Rapid urbanization and continuous development of infrastructure construction have led to an increased demand for deep braced excavations in urban built environments. One major concern with the construction of deep excavation support systems is the potential damage to nearby buildings and tunnels caused by excavation-induced ground movement. The ground movement behind the excavation is correlated with the extent of basal heaves and the magnitude of the wall deflections. Ground settlement is an important hydro-geological factor influencing the groundwater drawdown behind the excavation, due to possible leakage through the wall, flow along the wall interface, or poor connections between wall panels as a result of poor quality control. Therefore, assessing the distribution and magnitude of the ground surface settlement adjacent to a braced excavation is the most important consideration in the design phase. Numerical modeling is widely used, but it's time-consuming and requires considerable computational effort, especially three-dimensional computation. The use of empirical/semi-empirical methods to predict excavation-induced ground movement is more convenient[1-10].
Reliability-based analysis via the first-order reliability method (FORM) is increasingly employed in various geotechnical applications[11-13]to calculate the reliability index as well as the probability of failure. This method adopts the mean average and the standard deviation or the equivalent value to present uncertain parameters. The safety factor or safety margin is determined by measuring the shortest distance from the safety average to the directional standard deviation of the most likely failure combination of parameters on the limit state surface. However, natural soil properties vary spatially due to the complicated geological, environmental, and physical-chemical processes to which the soil has been subjected during its formation[14-15]. Several researchers have highlighted the effects of the spatial variation of soil properties on various geotechnical problems[16-21]. Reliability analysis considering spatial variability has been carried out by many researchers. Luo et al.[22] presented a simplified approach for the reliability analysis of basal heave in a braced excavation considering the spatial variability of the soil parameters using the first-order reliability method (FORM). Wang et al.[23] modeled the inherent spatial variability of the soil properties of drilled shafts by developing a reliability-based design (RBD) approach that integrated a Monte Carlo simulation (MCS)-based RBD with the random field theory. Cheon et al.[24] described the spatial variability of geotechnical properties for foundation design in deep water in the Gulf of Mexico, via a random field model that depicted spatial variations in the design of undrained shear strength. Li et al.[25] investigated the reliability of strip footing in the presence of spatially variable undrained shear strength with a non-stationary random field. Gong et al.[26] proposed a new framework considering the spatial variability of soil properties to analyze the probabilistic ability of a braced excavation in clay, which was modeled with the random field theory. Liu et al.[27] analyzed the reliability of slopes considering the spatial variability of the soil using a simplified framework that applied a strategy of variance reduction to enable more than one shear strength value to be considered in slope reliability problems based on Monte Carlo simulation and the multiple response surface method (MRSM). However, studies on the probabilistic assessment of ground surface settlement induced by the braced excavation that consider the uncertainties arising from the soil stiffness and strength parameters are limited. In addition, the influence of the spatial variability of soil properties, as well as the influence of groundwater drawdown, are scarcely investigated.
This paper adopts a framework combining a recently developed simplified LR model[28] to estimate the maximum ground surface settlement using the FORM EXCEL spreadsheet method to analyze the reliability. The variance reduction technique for considering soil spatial variability is employed to investigate the probability that a certain threshold ground surface settlement is exceeded. Some useful conclusions regarding the effects of spatial averaging, and the influence of several key design parameters such as the stiffness of the wall system, the magnitude of the threshold ground surface settlement, the coefficient of variation of the soil properties, as well as the magnitude of the groundwater drawdown are presented.
The developed logarithm regression (LR) model is a semi-empirical model proposed by Zhang et al.[28] for estimating the maximum ground surface settlement induced by the braced excavation considering groundwater drawdown in residual soils. It is based on the results of 746 plane strain finite element (FE) simulations using Plaxis 2D[29]. To reveal the increased stiffness of soils at small strain levels, the hardening small strain (HSS) model was adopted in the analysis. Many studies have utilized the HSS constitutive model in the modeling of excavation in soft/medium clay[30-32]. For the 746 FE model, the range of the excavation width (B) is 30~40 m, the excavation depth (He) is 14~20 m, the thickness of the soft clay (T) is 25~30 m, the system stiffness (ln S) is 7.3~8.8, the relative shear strength ratio of the soil (cu/σv′) is 0.25~0.35, the relative stiffness ratio of the soil is (E50/cu), and the groundwater drawdown (dw) is 0.3~12 m.
For simplicity, the physical and geometrical model is not shown in this paper. The diaphragm wall was inserted 5 m into the stiff clay layer, which was found to be adequate against basal heave failure. More model details can be found in Zhang et al.[28].
