Application of random set method in a deep excavation: based on a case study in Tehran cemented alluvium

Abstract

The design of high-rise buildings often necessitates ground excavation, where buildings are in close proximity to the construction, thus there is a potential for damage to these structures. This paper studies an efficient user-friendly framework for dealing with uncertainties in a deep excavation in layers of cemented coarse grained soil located in Tehran, Iran by non-deterministic Random Set (RS) method. In order to enhance the acceptability of the method among engineers, a pertinent code was written in FISH language of FLAC2D software which enables the designers to run all simulations simultaneously, without cumbersome procedure of changing input variables in every individual analysis. This could drastically decrease the computational effort and cost imposed to the project, which is of great importance especially to the owners. The results are presented in terms of probability of occurrence and most likely values of the horizontal displacement at top of the wall at every stage of construction. Moreover, a methodology for assessing the credibility of the uncertainty model is presented using a quality indicator. It was concluded that performing RS analysis before the beginning of every stage could cause great economical savings, while improving the safety of the project.

Keywords

uncertainty / reliability analysis / deep excavations / random set method / finite difference method

Introduction

In geotechnical engineering, the uncertainties such as the variability and uncertainty inherent in soil (or rock) properties have caught more attentions from researchers and engineers. To the point, two kinds of uncertainty associated with geotechnical problems exist, for which in the technical literature the terms ‘aleatory’ and ‘epistemic’ are commonly used. Aleatory uncertainty represents the natural randomness of a property. As an important feature, this kind of uncertainty is irreducible and is also known as natural variability or spatial variability in the case of soil parameters, which vary over space. Epistemic uncertainty is a type of uncertainty that originates from the lack of knowledge. As a result, this type of uncertainty can potentially be decreased by means of numerous observations and collecting information; however, it is not a reasonable practice in most cases. There are also other types of uncertainties concerning construction, manufacturing, maintenance and human errors [¹], which are not included in the latter categories and are usually not considered in models of engineering performance.

In dealing with uncertainties in any system, an uncertainty model is required. The uncertainty model consists of a collection of input uncertainty, model uncertainty, and the technique by which the system response is evaluated. Recent research on stochastic geotechnical modeling by means of random fields has provided very valuable theoretical insights (e.g., [²,³]), but is rarely applied in geotechnical practice mainly for two reasons: firstly, because the required samples for a given project are usually significantly in excess of what is available from a standard site investigation scheme in usual projects; Secondly, incorporating probabilistic concepts into numerical analyses of complex structures generally leads to a computational effort often not feasible from a practical point of view.

There are many probabilistic methods, standard reliability methods and procedures such as MC (Monte Carlo simulation), First Order Second Moment approximation (FOSM), First Order Reliability Method (FORM), Second Order Reliability Method (SORM), Latin Hypercube Sampling and Quasi Monte Carlo approaches which can be found in standard text books (e.g., [⁴‒⁹]). Although some general information on probabilistic parameters can be found in the literature (e.g., [¹⁰,⁵, ¹¹‒¹⁴]), geotechnical parameters for particular soils are given as intervals in most cases, with no information about the probability distribution across the interval and therefore a formal framework is required in order to import uncertainty in geotechnical systems.

When insufficient data are available for a particular project, as it is common in geotechnical projects, some other sources of information, such as previously published values for uncertain parameters can be utilized, but these alternative information sources often appear in a form not readily suitable for probabilistic analysis (e.g., [¹⁵]). Nevertheless, it is still possible to use probabilistic methods in these circumstances by making suitable assumptions on the statistics of the uncertainties. Because of these assumptions, required in order to construct a precise distribution, the numerical values obtained by probabilistic analysis (e.g., probability of failure) are quite sensitive to changes in the input distribution parameters (see, e.g., [¹⁶,¹⁷]).

In this paper, Random Set (RS) method is applied in reflecting the uncertainty of soil parameters in a reliability analysis of a deep excavation in Tehran, Iran cemented alluvium. There is very scarce researches on the benefits of applying RS in geotechnical problems such as deep excavation after Peschl [¹⁸], who has combined random sets and finite element method to analyze boundary value problems in geotechnical engineering. Since then, the work was continued in very limited studies and so there is a wide gap in applying and investigating the suitability of this mathematical model in literature. In this research, in order to increase the applicability of the mentioned approach, it is attempted to make a more user-friendly and easy-to-use framework for RS approach which encourages both engineers and owners to perform reliability analysis in deep excavation problems.

In order to solve the cumbersome procedure of changing the input parameters individually for all number of simulations, a pertinent code is written for the first time in FISH language of FLAC software. In this way, there is no need for the practical engineers to change the direct or indirect input variables, monitor the output data and record the desired information, every time in every single analysis. In the proposed technique, based on the provided code, all permutations are prepared using input intervals of each variable and the desired data are recorded simultaneously after each simulation. So, there would be great savings in both time and costs.

The results are presented in terms of probability of occurrence and most likely values of the horizontal displacement at top of the wall at every stage of construction. Additionally, a methodology is presented for assessing the credibility of uncertainty model by comparing the numerical results with the measured values using a quality indicator. Finally, the efficiency of RS method is evaluated with respect to the safety and cost of the project.

Random set theory

Random set theory provides a general framework for dealing with interval (or set-based) information and discrete probability distributions. In the following subsections, a brief introduction of mathematical concepts of this theory is presented. For further study and more details, one can refer to Peschl [¹⁸].

Basic concepts

Let X be a non-empty set containing all the possible values of a variable x. Dubois and Prade [¹⁹,²⁰] defined a random set on X as a pair (\mathcal{F},m), where \mathcal{F}=({{\displaystyle A}}_i:i=\mathrm{1},\mathrm{...},n) and m is a mapping, \mathcal{F}\to [\mathrm{0.1}], so that m(\phi )=\mathrm{0} and

(1){{\displaystyle \Sigma }}_{A\in \mathcal{F}}m(A)=\mathrm{1.}

where \mathcal{F}is called the support of the random set, the sets A_i are the focal elements (A_i\subseteqX) and m is called the basic probability assignment. Each set, A\in \mathcal{F}, contains some possible values of the variable, x, and m(A) can be viewed as the probability that A is in the range of x. Alternatively, the sets A_i could be ranges of a variable obtained from source number i with relative credibility m_i. Because of the imprecise nature of this formulation it is not possible to calculate the ‘precise’ probability, Pro, of a generic x∈X or of a generic subset E⊂X, but only lower and upper bounds on this probability (Fig. 1): Bel(E)≤Pro(E)≤Pl(E). In the limiting case, when i is composed of single values only (singletons), then Bel(E) = Pro(E) = Pl(E) and m is a probability distribution function.

