, , Tingqing, Wengng, 1

1a. School of Civil Engineering; 1b.Key Laboratory of New Technology for Construction of Cities in Mountain Area of Ministry of Education, Chongqing University, Chongqing 400045, P. R. China; 2. School of Civil and Environmental Engineering, Nanyang Technological University, Singapore 639798; 3. School of Civil Engineering, Chongqing Three Gorges University, Chongqing 404130, P. R. China)

Abstract: For braced excavations in deep deposits of soft clays or residual soils, the ground surface settlement behind the excavation is correlated with the extent of basal heave as well as the wall deflections and is also affected by the magnitude of the groundwater drawdown behind the retaining system. Reliability analysis based on a recently developed simplified logarithm regression model for estimation of the maximum ground surface settlement is presented. The first-order reliability method implemented with a variance reduction technique while considering soil spatial variability is employed to investigate the probability that certain ground surface settlement threshold is exceeded. This paper presents the effects of spatial averaging and the influence of several key design parameters including the stiffness of the wall system, the magnitude of the threshold ground surface settlement, the coefficient of variation of the soil properties, and the magnitude of the groundwater drawdown on the ground surface settlement. It is concluded that soil spatial variability results in a higher probability of failure (i.e., a lower reliability index), without considering it would result in an unreliable design. A larger characteristic length results in a lower probability of failure and a higher reliability index. When the spatial variability of both the and E50/cu are considered, the influence on is more significant.

Keywords: ground surface settlement; braced excavation; groundwater drawdown; spatial reliability; variance reduction

1 Introduction

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.

2 Review of the developed logarithm regression (LR) model

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 (lnS) 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].

(1)

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 settlementδvm. It is validated by a total of 19 well-documented actual case histories from various sites. The equation forδvm(mm) with the coefficient of determinationR2=0.924 5 takes the following form:

(E50/cu)-0.5479S-2.222 3(dw)0.101 3

(2)

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 ofdwfrom 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].

3 Reliability analysis considering spatial variability

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.

3.1 Brief introduction to spatial variability

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 functionΓ2calculated by the scale of fluctuationθand the characteristic lengthL, 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:

(3)

(4)

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 ofLthat should be used and examined the influence of differentLon the probability of excavation-induced basal-heave failure. For simplicity, the commonly adopted scale of fluctuation valuesθof 2, 5, 20, 50, 100 m[40-41], and the characteristic lengthsL=19, 26, 72 m are considered, which are closely associated with the excavation depth, the diaphragm wall depth, and the final strut depth.

Fig.1 Schematic diagram of the slip surface for braced

As shown in Fig. 1, the 1stL=19 m is the length ofod(the distance of the final strut to the bottom of the diaphragm wall), the 2ndL=26 m equals the length of the arccd, and the 3rdL=72 m is the length of the sliding surface (arcabcde). This method has been similarly adopted by Wu et al.[16]and Luo et al.[22].

3.2 Developed Excel spreadsheet

Fig.2 plots the FORM EXCEL Spreadsheet setup that implements the spatial variability for the calculation of the reliability indexand the probability of failurePfbased on the proposed estimation model of ground surface settlement. The spatial factors are inserted via Cells R3∶S5. The two variables ofcu/v′ andE50/cuare assumed to be normally distributed. Other parameters includingB,T,He, lnS, anddware assumed to be deterministic. In the example shown in Fig. 2,B=30 m,T=30 m, andHe=20 m are adopted in the spatial variability analysis for the detailed use of the developed spreadsheet[13]. The reliability indexis calculated in Cell O4, numerically expressed as Eq. (5)

=

(5)

wherexis the vector of random variables;mis the vector of mean values;σis the vector of standard deviation;Ris the correlation matrix; andFis the failure region. Cellg(x) contains the expression ofδvm-δvm_cr, which indicates that if the induced maximum ground surface settlement is greater than the threshold valueδvm_cr, it would be regarded as a failure or unsatisfactory performance. The column labeledxicontains the design point. For spatial variance,SD=Mean×COV×, in whichSDis the standard deviation, Mean is the mean value, COV is the coefficient of variation,is the standard deviation reduction factor. For random variables, the off-diagonal terms are zero. For Gaussian-distributed random variables, a direct relationship exists betweenandPf, i.e.,Pf=1-Φ(), in whichΦis the cumulative normal density function.

Fig.2 FORM EXCEL setup for evaluating the and

3.3 Influence of the cu/v′ and E50/cu of the soil

Fig.3 and Pf results from different spatial

3.4 Influence of ln(S)

Fig.4 shows the effects of ln(S) onandPffor the case ofB=30 m,He=20 m, ln(S)=8.176, anddw=4 m.increases as the system stiffness ln(S) becomes larger. It is reasonable thatincreases with a stiffer excavation supporting system. The system stiffness shows a significant influence onandPf; a larger ln(S) will result in a greaterand a smallerPf.

Fig.4 Influence of the logarithmic system stiffness ln(S) on and Pf for the case of B=30 m, He=20 m,

3.5 Influence of vm_cr

In this section, the choice of the threshold (critical) maximum ground settlementvm_crfor service ability considerations is considered. Typically, the thresholdvm_cris chosen as 0.75%-1.0% ofHe. Fig. 5 plots the effects ofθandδvm_cronandPfforB=30 m,He=20 m, ln(S)=8.176 andL=19, 26, 72 m, respectively. It indicates that bothθandδvm_crsignificantly influence the value ofandPf. However, the effects ofθonandPfare not as remarkable as that ofδvm_cr, especially whenθis greater than 20.tends to increase withδvm_cr, while the probability of failure is much lower when a greater threshold is exceeded. In addition,decreases with the increase ofθ. Furthermore,slightly increases withL, as indicated in Fig.5(a) and (b).

3.6 Influence of dw

Fig.6 compares the influence of different groundwater drawdowndwonandPffor the case ofB=30 m,He=20 m, ln(S)=8.176, andδvm_cr=200 mm. Greaterdwresults in a smaller, indicating that the greater the groundwater drawdown, the greater the probability thatδvmexceeds the thresholdδvm_cr. The magnitude of the groundwater drawdowndwshows a significant influence onandPf.

3.7 Influence of the COV of E50/cu

Fig.7 shows the influence of the coefficient of

Fig.5 Influence of θ, vm-cr and L on (a)reliability index and

Fig.6 Effects of dwon (a) and

variation, the COV ofE50/cuonandPffor the case ofB=30 m,He=20 m,dw=4.0 m, ln(S)=8.176, andδvm_cr=200 mm. Both the COV ofE50/cuandLhave a significant influence onandPf. However, whenθis greater than 50, the influence of the COV ofE50/cuonandPfis not as significant as that ofL.decreases with the increase of the COV ofE50/cu.

Fig.7 Effects of the COV of E50/cu on (a) and

3.8 Influence of the COV of cu/v′

Fig.8 Effects of the COV of on (a) and

5 Summary and conclusions

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).

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.

Acknowledgements

The authors would like to acknowledge the financial support from National Natural Science Foundation of China (Grant No. 52078086), Natural Science Foundation of Chongqing (No. cstc2018jcyjAX0632), Chongqing Engineering Research Center of Disaster Prevention & Control for Banks and Structures in Three Gorges Reservoir Area (No. SXAPGC18YB01).