Plaxis can not directly model the consolidation settlement when it is based on undrained parameters. Therefore, method A in Plaxis with c′ and φ is used in the analysis, simulating the long-term settlement incurred by groundwater drawdown, without considering the time effects. For an assumed cu/σ′v ratio, the effective friction angle φ is computed using the correlation proposed by Wroth et al.[33]
The groundwater drawdown simulation in this paper is implemented by changing the horizontal/vertical permeability ratio of the soil, kx/ky. The numerical analysis performed via Plaxis considers fully coupled flow-deformation, in which the groundwater drawdown of 12.0 m, 6.0 m 0.3 m can be realized. The use of the relative shear strength ratio and the stiffness ratios is based on Kung et al.[3], Zhang et al.[32], Xuan[34]).
A simple logarithm regression (LR) model based on the numerical results from 746 hypothetical cases[28], was developed to predict the maximum ground settlement δvm. It is validated by a total of 19 well-documented actual case histories from various sites. The equation for δvm (mm) with the coefficient of determination R2=0.924 5 takes the following form:
The index for the drawdown in the LR analysis was only 0.101 3, which is relatively small compared to the excavation depth, the relative shear strength ratio, and the system stiffness value. Based on Eq. (2), when other parameters are kept constant, an increase of dw from 0.3 m to 6.0 m will almost double the maximum ground surface settlement, which is consistent with the findings by Wen et al.[35].
Since the FE analysis and the proposed LR estimation model are unable to take into account the inherent spatial variability of soil properties, this section introduces a reliability-based method to estimate the braced excavation induced ground surface settlement considering groundwater drawdown by adopting the FORM spreadsheet method and implementing the spatial factors.
Spatial variability refers to the nonuniform distribution of basic soil properties such as permeability or the deformation modulus. The change in the spatial average of soil properties in a certain area is smaller than at a certain point, to some extent, and as the size of the area increases, the change in the soil properties decreases. A dimensionless variance reduction function Γ2 calculated by the scale of fluctuation θ and the characteristic length L, as proposed by Vanmarcke[36], was used to quantify the reduction in the point variance under local averaging. It is subsequently adopted by Vanmarcke to reveal spatial averaging for reliability analysis[37], by means of which the soil parameter variances can be reduced by multiplying a factor less than the unity, i.e. the variance reduction factor. This variance reduction technique has been successfully applied using different constant, triangular, and exponential models[37-38], among which the latter is more commonly assumed for geotechnical random field modeling, expressed as:
The reduced variance σΓ2 can be obtained through:
in which σ is the standard deviation of cu/σ′v or E50/cu. In this study, Γ is the standard deviation reduction factor.
For reliability analysis using the variance reduction technique, the characteristic length is of most importance. Schweiger et al.[39] found that for the analysis of supported excavations, the characteristic length is correlated to the length of the sliding surface. Luo et al.[22] investigated the value of L that should be used and examined the influence of different L on the probability of excavation-induced basal-heave failure. For simplicity, the commonly adopted scale of fluctuation values θ of 2, 5, 20, 50, 100 m[40-41], and the characteristic lengths L=19, 26, 72 m are considered, which are closely associated with the excavation depth, the diaphragm wall depth, and the final strut depth.
As shown in Fig. 1, the 1st L=19 m is the length of od (the distance of the final strut to the bottom of the diaphragm wall), the 2nd L=26 m equals the length of the arc cd, and the 3rd L=72 m is the length of the sliding surface (arc abcde). This method has been similarly adopted by Wu et al.[16]and Luo et al.[22].
Fig. 2 plots the FORM EXCEL Spreadsheet setup that implements the spatial variability for the calculation of the reliability index β and the probability of failure Pf based on the proposed estimation model of ground surface settlement. The spatial factors are inserted via Cells R3 :S5. The two variables of cu/σv′ and E50/cu are assumed to be normally distributed. Other parameters including B, T, He, ln S, and dw are assumed to be deterministic. In the example shown in Fig. 2, B=30 m, T=30 m, and He=20 m are adopted in the spatial variability analysis for the detailed use of the developed spreadsheet[13]. The reliability index β is calculated in Cell O4, numerically expressed as Eq. (5)
where x is the vector of random variables; m is the vector of mean values; σ is the vector of standard deviation; R is the correlation matrix; and F is the failure region. Cell g(x) contains the expression of δvm-δvm_cr, which indicates that if the induced maximum ground surface settlement is greater than the threshold value δvm_cr, it would be regarded as a failure or unsatisfactory performance. The column labeled xi contains the design point. For spatial variance, SD=Mean×COV×Γ, in which SD is the standard deviation, Mean is the mean value, COV is the coefficient of variation, Γ is the standard deviation reduction factor. For random variables, the off-diagonal terms are zero. For Gaussian-distributed random variables, a direct relationship exists between β and Pf, i.e., Pf =1-Φ(β), in which Φ is the cumulative normal density function.