Random set theory is one of the theories that deal with imprecise probabilities [²¹]; in this case, imprecise probabilities are expressed by the intervals [Bel(E), Pl(E)] where the belief function, Bel, of a subset E is a set function obtained through the summation of basic probability assignments of subsets A_i included in E; and the plausibility function, Pl, of subset E is a set function obtained through the summation of basic probability assignments of subsets A_i having a non-zero intersection with E. They are bounds for all possible probabilities of the event E.

Following Dempster [²²] and Shafer [²³], the lower bound Bel and the upper bound Pl of its probability measure are defined, for every subset E\in X, by

(2)\mathrm{Bel}(E)={{\displaystyle \Sigma }}_{{{\displaystyle A}}_i:{{\displaystyle A}}_{i\subseteq E}}m({{\displaystyle A}}_i),

(3)\mathrm{Pl}(E)={{\displaystyle \Sigma }}_{{{\displaystyle A}}_i:{{\displaystyle A}}_{i\cap E\ne \varphi }}m({{\displaystyle A}}_i).

Bounds on the system response

Random set theory provides an appropriate mathematical framework for combining probabilistic as well as set-based information, in which the extension of random sets through a functional relation is straightforward [²⁴]. Let f be a mapping {{\displaystyle X}}_{\mathrm{1}}\times \mathrm{...}\times {{\displaystyle X}}_N\to Yand {{\displaystyle x}}_{\mathrm{1}},\mathrm{...},{{\displaystyle x}}_Nbe variables whose values are incompletely known. The incomplete knowledge about the vector of basic variables x = (x₁,...,x_N) can be expressed as a random relation R, which is a random set (\mathcal{F},m) on the Cartesian product {{\displaystyle X}}_{\mathrm{1}}\times \mathrm{...}\times {{\displaystyle X}}_N\to Y. The random set (\mathcal{R},\rho ), which is the image of (F,m) through f is given by

(4)\mathcal{R}=\{{{\displaystyle R}}_j=f({{\displaystyle A}}_i),{{\displaystyle A}}_i\in F\};f({{\displaystyle A}}_i)=\{f(x),x\in {{\displaystyle A}}_i\},

(5)\rho ({{\displaystyle R}}_j)={{\displaystyle \Sigma }}_{{{\displaystyle A}}_i:{{\displaystyle R}}_j=f({{\displaystyle A}}_i)}m({{\displaystyle A}}_i).

If A₁,...,A_n are sets on {{\displaystyle X}}_{\mathrm{1}}\times \mathrm{...}\times {{\displaystyle X}}_N\to Y respectively and x₁,...,x_N are stochastically independent, then the joint basic probability assignment is given by

(6)m({{\displaystyle A}}_{\mathrm{1}}\times \mathrm{...}\times {{\displaystyle A}}_n)={{\displaystyle \prod }}_{i=\mathrm{1}}^n{{\displaystyle m}}_i({{\displaystyle A}}_i),{{\displaystyle A}}_{\mathrm{1}}\times \mathrm{...}\times {{\displaystyle A}}_n\in \mathcal{R}.

If the focal set A_i is a closed interval of real numbers: {{\displaystyle A}}_i=\{x|x\in [{{\displaystyle l}}_i,{{\displaystyle u}}_i]\}, then the lower and upper cumulative probability distribution functions, respectively F_*(x) and F^*(x), can be obtained given by Eqs. (7) and (8) at some point x which are illustrated in Fig. 2.

(7){{\displaystyle F}}_{*}(x)={{\displaystyle \Sigma }}_{i:x\ge {{\displaystyle u}}_i}m({{\displaystyle A}}_i),

(8)F^{*}(x)={{\displaystyle \Sigma }}_{i:x\ge {{\displaystyle l}}_i}m({{\displaystyle A}}_i).

In the absence of any further information, a random relation (random set model) can be constructed by assuming stochastic independence between marginal random sets (Eq. (6)).

The basic step in RSM is the calculation of the image of a focal element through the function f by means of Eqs. (4) and (5). The requirement for optimization in order to locate the extreme elements of each set {{\displaystyle R}}_j\in \mathcal{R} (Eq. (4)) can be avoided if it can be shown that the function f(A_i) is continuous in all {{\displaystyle A}}_i\in Fs and also no extreme points exist in this region, except at the vertices, in which case the methods of interval analysis are applicable, e.g., the Vertex method [²⁵],

(9)f({{\displaystyle A}}_i)=[{{\displaystyle l}}_i,{{\displaystyle u}}_i]

Assume each focal element A_i is an N-dimensional box, whose 2^N vertices are indicated as {{\displaystyle v}}_{k}k=\mathrm{1},\mathrm{...},{\mathrm{2}}^N. If the vertex method applies, then the lower and upper bounds R_j* and R_j* on each element will be located at one of the vertices:

(10){{\displaystyle R}}_{{{\displaystyle j}}_{*}}=\begin{array}{l}\min \\ k\end{array}\{f({{\displaystyle \vartheta }}_k):k=\mathrm{1},\mathrm{...},{\mathrm{2}}^N\},

(11){{\displaystyle R}}_j^{*}=\begin{array}{l}\min \\ k\end{array}\{f({{\displaystyle \vartheta }}_k):k=\mathrm{1},\mathrm{...},{\mathrm{2}}^N\}.

Thus function f(A_i), which represents a numerical model in the random set finite difference method (RS-FDM) framework, has to be evaluated 2^N times for each focal element A_i. The number of all calculations, n_c, required for finding the bounds on the system response is

(12){{\displaystyle n}}_c={\mathrm{2}}^N{{\displaystyle \Pi }}_{i=\mathrm{1}}^N{{\displaystyle n}}_i,

where N is the number of basic variables and n the number of information sources available for each variable.

Combination of random sets

An appropriate procedure is required if more than one source of information is available for one particular parameter in order to combine these sources. Suppose there are n alternative random sets describing some variable x, each one corresponding to an independent source of information. Then for each focal element A∈X,

(13)m(A)={\displaystyle \frac{\mathrm{1}}{n}}{{\displaystyle \Sigma }}_{i=\mathrm{1}}^n{{\displaystyle m}}_i(A).

