Numerical Modeling in Geomechanics

Conference on Numerical Modeling in Geomechanics CoNMiG-2017 Geotechnical Engineering: Reliability Aspects D K Baidya1 and A GuhaRay2 Professor, Department of Civil Engineering IIT Kharagpur 2 Asst Professor, BITS-Pilani Hyderabad campus, Secunderabad. 1

Abstract: The stability of geotechnical earth structures is often affected if the randomness of geotechnical variables is not incorporated in design. The Factor of Safety (FS) approach holds good only when the input parameters, namely, engineering properties of soil, location of ground water table and loading conditions etc., can be accurately assessed. The concept of reliability analysis is a well-established mathematical approach to account for these uncertainties of field variables. A few investigations highlighted the sensitivity (S) of random variables on different modes of failure. Also different approaches are available to determine the probability of failure (Pf) for variation of random variables. Contrarily, design approach incorporating both these effects has not yet been adequately addressed and more rigorous investigation is needed to examine the combined effects of sensitivity of random variables and failure of various earth structures. Also very little is known from the literature regarding any approach to generalize the “partial factor of safety” based upon variations of different geotechnical random variables. In the present study, a correction factor, named as the probabilistic risk factor (Rf) is formulated for different geotechnical random variables based on their variations. These risk factors reflect both reliability and sensitivity of random variables on failure probability, and at the same time may produce an economic design. Pf is calculated by Monte Carlo simulation and sensitivity analysis of each random variable is carried out by F-test analysis. The structure, redesigned by modifying the original random variables with the risk factors, is expected to perform satisfactorily with all the variations of random variables included into it. The proposed approach is applied to five different earth retaining structures viz. cantilever retaining wall, gravity retaining wall, reinforced retaining wall, sheet pile walls and finite slopes. It is observed that if the variation in some properties, like internal angle of friction and cohesion of soil be properly identified, significant reduction in cost can be achieved. The study also proposes various design charts for different geotechnical earth structures, under both static and dynamic loading conditions, for different variations of random variables, based upon the proposed design approach. This may help to simplify the use of the proposed method for the practicing engineers. Keywords: Geotechnical Random Variables, Reliability Analysis, Probability of Failure, Sensitivity Analysis, Earth Structures, Risk Factors 1. INTRODUCTION Uncertainty is evident in almost every field of engineering, and geotechnical engineering is no exception. Natural soils are heterogeneous and anisotropic due to their composition and complex depositional processes. The uncertainty in geotechnical engineering is mainly attributed to inherent or spatial variability, limited number of samples, testing and measurement errors, and the analytical models which relate the laboratory or in-situ property in terms of stability and deformation behavior of soil. D K Baidya and A GuhaRay