Fig. 3 presents β and Pf, which were calculated from several combinations of the spatial variability of cu/σ′v and E50/cu for L=19, 26 and 72 m, i.e., consideration of cu/σ′v variability only, consideration of E50/cu variability only, and consideration of the variability of both cu/σ′v and E50/cu. The case without considering any kind of spatial variability, which represents uniform ground conditions, is shown as a comparison. The increases of θ denote a more uniform ground, β converges to the value of 1.183 while Pf converges to the value of 11.845%. β is moderately higher than the case without considering spatial variability only when the spatial variability of cu/σ′v is considered. However, β is marginally influenced by the spatial variability of E50/cu. When the spatial variability of both cu/σ′v and E50/cu are considered, the influence on β is more significant, implying that neglecting the influence of soil spatial variability results in an unreliable design. A larger L results in a lower probability of failure and a higher reliability index.
Fig. 4 shows the effects of ln(S) on β and Pf for the case of B=30 m, He=20 m, ln(S)=8.176, and dw=4 m. β increases as the system stiffness ln(S) becomes larger. It is reasonable that β increases with a stiffer excavation supporting system. The system stiffness shows a significant influence on β and Pf; a larger ln(S) will result in a greater β and a smaller Pf.
In this section, the choice of the threshold (critical) maximum ground settlement δvm_cr for service ability considerations is considered. Typically, the threshold δvm_cr is chosen as 0.75%-1.0% of He. Fig. 5 plots the effects of θ and δvm_cr on β and Pf for B=30 m, He=20 m, ln(S)=8.176 and L=19, 26, 72 m, respectively. It indicates that both θ and δvm_cr significantly influence the value of β and Pf. However, the effects of θ on β and Pf are not as remarkable as that of δvm_cr, especially when θ is greater than 20. β tends to increase with δvm_cr, while the probability of failure is much lower when a greater threshold is exceeded. In addition, β decreases with the increase of θ. Furthermore, βslightly increases with L, as indicated in Fig. 5(a) and (b).
Fig. 6 compares the influence of different groundwater drawdown dw on β and Pf for the case of B=30 m, He=20 m, ln(S)=8.176, and δvm_cr= 200 mm. Greater dw results in a smaller β, indicating that the greater the groundwater drawdown, the greater the probability that δvm exceeds the threshold δvm_cr. The magnitude of the groundwater drawdown dw shows a significant influence on β and Pf.
Fig. 7 shows the influence of the coefficient of variation, the COV of E50/cu on β and Pf for the case of B=30 m, He=20 m, dw=4.0 m, ln(S)=8.176, and δvm_cr=200 mm. Both the COV of E50/cu and L have a significant influence on β and Pf. However, when θ is greater than 50, the influence of the COV of E50/cu on β and Pf is not as significant as that of L. β decreases with the increase of the COV of E50/cu.
Fig. 8 shows the influence of the COV of cu/σ′v on β and Pf for the case of B=30 m, He=20 m, ln(S)=8.176, dw=4.0 m, δvm_cr=200 mm. Both the COV of cu/σ′v and L significantly influence β and Pf. However, the influence of the COV of cu/σ′v on β and Pf is not as significant as that of L, especially when θ is greater than 20. β decreases with the increase of the COV of cu/σ′v.
A reliability-based framework that considers the spatial averaging effect of soil properties is proposed to assess the probability that threshold maximum ground surface settlement is exceeded by combining the FORM spreadsheet and the LR model proposed previously by Zhang et al.[28]. It is concluded that soil spatial variability results in a higher probability of failure (i.e., a lower reliability index).
The parametric analysis shows that the spatial variability of soil, the threshold ground settlement, the stiffness of the system, the level of groundwater drawdown, as well as the COV of cu/sv′ and E50/cu have a significant influence on the reliability index. When the spatial variability of both cu/σ′v and E50/cu are considered, the influence on β is more significant. A larger characteristic length results in a lower probability of failure and a higher reliability index. The proposed approach requires much less computational effort in dealing with the spatial variability of soil properties. It is expected that these conclusions will provide useful references and insights for the design of future excavation projects involving spatial variability.
For further study, a detailed characterization of geotechnical model uncertainties, especially from the perspective of the spatial variability of in situ soil properties, is indispensable. The authors are working on this by collecting borehole and bore log information regarding field instrumentation and tests.
2021, Vol. 43 Issue (1): 54-63
PDF
2021, Vol. 43 Issue (1): 54-63
PDF