Procedure of the RS-FDM

The merits of numerical modeling using the Finite Difference Method (FDM) in estimating deformations and internal forces of complex geomechanical structures are evident. The FDM has obtained significant fame over the last decades in solving geotechnical problems by introducing advanced soil constitutive models, which can describe the material behavior more accurately. Particularly in excavation problems-which in this work are the main concern- a very limited number of closed form solutions are available. This study takes advantage of both FDM and the concepts of random set analysis to accommodate a procedure in which a reliability analysis can be performed. Reliability analysis is of great importance especially in the case of scarce data, where a sufficiently accurate probability distribution function cannot be provided to perform a full probabilistic reliability analysis. The items that should be followed in the RS-FDM are summarized in the following steps:

• Preparation of FD model;

• Selecting the input parameters from different sources of information (random sets);

• Uncertainty reduction over the selected random sets;

• Employing a sensitivity scheme to identify the most effective variables (reducing computational effort);

• Computation of the calculation matrix;

• Finite difference calculations and determination of results in terms of bounds on discrete cumulative probability functions (CDF);

• Fitting the resulted CDF’s using the best-fit methods;

• Definition of suitable performance functions. The performance function can be evaluated with results (i.e. bounds on continuous distribution functions of the evaluated system parameters) from the finite difference calculations, in order to obtain a range for the probability of failure or unsatisfactory performance.

As it was stated previously, in this study an additional step is performed, which introduces RS into numerical simulations by writing a pertinent code.

Spatial correlation of soil parameters

The inherent spatial soil variability is the variation of soil properties from one point to another in space. Soil properties do not vary randomly in space but such variation tends to be gradual and follows a pattern that can be quantified using spatial correlation structures, where soil properties are treated as random variables. Various researchers (e.g., [²⁶‒³⁰,¹⁰]), have pointed out that soil properties should be modeled as spatially correlated random variables or ‘random fields’, because the use of perfectly correlated soil properties gives rise to unrealistically large values of failure probabilities for geotechnical structures (see, e.g., [²]). The random field model is now widely used to model the correlated structure of soil properties.

The finite difference code FLAC2D [³¹] used in the proposed framework requires the soil profile to be modeled using homogeneous layers with constant soil properties. For this reason, soil properties have to be defined not only for a certain point in space, but for the entire soil layer which is used in the calculation. Due to the fact of spatial averaging of soil properties the inherent spatial variability of a parameter, as measured by its standard deviation, can be reduced significantly if the ratio of Q/L is small, where Q is the spatial correlation length and L is a characteristic length, e.g., that of a potential failure surface. The variance of these spatial averages can be correlated to the point variance using the variance reduction factor, G², as discussed by Vanmarcke [³⁰] through s_G=G.s, where s is the standard deviation of field data (point statistics) and s_G is the standard deviation of the spatial average of the data over volume V. The variance reduction factor depends on the averaging volume and the type of correlation structure in the form

(14){\Gamma }^{\mathrm{2}}\{\begin{array}{l}\left[{\displaystyle \frac{\mathrm{\Theta }}{L}}\left(\mathrm{1}-{\displaystyle \frac{\mathrm{\Theta }}{\mathrm{4}L}}\right)\right]{\displaystyle \frac{\mathrm{\Theta }}{L}}\le \mathrm{2}\\ \mathrm{1}{\displaystyle \frac{\mathrm{\Theta }}{L}}\rangle \mathrm{2}\end{array}.

This leads to a reduction in inherent spatial variability as the size of the averaged length or volume increases. Above expression for the variance reduction has been adopted because of its simplicity but other formulations are possible and have been suggested by many scientists (e.g., [³²,¹]).

In this study, an alternative approach introduced by Peschl [¹⁸] based on Vanmarcke [³⁰] method has been adopted that applies the variance reduction technique for the random set theory. If n sources of information are assumed, the function of the spatial average of the data x_i,G can be calculated from the discrete cumulative probability distribution of the field data x_i

(15){{\displaystyle x}}_{i,\Gamma }=x^{\text{'}}-(x^{\text{'}}-{{\displaystyle x}}_i).\Gamma ,x^{\text{'}}={\displaystyle \frac{\mathrm{1}}{n-\mathrm{1}}}.{{\displaystyle \Sigma }}_{i=\mathrm{1}}^{n=\mathrm{1}}{\displaystyle \frac{({{\displaystyle x}}_{i+\mathrm{1}}+{{\displaystyle x}}_i)}{\mathrm{2}}}.

Figure 3 shows schematically upper and lower discrete cumulative probability distribution of field data, F^*(x) and F_*(x), respectively, and the discrete upper and lower function of the spatial average of the data, F^*(x_G) and F_*(x_G). The proposed method reduces the aleatory type of uncertainty of a specific parameter in qualitatively the same way as proposed by Vanmarcke [³⁰], but does not affect the epistemic uncertainty.

Soheil commercial complex case study

Subsurface condition

The project site of Soheil commercial complex is located in northern regions of Tehran, Iran which is surrounded by some buildings and one major street. The city of Tehran is located at the foot of the southern slopes of the Alborz Mountains Range and sits on an alluvial plain formed over time by flood erosion of the mountains. As a result of this process, large and small particles have settled on high and low elevations, respectively, resulting in varying geological conditions. Rieben [³³] divided the Tehran coarse-grained alluvia into four categories, identified as A, B, C, and D, where A is the oldest and D is the youngest. The site of Soheil complex is located on A alluviums of Tehran, which are characterized mostly by their cementation and hard particles.

The behavior of the subsoil is characterized by soil parameters established from a number of laboratory and in situ tests such as Standard Penetration Test (SPT), Plate Load Test (PLT) and in situ direct shear test. Subsurface investigation was carried out to evaluate the subsurface conditions and design of excavation stabilization work. Twenty-two soil borings, including fourteen boreholes and eight test pits, with a total length of 818 m were drilled (or dug) within the soil investigation works. According to the subsurface investigations, the soil profile of the investigation site consists of three main layers: a 1‒7 m thick fill layer in the shallow depths which contains fine to coarse-grained materials, a 16.0 m thick very dense cemented clayey sand layer with occasional clay lenses, and a thick layer of cemented clayey gravel at 18.0 m to the final depth of borings. Again, some cohesive clay lenses were observed in this layer. In accordance with the data gained from in situ and laboratory tests performed during the subsurface investigation, the soil model is given in Fig. 4. No groundwater was observed during the construction stages.

