
figure 1

Force of struts with deformation.

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The example of braced excavation is taken from Schweiger [6]. As shown in Fig. 1, the model is 90 m in width and 60 m in height. The depth of the excavation is 26 m with a retaining wall. Three struts are applied to minimize wall movements. The process of excavation is divided into four stages and the depth of each excavation is 4 m. Three struts placed at 4.8 m, 9.3 m, and 14.35 m are installed during excavation. The soil type is sand. The Poisson’s ratio, cohesion, angle of internal friction, and density of soil are 0.3, 0 Pa, 35°, and 1900 kg/m3, respectively. The stiffness of struts is chosen as 104 kN/m. The Young’s modulus, Poisson’s ratio, and density of retaining wall are 30 GPa, 0.15, and 2400 kg/m3, respectively. These parameters are summarized in Table 1.

Table 1. Parameters of the numerical model and random field.
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To simulate the struts, a 1 m-line is defined in geometry for each strut. The struts are active for two conditions. First, the depth of the excavation exceeds the location of a strut. Second, the maximum horizontal deflection is reached. For the second condition, it can be achieved by defining a function as shown in Fig. 1. It means that when the deformation of the wall deflection exceeds 25 mm, the struts start to work. The force provided by the strut is linearly increased by the deformation and the gradient is the stiffness of the strut.

According to the geostatic equilibrium, the in-situ stress in the horizontal and vertical directions are 24 kPa and 35 kPa, respectively. To achieve the second method mentioned before, the in-situ stress is applied by external stress in the whole soil domain and corresponding boundary AB. The boundary load on the retaining wall of boundary AB is 24 kPa and it is 35 kPa for EB and AH. The bedrock is below FG and a fixed constraint is applied on FG. According to the symmetry of the structure, only the right half of the domain is built, and symmetry boundary condition is used on EF. Extrusion operators are applied on the boundaries of BD and CD ensuring that the normal displacement between the retaining wall and the soil stays in contact, with the tangential displacement being unconstrained at the same time. The geometry and boundary conditions of the numerical model are shown in Fig. 2.

Fig. 2.
figure 2

Geometry and boundary conditions of the numerical model.

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2.2 Random Field

Young’s modulus E of soil is chosen to consider the spatial variability of strength properties. According to the previous studies [7, 8], E is generally the lognormal distributed. Therefore, lnE (Natural logarithm of E) is subject to normal distribution. The mean value μln and the standard deviation σln of lnE are calculated as:

$$ \mu_{\ln } = \ln \mu - \frac{1}{2}\sigma_{{}}^{2} $$
(4)
$$ \sigma_{\ln }^{2} = \ln (1 + \frac{{\sigma_{{}}^{2} }}{{\mu_{{}}^{2} }}) $$
(5)

where μ and σ are the mean value and standard deviation of E. It is reported that the coefficient of variation (COV) of E ranges from 2%–60%. Therefore, for sandy soil, μ and σ are assumed to be 20 MPa and 10 MPa, respectively (Table 1).

The exponential covariance function C(x) [8,9,10] is adopted to generate random fields of InE:

$$ C({\mathbf{x}}) = \sigma_{\ln }^{2} \exp \left\{ { - \left[ {\frac{{(x_{1} - x_{2} )^{2} }}{{l_{x}^{2} }} + \frac{{(z_{1} - z_{2} )^{2} }}{{l_{z}^{2} }}} \right]^{1/2} } \right\} $$
(6)

where lx and lz are the horizontal and vertical correlation lengths of InE, respectively; x = [(x1, z1), (x2, z2)] denotes the coordinates of the two points in the domain. The correlation length of InE ranges from 1–100 m varies from site to site [11]. In this study, the horizontal and vertical correlation lengths are assumed to be 9 m and 6 m, respectively.

3 Results and Discussions

3.1 Deterministic Results

The deterministic results of braced excavation are shown in Fig. 3. The total displacement at the final stage is shown in Fig. 3(a). The deformation is magnified by 238 times for better visualization. The final maximum displacement is around 25 mm and in the interface between retaining wall and soil. Figure 3(b) is the plastic region of the soil at the final stage. The plastic region is adjacent to the retaining wall indicating the high-risk zone of braced excavation. Figure 3(c) shows the wall deflection in each stage. Note that the struts will be installed if the wall deflection exceeds 25 mm. It can be inferred that the 4.8 m-strut is activated at the second stage and the other two struts are valid at the fourth stage. Figure 3(d) shows surface settlement over stages. The struts can also control the settlement of braced excavation. At the fourth stage, the settlement is reduced significantly because of the installation of two struts.

Fig. 3.
figure 3

Deterministic results: (a) final total displacement; (b) final plastic region; (c) wall deflection and (d) settlement.

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3.2 Uncertainty of Braced Excavation by Stages

Figure 4 displays the uncertainty of wall deflection during excavation. The blue shadows are the confidence intervals of each percentage. At the first stage, the uncertainty of wall deflection is noteworthy. It gradually decreases with depth. At the second stage, the uncertainty is considerably reduced because of the installation of the first strut. At the third stage, the uncertainty away from the strut is improved but it is still very small compared to the first stage. At the final stage, another two struts are active, and the uncertainty of wall deflection is trivial. Only the lower part of the retaining wall suffers from uncertainty. To conclude, the uncertainty of wall deflection is significantly reduced after the first strut. The uncertainty of wall deflection above the depth of struts is well-controlled at the final stage of excavation.

The uncertainty of settlement during excavation is shown in Fig. 5. As the excavation goes on by stages, the uncertainty of surface settlement increases. It implies that the effectiveness of struts on the reduction of uncertainty for settlement is limited. In addition, the mean results of the settlement are larger than the deterministic results. It gradually closed to the deterministic results. The two results almost overlapped at the final stage. It illustrates that the deterministic results may underestimate the settlement of braced excavation. Controlling the settlement during excavation based on the deterministic result may have potential risks. It is necessary to consider spatial variability of soil strength properties for braced excavation.

Fig. 4.
figure 4

Uncertainty of wall deflection during excavation: (a) Stage 1; (b) Stage 2; (c) Stage 3 and (d) Stage 4.

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Fig. 5.
figure 5

Uncertainty of settlement during excavation: (a) Stage 1; (b) Stage 2; (c) Stage 3 and (d) Stage 4.

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4 Conclusions

In this study, the numerical model of a braced excavation is firstly established by an elastic-plastic model with Drucker-Prager failure criterion in COMSOL Multiphysics. The staged excavation with struts is considered. Random field theory is used to simulate the spatial variability of Young’s modulus. The uncertainty of braced excavation on ground settlement and deflection of retaining wall by stages are studied by Monte Carlo simulation based on 500 random fields. Major conclusions are summarized below:

  1. (1)

    The uncertainty of wall deflection during excavation is significantly reduced after the first strut. The uncertainty of wall deflection above the depth of struts is well-controlled at the final stage of excavation.

  2. (2)

    The deterministic result may underestimate the settlement of braced excavation. Controlling the settlement during excavation based on the deterministic result may have potential risks. It is necessary to consider spatial variability of soil strength properties for braced excavation.