Conference on Numerical Modeling in Geomechanics CoNMiG-2017 The increasing frequency of landslides and failures of earth structures and their adverse impact have led the geotechnical engineers to recognize the importance of probabilistic approaches for analysis of geotechnical structures. Conventional design of geotechnical structures is based on limit equilibrium methods (LEM) and on the concept of Factor of Safety (FS). In Working Stress Design method, engineering design is mostly dependent on assigning a single global factor of safety (FSg) either to the material properties or to the applied loads. In Limit State Design approach, separate partial safety factors (FSp) are assigned for loads and resistances. Factor of Safety, which is generally used in conventional design procedures to include implicitly all the sources of uncertainty in geotechnical property evaluation, does not truly account for involved risk. This method holds good only when the input parameters, namely, engineering properties of soil, location of ground water table and loading conditions etc., required for design can be accurately assessed. But variation of site data from the estimated value is more common than the exception. The concept of reliability analysis is a well-established mathematical approach to account for these uncertainties of field variables. A number of approaches have been developed for assessing reliability of geotechnical structures in terms of reliability index (β) and probability of failure (Pf). Casagrande's well known Terzaghi Lecture (1965) was on “calculated risk” where the author proposed that very careful consideration of risk in geotechnical studies was quite necessary. Harr (1984), Kulhawy (1992), Lacasse and Nadim (1997), Duncan (2000) demonstrated the importance of soil variability on stability assessment of different geotechnical structures. In the area of earth retaining walls, researchers focused on cantilever, gravity and reinforced earth walls. These subjects were addressed by Ang and Tang (1984), Basha and Babu (2009, 2010a, 2010b, 2014), Biernatowski and Puła (1988), Castillo et al. (2004), Chalermyanont and Benson (2004, 2005), Hoeg and Murarka (1974), Zevgolis and Bourdeau (2010). In the area of sheet pile walls, research on soil deformation and stability by Basma (1990), Becker (1996a, 1996b), Cherubini (2000), Nataraj and Hoadley (1984), Pane and Tamagnini (1997), Phoon et al. (2003), Rowe (1955), Sandford (1992) were reported. For slope stability, in particular, there are a number of publications which dealt with advanced applications such as reliability associated with progressive failure modes and system reliability (Bhattacharya et al., 2003; Ching et al., 2009; Christian et al., 1994; Griffiths et al., 2009; Li and Lumb, 1974; Navarro et al., 2010). A few approaches for calculating sensitivity of random variables on overall failures of structures have also been reported in literature (Andres, 1997; Frey and Patil, 2002; Helton and Davis, 2002; Oakley and O’Hagan, 2004; Saltelli et al., 1999). In geotechnical engineering, the differential analysis method for sensitivity analysis has been applied (Babu and Basha, 2006, 2008a, 2008b) to estimate the sensitivity of geotechnical random variables on failure probabilities of different structures. Though significant advancement took place in application of reliability theory to geotechnical engineering during last few decades, still there is ample scope for improvement. The literature recommends the use of partial safety factors for different geotechnical variables instead of applying a lumped factor of safety to the structure as a whole. But taking into account the uncertain nature of soil properties, it may be noted that these partial safety factors may vary D K Baidya and A GuhaRay

Conference on Numerical Modeling in Geomechanics CoNMiG-2017 depending upon the amount of uncertainties, and hence may lead to under-estimation or overestimation of various parameters/ element dimensions. Very little is known from the literature regarding any approach to generalize this “partial factor of safety” based upon variations of different geotechnical random variables. Past research works also highlight a few studies for calculating the severity of random variables on different modes of failure. Also different approaches are available to determine the failure probability (Pf) for variation of the random variables. Contrarily, design approach incorporating both the effects has not yet been adequately addressed and more rigorous investigation is needed to examine the combined effects of sensitivity of random variables and failure of various earth structures. Also, design charts considering the combined effects of these factors may be given due consideration for the benefit of the practitioners. Hence, the broad objective of the present study is to develop an approach which will reflect both reliability, and sensitivity of random variables on failure probability, and at the same time produce an economic design. The study aims to design a risk factor for different geotechnical random variables based on their variations. Design charts are also proposed for different geotechnical earth structures, under both static and dynamic loading conditions, for different variations of random variables, based upon the proposed design approach. 2. ASSESSMENT OF UNCERTAINTIES 2.1.Probability Distribution Function for Input Variables From the past literature, it has been seen that many researchers (Duncan, 2000; Harr, 1984; Kulhawy, 1992; Lacasse and Nadim, 1997) suggested coefficient of variations of input geotechnical parameters. These geotechnical parameters were modeled by distribution functions, the most common being those functions having the exponential form such as Normal, Lognormal, Exponential and Gamma distribution, or the non-exponential form such as the Beta distribution. From Figure 1, it is evident that if a data set follows normal distribution, it can also be modeled by Beta distribution provided its lower and upper bound be determined appropriately, whereas the Log normal distribution curve tends to differ to some extent from the histogram. Hence it is important to use judgment for the choice of a distribution for a geotechnical parameter (Duncan, 2000). In absence of a large number of geotechnical data, one may opt for the exponential distribution forms. Many investigators (Duncan, 2000; Harr, 1984; Kulhawy, 1992; Lacasse and Nadim, 1997) have suggested coefficient of variation of a number of geotechnical parameters and in-situ tests, which are summarized in Table 1. It is quite evident from Table 1 that such a high variation of geotechnical parameters associated with mean design parameter cannot be neglected in design.