Construction details

This case history involves construction of a high-rise building with a five story basement car park. The depth of entire excavation varies in different sides of the site plan, due to the ground slope and the existing basements of adjacent structures, but the depth of excavation in the desired section is about 21 m. Because of the neighboring structures and the depth of the foreseen foundation excavation, the necessity for constructing a retaining system has risen. Table 1 summarizes the geometric configuration of the wall and Table 2 presents other design details of the excavated wall.

Soil constitutive model

The soil model was a modified Duncan-Chang hyperbolic model [³⁴,³⁵]. This model is a nonlinear model that includes the influence of the stress level on the stiffness, strength, and volume change characteristics of the soil. With this soil model it is possible to simulate the hysteresis behavior of the soil. The expression that gives the tangent Young’s modulus, E_t, for the hyperbolic model is

(16){{\displaystyle E}}_t={{\displaystyle \left[\mathrm{1}-{\displaystyle \frac{{{\displaystyle R}}_f(\mathrm{1}-\sin \varphi )({{\displaystyle \sigma }}_{\mathrm{1}}-{{\displaystyle \sigma }}_{\mathrm{3}})}{\mathrm{2}(c\cos \varphi +{{\displaystyle \sigma }}_{\mathrm{3}}\sin \varphi )}}\right]}}^{\mathrm{2}}{{\displaystyle Kp}}_a\left({\displaystyle \frac{{{\displaystyle \sigma }}_{\mathrm{3}}}{{{\displaystyle p}}_a}}\right),

where s₁ and s₃ have initial values of gz and K₀gz (z = depth), respectively, and are updated as the loading and unloading takes place in increments; p_a = atmospheric pressure, K= the initial tangent modulus factor, c= cohesion of soil, \varphi= internal friction angle of soil and R_f= the failure ratio. The unload-reload E_ur modulus is given by

(17){{\displaystyle E}}_{\mathrm{ur}}={{\displaystyle K}}_{\mathrm{ur}}{{\displaystyle p}}_a{{\displaystyle \left({\displaystyle \frac{{{\displaystyle \sigma }}_{\mathrm{3}}}{{{\displaystyle p}}_a}}\right)}}^n,

where K_ur= unload-reload modulus factor and n= the stress influence exponent for the tangent Young’s modulus. At the point of unloading on the stress-strain curve, the modulus changes from E_t from Eq. (16) to E_ur from Eq. (17). To decide whether an element is on the loading or unloading path, a stress state (SS) coefficient is calculated at each step [³⁶]:

(18)SS={\displaystyle \frac{{{\displaystyle \sigma }}_{\mathrm{1}}-{{\displaystyle \sigma }}_{\mathrm{3}}}{{{\displaystyle ({{\displaystyle \sigma }}_{\mathrm{1}}-{{\displaystyle \sigma }}_{\mathrm{3}})}}_f}}\sqrt[\mathrm{4}]{{\displaystyle \frac{{{\displaystyle \sigma }}_{\mathrm{3}}}{{{\displaystyle p}}_a}}},

If the current value of SS is larger or equal to the highest past value of SS (SS max-past) then E_t is used. If SS<SS_max, the unloading modulus is then used. This hyperbolic model was coded in FLAC2D V7.0 using FISH language.

Modeling

According to Lim and Briaud [³⁷], the bottom of the mesh is best placed at a depth where the soil becomes notably harder. The distance from the bottom of the excavation to the hard layer, is called D_b. They showed that D_b affects the vertical movement of the ground surface at the top of the wall but comparatively has very little influence on the horizontal movement of the wall face. For nearly all analyses a value of D_b equal to 63 m or about 3 times the wall height was used.

Considering the parameters H, W_e, B_e, and D_b as defined in Fig. 5, it was found in separate studies (e.g., [³⁷,³⁸]) that W_e = 3D_b and B_e = 3(H+D_b) were appropriate values for W_e and B_e; indeed, beyond these values, W_e and B_e have little influence on the horizontal deflection of the wall due to the excavation of the soil. For the instrumented wall to be simulated, H, D_b, B_e, and W_e were assigned 21 m, 63 m, 240 m, and 30 m, respectively.

FLAC2D V7.0 was used to carry out the finite difference based simulations of the excavated wall considering it as a plane strain problem and accounting for the long term behavior using drained conditions. Finno et al. [³⁹] compared the three dimensional effects for supported excavation. Based on their studies, they found that the 2D and 3D calculation of movement near the center of a “long wall” are similar for an excavation with a rigid layer below the excavation button. Thus, in order to compare the random set results with the field measurements, the horizontal displacement measured at the top and center of the studied wall was selected in current study. The computational steps have been defined as follows (according to the real construction process):

Step 1: initial stresses;

Step 2: activation of superstructures (i.e., buildings or streets), reset displacements after this step;

Step 3: first excavation step, shotcrete and nailing (i.e., stage 1 of construction);

Steps 4 to 9: excavation steps, shotcrete and pre-stressing the strands with 200 kN/m (i.e., stages 2 to 7 of construction).

Methodology

Sensitivity analysis

Reducing the number of basic variables, which are actually considered in the formal RS-FDM analysis, is accomplished by means of a sensitivity analysis. This is essential in order to reduce the number of required finite difference runs. For instance, 1024 finite difference runs are required if 5 basic variables with two sets are taken into account (see Eq. (12)), while this amount is decreased to 256 runs in case of 4 basic variables.

A relatively simple sensitivity method given by U.S. EPA: TRIM [⁴⁰] is used, in which three major coefficients, namely sensitivity ratio (Eq. (19)), sensitivity score (Eq. (20)) and relative sensitivity (Eq. (21)) of each input variable with respect to any system response are calculated. Figure 6 illustrates the parameters used in Eqs. (19)‒(21). The ratio of the change in model output per unit change of an input variable is called sensitivity ratio. In the process of random set analysis, sensitivity analysis is recommended to be carried out over both a small and a large amount of change in input variables which are called local and range intervals respectively (Fig. 6). The total number of 4N+1 calculations are needed to accomplish the sensitivity analysis.

When the sensitivity ratio is weighted by some characteristic of the input variable (e.g., standard deviation over mean, s/m) the sensitivity score will be obtained, which makes the sensitivity ratio independent from the units of the variable. In this work, the ratio of total range over reference value, x, is used instead of s/m. After the calculation of the sensitivity score of each variable, h_SS,i, the sensitivity score of each input variable on a system response (e.g., displacement, A) at all construction steps can simply be added up to be representative h_SS,Ai for the whole construction sequence. Then the relative sensitivity of the system response A is obtained as follows:

(19){{\displaystyle \eta }}_{SR}=\left|{\displaystyle \frac{\left[{\displaystyle \frac{f({{\displaystyle x}}_{L,R})-f(x)}{f(x)}}\right]}{\left[{\displaystyle \frac{{{{\displaystyle x}}_{L,R}}^{-x}}{x}}\right]}}\right|,

(20){{\displaystyle \eta }}_{SS}=({{\displaystyle \eta }}_{{{\displaystyle SR,x}}_R^{\mathrm{lo}}}+{{\displaystyle \eta }}_{{{\displaystyle SR,x}}_R^{\mathrm{up}}}+{{\displaystyle \eta }}_{{{\displaystyle SR,x}}_L^{\mathrm{lo}}}+{{\displaystyle \eta }}_{{{\displaystyle SR,x}}_L^{\mathrm{up}}}).{\displaystyle \frac{({{\displaystyle x}}_R^{\mathrm{lo}}-{{\displaystyle x}}_R^{\mathrm{up}})}{x}},

(21){{\displaystyle \alpha }}_A({{\displaystyle x}}_i)={\displaystyle \frac{{{\displaystyle \eta }}_{SS,{{\displaystyle A}}_i}}{{{\displaystyle \Sigma }}_{i=\mathrm{1}}^N{{\displaystyle \eta }}_{SS,{{\displaystyle A}}_i}}}.

For the 9 variables shown in Table 3, 37 calculations are required to obtain a sensitivity score for each variable. In this case the horizontal displacement of the top of the diaphragm wall, u_x,A, at the final construction step is evaluated. At this point a decision can be made as to which variables should be used in further calculations and which can be treated as deterministic values as their influence on the result is not significant. The defined threshold value has been chosen in this case as approximately 5%. Based on the results of the sensitivity analysis presented in Fig. 7 the following parameters were considered in the random set model: cohesion and stiffness for the middle layer (i.e., C2 and K2), and stiffness of the third layer (i.e., K3), which implies that 64 simulations are required (Eq. (12)).

In this step, it is recommended to investigate the correlation coefficient of random variables. The correlation coefficient (r_X,Y) represents the degree of linear dependence between two random variables. The two random variables can be considered to be statistically independent if the correlation coefficient is less than±0.3; they can be considered to be perfectly correlated if the correlation coefficient is greater than±0.9 [⁶]. There is various researches on how one can determine whether variables are correlated each other or not in the literature (e.g., [⁴¹,⁴²]), in which the covariance matrix of input variables are computed to confirm they are correlated. The cohesion and stiffness parameters of third layer are assumed to be uncorrelated mainly for two reasons: firstly, the statistical information on evaluating the correlation of stiffness-cohesion is very few in the current project and secondly, there is no obvious correlation between stiffness and cohesion of c-j soils reported in the literature, because they depend on different factors (e.g., [⁴³,⁴⁴]).

Application of random set method

The material parameters for the soil layers which were treated as basic variables are summarized in Table 3. The parameters were established not only from laboratory and in situ tests (geotechnical report) but also from previous experience of the region (expert knowledge). Both sources have been equally weighted in this particular case. It should be noted that K_ur was select as K_ur=3K, based on in situ Plate Load Tests, n constant was assumed to be equal to 0.5 and R_fwas 0.93 for all three layers.

Two published sources of information were available and these interval estimates were combined using the averaging procedure in Eq. (13). As an example, the random sets for the friction angle of layer one, modulus factor and cohesion of layer two and the friction angle of layer three are depicted in Fig. 8. It follows from Fig. 8 that as a result of considering spatial correlation the discrete cumulative probability function becomes steeper. Typical values of spatial correlation lengths, Q, for soils as given by Li and White [⁴⁵] are in the range 0.1 to 5 m for Q_y and from 2 to 30 m for Q_x. Based on the subsurface conditions in the project and some specific regulations in the region of interest (e.g., the minimum distance between the boreholes must be at least 15 m base on the uniformity level of subsurface soil), the spatial correlation length for this study is assumed to be 10 m. The characteristic length, L, has been taken as 30 m, which is based on a stability analyses investigating potential failure mechanisms for this problem.

After the identification of input basic variables, the combination of different sources and extremes of the parameters based on a random set model have to be calculated. Let x∈X be a vector of set value parameters, in which: x=(C2, K2, K3) and a random relation is defined on the Cartesian product C2\timesK2\timesK3. As a result, according to combination calculus, the pairs generated by the Cartesian product are given in the following vector:

(22)C\mathrm{2}\times K\mathrm{2}\times K\mathrm{3}=\{{{\displaystyle ({{\displaystyle C\mathrm{2}}}_{\mathrm{1}},{{\displaystyle K\mathrm{2}}}_{\mathrm{1}},{{\displaystyle K\mathrm{3}}}_{\mathrm{1}},)}}_{\mathrm{1}},{{\displaystyle ({{\displaystyle C\mathrm{2}}}_{\mathrm{2}},{{\displaystyle K\mathrm{2}}}_{\mathrm{1}},{{\displaystyle K\mathrm{3}}}_{\mathrm{1}},)}}_{\mathrm{2}},\mathrm{...},{{\displaystyle ({{\displaystyle C\mathrm{2}}}_{\mathrm{2}},{{\displaystyle K\mathrm{2}}}_{\mathrm{2}},{{\displaystyle K\mathrm{3}}}_{\mathrm{2}},)}}_{\mathrm{8}}\}.

Here the index of parameters denotes the relevant set number and the index of pairs signifies one combination of basic variables. Because there are two sets for each basic variable, 8 combinations will be produced. For each combination, an interval analysis is required, by which the deterministic input parameters of the worst and the best case of each combination are being realized. As an example, the deterministic input values of such analysis for the case of (C2₁, K2₂, K3₁) are presented in Table 4.