D K Baidya and A GuhaRay

Conference on Numerical Modeling in Geomechanics CoNMiG-2017

Figure 1: Distribution Fit to Input Parameters

Table 1: Coefficient of Variation (COV) for geotechnical properties and in-situ tests Property or in-situ test results

Coefficient of Variation (COV)

Source

Unit Weight (γ)

3-7%

Harr (1984), Kulhawy (1992)

Buoyant Unit weight (γb)

0-10%

Lacasse and Nadim (1997), Duncan (2000)

Effective Stress Friction Angle (φ′)

2-13%

Harr (1984), Kulhawy (1992)

Undrained Cohesion (c) for clay

20-50%

Lumb (1974), Singh (1971)

Undrained Cohesion (c) for sand

25-30%

Lumb (1974)

Undrained Shear Strength (Su)

13-40%

Harr (1984), Kulhawy (1992), Lacasse and Nadim (1997), Duncan (2000)

Undrained Strength Ratio (Su/σ′v)

5-15%

Lacasse and Nadim (1997), Duncan (2000)

Compression Index (Cc)

10-37%

Harr (1984), Kulhawy (1992), Duncan (2000)

Coefficient of Consolidation (cv)

33-68%

Duncan (2000)

Coefficient of permeability of partly saturated clay (k) SPT blow count (N) D K Baidya and A GuhaRay

130-240% 15-45%

Harr (1984), Benson et al. (1993) Harr (1984), Kulhawy (1992)

Conference on Numerical Modeling in Geomechanics CoNMiG-2017 3. RELIABILITY APPROACH Reliability and probabilistic design methods have attracted increasingly more interest in recent years (Baecher and Christian, 2003; Harr, 1987). The concept of reliability analysis is explained with the help of Figure 2.

Figure 2:Safe domain and Failure domain in 3D state space If all loads are represented by the variable Q, total resistance by R and the joint probability function by fRQ, then the space of state variables is a three-dimensional space as shown in Figure 2. Both R and Q are considered to be normally distributed. The margin of safety MisM = R – Q. If both Qand Rare uncertain, M is also uncertain. The mean and standard deviation of M is given by  M   R  Q

 M   R2   Q2  2QR R Q where ρRQis the correlation coefficient between Q and R. The reliability index is then calculated by  

M M

Probability of failure can be calculated by Pf   (  ) where  is the cumulative distribution function for a standard normal variable. Figure 3 illustrates the relationship between reliability index β and probability of failure (Pf) for factor of safety (FS) for normal distribution and lognormal distribution (Baecher and Christian, 2003). USACE (1997) states that for the good performance of the system, β ≥ 3.0. From Figure 3, it is evident that for β2, assumption of a normal distribution may lead to over-estimation of Pf. However, in most cases, a normal distribution is assumed. Methods generally used to evaluate β are (i)

First Order Second Moment Method (FOSM)


Conference on Numerical Modeling in Geomechanics CoNMiG-2017 (ii) (iii) (iv) (v) (vi) (vii)

Mean Value First Order Second Moment Method (MVFOSM) Taylor’s Series Expansion Method Hasofer-Lind Method Point Estimate Method (PEM) Response Surface Methodology (RSM) Monte Carlo Simulation (MCS)

10

Probability of Failure (Pf)

10 10 10 10 10 10

0

Normal Lognormal (COV = 5%) Lognormal (COV = 10%) Lognormal (COV = 15%)

-1

-2

-3

-4

-5

-6

0

0.5

1

1.5

2 2.5 3 Reliability Index (  )

3.5

4

4.5

5

Figure 3: Relationship between β and Pf (Baecher and Christian, 2003)

4. SENSITIVITY ANALYSIS Sensitivity analysis reflects the impact of different sources of uncertainty of the input parameters on the model outputs. Sensitivity can be determined mathematically, statistically or even graphically. Generally, the approaches can be categorized into two main groups – local methods and global methods (Saltelli et al., 1999). A large number of sensitivity analysis methods are available in literature, out of which only a few methods including partial differentiation, partial correlation and regression techniques have been commonly used by investigators. According to Andres (1997), a good sensitivity analysis tool should have the capability to generate repeatable results using a different sample set to evaluate model sensitivities. The effectiveness of a sensitivity analysis method refers to its ability to correctly identify the influential parameters controlling a model’s performance. 4.1.Local Methods The local sensitivity analysis methods aim at evaluating sensitivity at a particular point in the parameter hyperspace. The most frequently used local parameter methods are the nominal range and differential analysis methods (Frey and Patil, 2002; Helton and Davis, 2002). Nominal range D K Baidya and A GuhaRay

Conference on Numerical Modeling in Geomechanics CoNMiG-2017 sensitivity analysis calculates the percentage change of outputs due to the change of model inputs with respect to their nominal values. Differential analysis utilizes partial derivatives of the model outputs Y with respect to an input Y factor Xii.e. where the subscript X0 indicates that the derivative is taken at some fixed X i X 0 point in the space of the input. The differential analysis method for sensitivity analysis has been applied by Babu and Basha (2006, 2008a, 2008b) to estimate the sensitivity of geotechnical random variables on failure probabilities of different structures. These methods are straightforward and easy to implement and they meet modest computational demands. The major drawback of these methods is their inability to take into account parameter interactions, thus under-estimating true model sensitivities. 4.2.Global Methods Global sensitivity analysis methods such as regional sensitivity analysis (Hornberger and Spear, 1981), variance based methods (Saltelli et al., 2000), regression based approaches (Helton and Davis, 2002), and Bayesian sensitivity analysis (Oakley and O’Hagan, 2004) attempts to vary all model input parameters within feasible predefined ranges and potentially accounts for parameter interactions. ANOVA (Analysis of Variance) using iterated fractional factorial design sampling, aims at decomposition of the variance into partial variances, when all the inputs are varying. The method assumes the model response to be normally distributed. The model response variable Y is decomposed into first and second order effects of two factors A and B according to 2-way ANOVA model. Yijk     i   j     ij   ijk

where i and j are the levels of factors A and B respectively, αi and βj are the main effects of i-th and j-th level of A and B respectively, (α x β)ij represents the interaction between A and B, εijk is the error term, reflecting the effects other than those considered in the previous terms. The F-test is used to evaluate the statistical significance of differences in the mean responses among the levels of each parameter or parameter interaction. The F values are calculated for all parameters and parameter interactions. Higher the F values indicate more significant differences are and hence more sensitivity of the parameter or parameter interaction. 5. FORMULATION OF PROBABILISTIC RISK FACTORS Previous literature illustrates a number of methods to find out Pf of a structure. Also sensitivity analysis has been carried out to identify the effect of random variables on these failure modes. But it may be noted that a variable having a high sensitivity value may have little contribution on the total probability of failure of the structure, and vice-versa. This case generally arises when a D K Baidya and A GuhaRay

Conference on Numerical Modeling in Geomechanics CoNMiG-2017 failure mode is highly affected by a little perturbation of a particular random variable, but this failure mode has negligible effect on the overall failure of the structure. In this case, this random variable does not require a high safety factor. Alternatively, a variable may show low sensitivity on a particular failure mode, but if this failure mode contributes significantly to the overall failure, it will be unwise to assign a very low safety factor to this variable. Hence, a methodology is required to analyse and identify: (i) (ii) (iii)