Similarly, the construction of all 64 required realizations (see Eq. (12)) and relevant input files for a deterministic finite difference analysis are accomplished. The next step is to determine the probability of the assignment of each realization. Assuming that the random variables are stochastically independent, which was discussed before, the joint probability of the response focal element obtained through function f(x) (in this case, the finite difference model) is the product of the probability assignment m of input focal elements by each other. For instance, in the above case since the mass probability of each set equals to 0.5 it results in:

(23)m(f({{\displaystyle C\mathrm{2}}}_{\mathrm{1}},{{\displaystyle K\mathrm{2}}}_{\mathrm{1}},{{\displaystyle K\mathrm{3}}}_{\mathrm{1}}))=m({{\displaystyle C\mathrm{2}}}_{\mathrm{1}}).m({{\displaystyle K\mathrm{2}}}_{\mathrm{1}}).m({{\displaystyle K\mathrm{3}}}_{\mathrm{1}})=\mathrm{0.5}\times \mathrm{0.5}\times \mathrm{0.5}=\mathrm{0.125.}

In order to reduce the computational effort and subsequently the cost of project, the whole process of selecting different permutations of input basic variables were encoded in model using FISH language of FLAC2D software. Following steps were considered in the algorithm of the written code:

1) In the first step, the values of different sets of basic variables are entered in the code using table commands (number of information sets as rows and lower and upper values of each sets as columns).

2) Creating a primary m×n matrix; m and n represent the rows and columns of the matrix, respectively. The column of the matrix presented in Table 5 includes two parts: permutation of information sets (characterized by number of sets) and of lower/upper values of the parameters selected in the first part. The numbers 1 and 2 in the first part of the columns represents the number of selected input set and then numbers 1 and 2 in the second part denotes the lower and upper values of the selected parameter, respectively.

3) After creating the first combination of input variables, the first analysis is performed automatically.

4) The next step is to create other combinations of input variables. In this phase, the new row of the matrix will be compared with the previous rows using loop commands, in order to check whether the new row is identical to previous rows or not. If yes, the new row is deleted from temporary matrix and another row will replace it. This will continue until a new row is obtained. Again, the analysis is performed based on the new permutation automatically.

5) Finally, the desired responses for each combination are recorded in a text file using pertinent codes.

Results

The horizontal displacement at top of the excavated wall, is often used as measure to assess the likelihood of damage for the adjacent structures and facilities. The value tolerated depends on a number of factors such as the structure of the building, soil type and height of the excavation wall. A value of about H/500 for sandy soils is used here as a limiting value for the evaluation of the limit state function [⁴⁶], in order to obtain the reliability in terms of serviceability, where H is the wall height.

To construct the Belief and Plausibility distribution function (i.e., lower and upper bounds) of a required response from deterministic FD calculations, the following procedure is pursued. The U_x,A values pertinent to all 8 simulations (A is the point at top of the excavation), given in Table 4, are sorted out to obtain the minimum and maximum of crown displacements which determine the focal element extremes of U_x,A corresponding to the combination (C2₁, K2₂, K3₁). The displacement values of those simulations between the extremes are discarded. According to Eq. (13), the probability assignment of this focal element is 0.125, which constitutes one step in the cumulative distribution function depicted in Fig. 9. Similarly, this process is repeated for other combinations (e.g., (C2₂, K2₁, K3₂)) to calculate the extremes of all focal elements of U_x,A. Finally, the left and right extremes of all focal elements are arranged in ascending order to obtain the upper and lower bounds for U_x,A, respectively.

Figure 9 depicts the calculated cumulative distribution functions (CDF) of the horizontal displacement of point A (Fig. 5) after different stages of construction (i.e., stages 3 to 7). These discrete CDFs were fitted using best-fit methods in order to achieve a continuous function (dotted line in Fig. 9). It turned out that Beta distribution ranks first among the others and has been fitted to both lower and upper discrete CDF’s of system response. The probability of exceeding the limiting value of U_x,A for different stages (P_f) are presented in Table 6. Accordingly, the minimum and maximum probabilities of unsatisfactory performance (corresponding to upper and lower values of input data) are both zero in every construction phase, except for the upper CDF of final depth of the excavation. In this particular case, based on the table presented by US Army Corps of Engineers [⁴⁷] on qualitative evaluation of performance level (Table 7), the calculated probability clearly indicate that level of displacement performance is assessed as “Hazardous” performance. Thus, it can be concluded that for all previous construction stages (i.e., up to the depth 18 m), the applied design of the excavated wall was so conservative and cost savings could be performed by applying changes in structural elements (i.e., changes in nail/strands length, spacing, rows and prestressing forces of strands), during the construction stages.

One can demonstrate the validity of the numerical calculations against the observations using the quality indicator shown in Fig. 9. The quality indicator consists of three parts: first in the middle, the green color shows the area of the most likely values of the system response. The full red color identifies the theoretical zone of an unlikely system response. Outside the most likely values zone, the green color zone in the quality indicator starts to fade and convert into the red one. This transition zone can be called an alarm zone, because it indicates that the actual ground condition is gradually moving away from the assumed conditions invoked in the RS analysis.

The results generated by RS-FDM along with the quality indicator defined in this manner can be considered as a useful tool for decision making. In addition, when the measured horizontal displacements of the excavated wall are within the acceptable predicted range (i.e., green zone) one can assume (at least from a practical point of view) that the design of the support elements are also reliable.

Strictly speaking if the numerical model is appropriate and the parameters chosen cover the “true” uncertainty, in-situ measured values must fall inside the range of RS-FDM results, if not, this would be a clear indication that either the model itself is not appropriate or the range of parameters is not representative for the ground conditions. Figure 9 indicates that the predicted horizontal displacements at top of excavation lies within the range of the most likely values in stages 5, 6, and 7, while it is located near the alarm zone in stages 3 and 4. It can be concluded that the soil parameters in the upper half of the second layer are overestimated and it was better to divide this layer into two weak and strong separate layers. In this manner, the uncertainty of soil parameters could be captured more reasonably and hence the reliability of the design would have been improved.

Generally, the most likely values are defined as values with the highest probability of occurrence, where the slope of the corresponding cumulative distribution function is steepest. For the purpose of simplification, it is assumed that the most likely results are those values, whose measure of their belief degree are less than 50% and their corresponding plausible likelihood of occurrence are larger than 50% as shown in Fig. 10. For instance, the most likely horizontal movement at top of excavation in final stage is in a range from 9.5 to 32.6 mm.

The other definition is presented by Tonon et al. [²⁴]. The mean value of the true system response obtained by random set bounds is within the following range given by Tonon et al. [²⁴]:

(24)\mu =[{{\displaystyle \Sigma }}_{i=\mathrm{1}}^N{{\displaystyle m}}_i.\inf ({{\displaystyle A}}_i):{{\displaystyle \Sigma }}_{i=\mathrm{1}}^N{{\displaystyle m}}_i.\sup ({{\displaystyle A}}_i)],

where inf(A) and sup(A) denote lower and upper extremes of focal element A, respectively. The intervals obtained from both the most likely range definition and those calculated from Eq. (24), indicate a good conformity and they have been tabulated in Table 8.