All potential failure modes of a system The effects of these failures on the system How to correct and or mitigate the failures or effects on the system

In risk management, risk is the combination of severity of the damage and the probability that it will occur. One tool often employed in risk management is Failure Mode and Effect analysis (FMEA) (Rausand et al., 2004; Stamatis, 1995) and one of its common approaches is to calculate the Risk Priority Number (RPN) which is defined as RPN= Severity x Occurrence x Detection This concept of RPN may be extended in geotechnical risk management if severity and occurrence can be identified analogous to sensitivity and probability of failure respectively. The third factor (detection) may be neglected in this case because it is quite obvious to assume that the failure will not escape detection. Thus a new factor Probabilistic Risk Factor (Rf)for each random variable may be introduced which will take into account the sensitivity Si or severity of each variable and the Pf of each mode of the structure. For each random variable, it may be defined as the product of the normalized probability of failure (Pf') and normalized sensitivity (S') values and is given by n

R f (i )  1   Pf  ( j )  S (i) j 1

where i represents random variable and j represents number of modes of failure. The original values of the random variables, when modified by these Rf values, yields corrected values of the random variables, which have variations included into them. With the modified soil properties so obtained, the structure is redesigned in an iterative process as shown in the flow chart (Figure 4). The scheme for risk factor based design optimization of geotechnical structures, presented in this flow-chart is coded in a program written in MATLAB 2015a. Finally this process yields the structure, which satisfies all the stability requirements. Hence, it is seen that for a site having less variation of soil properties, it is not necessary to apply a global factor of safety of 1.5 to the entire structure. Instead partial risk factors, designated as probabilistic risk factor (Rf) in this study, may be assigned depending upon site conditions and variations of soil properties, which may prove to be cost effective for the structure. 6. ANALYSIS OF A GRAVITY RETAINING WALL A typical gravity retaining wall, as shown in Figure 5, is considered for the investigation. The geotechnical random variables considered are internal angle of friction of backfill (φ1) and D K Baidya and A GuhaRay

Conference on Numerical Modeling in Geomechanics CoNMiG-2017 foundation soil (φ2), unit weight of backfill (γ1) and foundation soil (γ2), and cohesion of backfill (c1) and foundation soil (c2). Start

Identify Random Variables and soil probabilistic characteristics Design/Model the Structure Deterministic Safety Factors by conventional Limit Equilibrium Method/ Finite Element Method Identify Potential Failure Modes Determine Probability of Failure (Pf) Identify Severity Effect (S) of Random Variables on each Failure Mode by F-Statistics value (ANOVA) Compute Rf = Pf x S Redesign with Reduced Random Variables No

Is Structure Safe? Yes Design Recommendations

End

Figure 4: Risk Factor based optimisation Flow Chart The wall is analysed for base sliding, overturning about its toe, location of resultant of applied loads i.e. eccentricity and foundation bearing capacity failure. The limit equilibrium equations and deterministic FS values for the different failure modes are presented in Table 2.


Conference on Numerical Modeling in Geomechanics CoNMiG-2017 a=0.4

GL φ1 =350 γ1=18kN/m3 c1= 0kN/m2

W5

PA

H = 4.0 W6

W2

δ1 90-α

W1 GL

W3

D = 1.5

t=0.7

α W4

Lt =0.35 bf= 0.5 tan δ2

bb=0.9

Lh=0.7

B =2.75

φ2 =220 γ2=19 kN/m3 c2=30 kN/m2

Figure 5: Cross-sectional view of a Gravity Retaining Wall (Static Analysis) Table 2: Limit Equilibrium Equations and Deterministic Factor of Safety Sliding (FSsli) tan  2  FV 

F

Overturning (FSot)

M M

2 Bc2 3

H

1.501

Eccentricity (FSecc)

R o

3.3

Bearing (FSbc)

B 6e

qult qmax

2.43

3.204

From Table 2, it can be seen that the wall is safe against all modes of failure in deterministic analysis. The minimum FS is found 1.501 for sliding mode of failure which is just safe. The cross-sectional area of the gravity retaining wall required is 6.725m2. For the probabilistic analysis of the retaining wall, the performance functions are defined as gi(x) = (FS)i-1, where i denotes different failure modes. Failure occurs when gi(x)< 0. Analysis of Pf (j) (j=individual failure modes) are carried out by MCS by simulating 30,000 variables in MATLAB 2015a. Table 3 gives the probability of failure for different modes for COV of φ1 = 13%, γ1 = 7%, φ2 = 13%, γ2 = 7%, c2 = 50%. Table 3: Probability of Failure by MCS Modes of Failure Pf


Sliding Overturning Eccentricity Bearing 0.0163

0.000

0.000

0.001

Conference on Numerical Modeling in Geomechanics CoNMiG-2017 For sensitivity analysis, 1σ variation (68.27% of the values lie within 1σ of the mean µ2) i.e. µ1=µ2-1σ and µ3=µ2+1σ of a particular random variable (φ1) is considered, against a large number of simulated values for other 4 input random variables (γ1, γ2, φ2, c2), which are analyzed at their maximum COV values. Sensitivity analysis is carried out for COV of φ1 = 13%, γ1 = 7%, φ2 = 13%, γ2 = 7%, c2 = 50%. Figure 6 shows the relative importance of the random variables based on ANOVA F-statistics.

Figure 6: Relative Importance of Parameters based on ANOVA F-Statistics for different modes of failure From Figure 6, it is evident that variation in φ1 has a significant effect on all the modes of failure, and is the most important and governing parameter for overturning and eccentricity modes of failure. The variation in soil unit weight, γ1 and γ2 has insignificant influence on all the modes of failure. COV in internal friction angle of foundation soil, φ2 is significant on bearing mode of failure and to some extent on sliding mode. It is observed that the effect of variation in c2 on sliding and bearing mode of failure is the maximum. The probability of failure of different modes and sensitivity of the random variables on these modes are coupled mathematically. The original values of the random variables, when modified by these Rf values, yields modified values of the random variables (Table 3), which have variations included into them. Finally the structure is redesigned with the modified soil properties so obtained.


Conference on Numerical Modeling in Geomechanics CoNMiG-2017 Table 3: Risk Factors and Modified Values for Random Variables Parameters φ1(degrees) γ1 (kN/m3) φ2(degrees) γ2 (kN/m3) c2(kN/m2)

Mean Values of Random Variables 35 18 22 19 30

Normalised S x Pf

Rf

Modified Mean Values of Random Variables

0.279674939 0.016958802 0.04873781 4.3946E-05 0.654584502

1.30 1.05 1.05 1.00 1.70

26.93 17.15 20.95 19.00 17.65

Since φ1 is found to be the most sensitive parameter, the variation of Rf of different random variables with variation of φ1, keeping other variables constant, is presented in Figure 7. From Figure 7, the different values of Rf may be chosen depending upon coefficient of variation of φ1. From the figure, it is evident that as COV of φ1increases from 5% to 20%, Rf for φ1 increases from 1.1 to 1.7, while that of γ1 and γ2 remains almost constant. Rf of both φ2 and c2 decreases with increase in COV of φ1, the rate of decrease being much faster in case of c2 (1.95 to 1.30).

Figure 7: Variation of Rf with variation of φ1

From Table 4, it is evident that if the variation in internal angle of the backfill soil φ1 remains within 5%, significant reduction in cross-sectional area (9.22%) may be achieved. But if the variation is more than approx7-8%, the structure needs to be modified.