Figure 11 shows the most likely value ranges obtained by RS-FDM and the allowable and measured horizontal displacements for comparison purposes. It can be observed that even the worst condition of the design leads to the displacements which are near the allowable displacements. At this stage it can be deducted that such an analysis can be used to change the configuration of the wall (i.e., length, spacing and specifications of structural elements) before the beginning of the next stage of construction, in order to achieve a more cost effective design.

In order to see how the calculated range of results fit into data published in the literature, maximum horizontal wall displacements are evaluated and compared to the database of some 300 case histories of wall and ground movements due to deep excavations worldwide presented by Long [⁴⁸]. Figure 12 depicts normalized maximum lateral wall movement vs. wall height (dh_max/H vs. H). Long [⁴⁸] reported that cases with a value of dh_max/H>0.3% are likely to experience problems. The values of the most likely range of the normalized maximum lateral wall movement for the final excavation step for the analysis presented here are included in the diagram and it can be seen that the upper and lower value does not exceeds the threshold value of 0.3% which is in line with the probabilities presented earlier.

Conclusions

In uncertainty and reliability analysis of excavation problem involved in this work, the random set approach was selected as a non-deterministic non-probabilistic method. Since at the beginning of the excavation projects usually there is a lack of sufficient information and thus a probability distribution for input parameters is not available, the required data are derived from different independent sources which are to be combined. This data also suffers from ambiguity and imprecision. According to these conditions, previous researchers had already recognized RS as an appropriate mathematical framework to deal with such kind of information. When a large number of uncertain variables exists and each are estimated from several sources (numerous focal elements), the computational effort of the random set approach increases exponentially, which is not favorable. However, it is fortunately not the case in the majority of excavation projects and insignificant input variables are identified in order to reduce the number of the underlying uncertain variables by performing a sensitivity analysis scheme.

This work demonstrated the merits of applying the Random Set approach as a beneficial tool in the context of engineering judgment and decision making. For instance: 1) RS-FDM results simplify the interpretation of the numerical results against measurements which leads to improved safety of the project, 2) the RS-FDM framework can be placed within the observational method design procedure as a supplementary analyzing tool, and 3) spatial correlation is taken into account in a more rigorous way.

From a practical point of view RS-FDM has provided a simple framework to predict the system response within a range. This range is in the form of a p-box which incorporates imprecise probabilities concepts. Working with a range of probabilities seems to be more acceptable for geotechnical engineers in practice. Furthermore, employing the RS-FDM framework helps increasing the credibility of numerical analysis results since its computational results, shown in a real project, could successfully capture the range of the ground behavior.

In the case of abundant sources of information (many focal elements) and a large number of uncertain basic variables, a rather high computational effort is indispensable in RS-FDM. In this case, a suitable code was written facilitate the simulation procedure by omitting the need to run several deterministic models one by one.

The proposed approach offers an alternative way of analysis when insufficient information is available for treating the problem by classical probabilistic methods. The applicability of the RS method for solving practical boundary value problems has been shown by analyzing the excavation for a deep excavation in cemented soil of Tehran alluvium, presenting comparison with field measurements.

It was observed that there could be great saving in project cost, where there was a notable difference between the upper bound of the response (i.e., worst scenario) and the allowable horizontal displacements. In addition, by monitoring the measured values and the most likely values obtained by RS-FDM, the engineer could introduce the uncertainty of input parameters into the model more satisfactorily by modifying the input parameters, thus obtaining more reliable outputs (i.e., probability of unsatisfactory performance and most likely values) in RS analysis and hence improving the safety of design.

References

[1] Lacasse S, Nadim F. Uncertainties in characterizing soil properties. In: Shackelford C D, Nelson P P, Roth M J S, eds. Proceedings of Conference on Uncertainty in the Geologic Environment: From Theory to Practice (Uncertainty ‘96). American Society of Civil Engineers, 1996, 49–75

[2] Griffiths D V, Fenton G A. Influence of soil strength spatial variability on the stability of an undrained clay slope by finite elements. Geo-Denver Conf. on Soil Stability, Denver, Colorado, United States, 2000, 184–193

[3] Fenton G A, Griffiths D V. Bearing capacity of spatially random c–ϕ soils. In: Proceedings of the International Conference on Computer Methods and Advances in Geomechanics, Rotterdam, Balkema, 2001, 1411–1415

[4] Ang A H S, Tang W H. Probability Concepts in Engineering Planning and Design: Volume I-Basic Principles. New York: Wiley & Sons, 1975

[5] Harr M E. Reliability Based Design in Civil Engineering. New York: Dover publications, 1996

[6] Haldar A, Mahadevan S. Probability, Reliability, and Statistical Methods in Engineering Design. New York: Wiley & Sons, 2000

[7] Vu-Bac N, Lahmer T, Keitel H, Zhao J, Zhuang X, Rabczuk T. Stochastic predictions of bulk properties of amorphous polyethylene based on molecular dynamics simulations. Mechanics of Materials, 2014a, 68: 70–84

[8] Vu-Bac N, Lahmer T, Zhang Y, Zhuang X, Rabczuk T. Stochastic predictions of interfacial characteristic of polymeric nanocomposites (PNCs). Composites. Part B, Engineering, 2014b, 59: 80–95

[9] Vu-Bac N, Silani M, Lahmer T, Zhuang X, Rabczuk T. A unified framework for stochastic predictions of mechanical properties of polymeric nanocomposites. Computational Materials Science, 2015, 96: 520–535

[10] Rackwitz R. Reviewing probabilistic soils modeling. Computers and Geotechnics, 2000, 26(3–4): 199–223

[11] Phoon K, Kulhawy F H. Characterization of geotechnical variability. Canadian Geotechnical Journal, 1999, 36(4): 612–624

[12] Duncan J M. Factors of safety and reliability in geotechnical engineering. Journal of Geotechnical and Geoenvironmental Engineering, 2000, 126(4): 307–316

[13] Lee I K, White W, Ingles O G. Geotechnical Engineering. Boston: Pitman, 1983