Conference on Numerical Modeling in Geomechanics CoNMiG-2017 Table 4:Modification of the Structure Dimensions Deterministic (m) 5.5 H 0.35 Lt 0.7 Lh 0.5 bf 0.9 bb 0.3 a 0.7 t 6.725 Area (m2) – % Savings

5% 5.5 0.35 0.7 0.5 0.7 0.3 0.7 6.105 + 9.22%

COV of φ1 10% 13% 5.5 5.5 0.35 0.35 0.7 0.7 0.5 0.6 1.0 1.0 0.3 0.4 0.7 0.7 7.035 7.895 - 4.61% -17.4%

20% 5.5 0.35 0.8 0.75 1.2 0.5 0.7 9.6 -42.75%

A comparative study for the different wall heights are presented in Tables 5. Chen (1995), Das (1999) and Ranjan and Rao (2000) recommended cross-sectional area for different heights of gravity retaining wall based on deterministic analysis. Babu and Basha (2007) proposed crosssectional area based on target reliability based design approach for 5% and 10% COV of φ. It may be noted from Table 5 that the cross-sectional area required for COV of φ = 5% is less than that obtained by deterministic method, while that obtained for COV of φ = 10% is on the higher side. Table 5: Comparative Study of cross-sectional area (m2) for COVφ=5% and 10% Ht of Wall (m)

Chen (1995)

3.0 4.0 5.0

2.12 3.56 7.686

Das (1999)

2.014 3.11 6.27

Ranjan & Rao (2000) 1.94 3.255 6.395

Babu and Basha (2007) COVφ COVφ =5% =10% 2.22 2.51 3.95 4.47 6.74 7.62

Present Study COVφ =5% 1.96 2.93 5.46

COVφ =10% 2.3 3.376 6.86

Based on the present approach, some design recommendations are proposed for design of gravity retaining walls. The design recommendations are: (i) Length of top of stem, a = minimum 0.3m (ii) Thickness of base slab, t = 0.12H (iii) Length of Toe, Lt = 0.06H (iv) Length of Heel, Lh = 0.14H (v) Base width, B = 0.4 - 0.48H for COVφ = 5% and B =0.5 - 0.6H for COVφ=10% For length of front and back part of bottom of stem, bfand bb, there may be a number of combinations which can be made. The combinations should be made such that the resultant falls within middle-third of the base.


Conference on Numerical Modeling in Geomechanics CoNMiG-2017 Hence it may be seen that the area required by the deterministic process is on the higher side for COV of φ= 5% and hence is uneconomic. On the other hand, if COV of φ increases, this crosssectional area is inadequate for the stability of the wall. 7. PROBABILISTIC RISK FACTOR BASED APPROACH APPLIED TO OTHER EARTH RETAINING STRUCTURES The probabilistic risk factor approach is applied to different earth retaining structures for various loading and water table conditions. The results obtained are presented briefly in the following sections. 7.1. Cantilever Retaining Walls The analysis is carried out for four modes of failure of a cantilever retaining wall viz. sliding, overturning, eccentricity and bearing capacity failure. Limit state functions for all modes of failure are developed. The proposed procedure employs Monte Carlo Simulation to obtain the probability of failure. Sensitivity analysis by F-test method is conducted to assess the effect of uncertainties in design parameters on failure of cantilever retaining walls. For the chosen parameters and variability associated with geotechnical properties of the cantilever retaining wall, the internal angle of friction (φ1) of the backfill soil is found to be the most sensitive parameter compared to the other four geotechnical random variables γ1, φ2, γ2 and c2. Sliding and eccentricity modes of failure are found to be the most potential failure modes. In the analysis of the retaining wall, it is evident that overturning failure mode contributed minimum to the global failure probability, although it is most affected by variation of φ1.A range of risk factors and design guidelines for different elements of cantilever retaining wall are proposed based on the present approach. The value of risk factor comes out to be the highest for φ1. As COV of φ1 increases, Rf for φ1 increases, that of γ1 and γ2 remains almost constant (=1), whereas Rf decreases in case of c2. Hence, if the backfill material is well engineered and if the coefficient of variation is less, then an optimal structure can be designed both in terms of economy and safety. Table 6:Comparative Study of cross-sectional area for COVφ1=5% and 10% Ht of Wall (m)