[14] Kulhway F H. On the evaluation of static soil properties. In: Proceedings of a Specialty ASCE Conference on Stability and Performance of Slopes and Embankments-II, Volume 1. Berkley, California, 1992, 95–115

[15] Peschl G M, Schweiger H F. Reliability analysis in geotechnics with finite elements – comparison of probabilistic, stochastic and fuzzy set methods. In: Proceedings of the 3rd International Symposium on Imprecise Probabilities and Their Applications (ISIPTA_03). Canada: Carleton Scientific, 2003, 437–451

[16] Ben-Haim Y, Elishakoff I. Convex Models of Uncertainty in Applied Mechanics. Amsterdam: Elsevier, 1990

[17] Oberguggenberger M, Fellin W. From probability to fuzzy sets: the struggle for meaning in geotechnical risk assessment. In: Proceedings of the International Conference on Probabilistics in Geotechnics – Technical and Economical Risk Estimation, Essen: VGE, 2002, 29–38

[18] Peschl G M. Reliability Analyses in Geotechnics with the Random Set Finite Element Method. Dissertation for PhD degree. Graz University of Technology, 2004

[19] Dubois D, Prade H. Consonant approximation of belief functions. International Journal of Approximate Reasoning, 1990, 4(5–6): 419–449

[20] Dubois D, Prade H. Random sets and fuzzy interval analysis. Fuzzy Sets and Systems, 1991, 42(1): 87–101

[21] Walley P. Statistical Reasoning with Imprecise Probabilities. London: Chapman and Hall, 1991

[22] Dempster A P. Upper and lower probabilities induced by a multi-valued mapping. Annals of Mathematical Statistics, 1967, 38(2): 325–339

[23] Shafer G. A Mathematical Theory of Evidence. Princeton: Princeton University Press, 1976

[24] Tonon F, Bernardini A, Mammino A. Determination of parameters range in rock engineering by means of Random Ret Theory. Reliability Engineering & System Safety, 2000, 70(3): 241–261

[25] Dong W, Shah H C. Vertex method for computing functions of fuzzy variables. Fuzzy Sets and Systems, 1987, 24(1): 65–78

[26] Cornell C A. First-order uncertainty analysis of soils deformation and stability. In: Proceedings of the International Conference on Applications of Statistics and Probability (ICASP1), 1971, 129–144

[27] Alonso E E, Krizek R J. Stochastic formulation of soil profiles. In: Proceedings of the International Conference on Applications of Statistics and Probability (ICASP2). Aachen, Germany, 1975, 9–32

[28] Lumb P. Spatial variability of soil properties. In: Proceedings of the 2nd International Conference on the Application of Statistics and Probability to Soil and Structural Engineering (ICASP). Aachen, Germany, 1975, 397–421

[29] Vanmarcke E H. Probabilistic modeling of soil profiles. Journal of the Geotechnical Engineering Division, ASCE, 1977, 103: 1227–1246

[30] Vanmarcke E H. Random Fields – Analysis and Synthesis. Cambridge, MA: MIT-Press, 1983

[31] Itasca. FLAC2D (Fast Lagrangian Analysis of Continua) version 7.0 user’s manual. Minneapolis, MN, USA: Itasca Consulting Group, 2011

[32] Christian J T, Ladd C C, Baecher G B. Reliability applied to slope stability analysis. Journal of Geotechnical Engineering 1994, 120(12): 2180–2207

[33] Rieben E H. Geological observation on alluvial deposits in northern Iran. Geological Organization of Iran, Report 9, 1966 (in French)

[34] Duncan J M, Byrne P M, Wong K S, Marby P. Strength, stress-strain and bulk modulus parameters for finite element analysis of stresses and movements in soil mass. Rep. No. UCB/GT/80-01, University of California, Berkeley, California, 1980

[35] Seed R B, Duncan J M. SSCOMP: a finite element analysis program for evaluation of soil structure interaction and compaction effects. Report No. UCB/GT/84-02, University of California, Berkeley, California, 1984

[36] Briaud J L, Lim Y. Tieback walls in sand: numerical simulations and design implications. Journal of Geotechnical and Geoenvironmental Engineering, 1999, 125(2): 101–110

[37] Lim Y, Briaud J L. Three dimensional nonlinear finite element analysis of tieback walls and of soil nailed walls under piled bridge abutment. Report to the Federal Highway Administration and the Texas Department of Transportation, Department of Civil Engineering, Texas A&M University, College Station, Texas, 1996

[38] Dunlop P, Duncan J M. Development of failure around excavated slopes. Journal of the Soil Mechanics and Foundations Division, ASCE, 1970, 96: 471–493

[39] Finno R J, Blackburn J T, Roboski J F. Three dimensional effects for supported excavations in clay. Journal of Geotechnical and Geoenvironmental Engineering, 2007, 133(1): 30–36

[40] U.S. EPA. TRIM FaTE Technical Support Document Volume I: Description of Module. EPA/43/D-99/002A, Office of Air Quality Planning and Standards, 1999

[41] Vu-Bac N, Rafiee R, Zhuang X, Lahmer T, Rabczuk T. Uncertainty quantification for multiscale modeling of polymer nanocomposites with correlated parameters. Composites. Part B, Engineering, 2015, 68: 446–464

[42] Vu-Bac N, Lahmer T, Zhuang X, Nguyen-Thoi T, Rabczuk T. A software framework for probabilistic sensitivity analysis for computationally expensive models. Advances in Engineering Software, 2016, 100: 19–31

[43] Baecher G B. Geotechnical Error Analysis. Special Summer Course 1.60s. Cambridge: Massachusetts Institute of Technology, 1985

[44] Kulhawy F H, Birgisson B, Grigoriu M D. Reliability based foundation design for transmission line structures: transformation models for in situ tests. Electric Power Research Institute, Palo Alto, California: Report EL-5507(4), 1992

[45] Li K S, White W. Reliability index of slopes. In: Proceedings of 5th International Conference on Application of Statistics and Probability in Soil and Structural Engineering, 1987, (2): 755–762

[46] Federal Highway Administration (FHWA). Soil nail walls reference manual. U.S. Department of Transportation Federal Highway Administration, 2015

[47] US Army Corps of Engineers (USACE). Introduction to probability and reliability methods for use in geotechnical engineering. Engineering Technical Letter No. 1110-2-547, 1997

[48] Long M. Database for retaining wall and ground movements due to deep excavations. Journal of Geotechnical and Geoenvironmental Engineering, 2001, 127(3): 203–224

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