Das (1999)

3.0

1.440

4.0

Bowles (1996)

1.44

Saribas & Erbatur for min. cost (1996) 1.257

Saribas & Erbatur for min. weight (1996) 1.257

2.380

2.380

1.952

5.0

3.550

3.550

6.0

4.950

7.0 8.0

Babu and Basha (2008)

Present Study

COVφ1 =5%

COVφ1 =10%

COVφ1 =5%

COVφ1 =10%

1.341

1.395

1.29

1.29

1.853

1.984

2.080

1.90

1.90

2.868

2.573

2.725

2.875

2.64

2.80

4.950

3.976

3.578

3.564

3.780

3.83

4.07

6.580

6.580

-

-

4.501

4.795

4.86

5.2

8.440

8.440

-

-

5.536

5.920

6.00

6.44


Conference on Numerical Modeling in Geomechanics CoNMiG-2017 7.2. Gravity Retaining Walls under Seismic Loading conditions The study also highlights the importance of the soil-wall friction angle δ in the stability analysis of gravity retaining walls under seismic loading. As δ decreases from 2φ/3to φ/2, Pf of the structure increases significantly, thereby increasing the cross-sectional area required for stability of the retaining wall. It is seen that pseudo-dynamic method gives more realistic results than pseudo-static method of analysis, as the former considers the effect of shear and primary wave velocities and time in the analysis. Hence designs based on pseudo-dynamic methods are rational and give more economic designs. Of course, these issues ignore amplification and damping effects. It is also observed that the factors H/λ and H/η have significant effect on the stability of the retaining wall under earthquake conditions. The minimum value of H/λ required to keep Pf within a desired value depends upon the vertical seismic acceleration coefficient kv. Also for same variation of the random variables, H/λ has a comparatively lesser effect than H/η on Pf of a structure. It is observed that the percentage savings in cross-sectional area in pseudo-static case is significantly greater than that of pseudo-dynamic case. If the variation of internal angle of friction of backfill and foundation soil (φ1 andφ2) is more than approx 7-8%, the structure needs to be modified for safety. Similarly, the dimension of the structure needs to be increased if COV of c2 rises above 20% to maintain stability. A comparative study is also presented, which shows that consideration of variability of random variables may also lead to optimization of the structure and result in considerable economy. The risk factor based approach is applied to a series of 54 retaining walls in Tokyo Metropolitan and Yokohama Municipality. A range of Rf values is proposed corresponding to different seismic accelerations for the retaining walls built in Tokyo Metropolitan and Yokohama Municipality. The generalized design recommendations proposed, based on these Rf values, attempts to redesign the retaining walls based upon different earthquake intensities. 7.3. Sheet Pile Walls A method for obtaining the penetration depth of cantilever sheet pile walls for different soil and water table conditions by probabilistic risk factor based approach is proposed for cantilever and anchored sheet pile walls. Pf is determined by (i) Monte Carlo Simulation by coding the Limit Equilibrium Equations in MATLAB 7.1 and (ii) fitting Response Surface Model to FS obtained by Finite Element Analysis in PLAXIS 2D-V8. Flexibility of the sheet pile wall is incorporated in the Finite Element analysis. From the analysis, the variation in internal friction angle of backfill and foundation soil and cohesion of the foundation soil is found to be the most sensitive parameter for cantilever sheet pile wall. In Case 3, for high variation of c2 (COV> 20%) and for h1/H< 0.5, the embedment depth D becomes so high that it is not feasible to use a cantilever sheet pile walls. For cohesive backfill, variation of c1 has significant effect on the stability of the sheet pile wall. For cantilever sheet pile walls, the required depth of embedment is much less for a cohesive backfill in comparison to a cohesionless backfill. But a cohesionless backfill is generally preferred as a backfill material for a sheet pile wall because it has many advantages such as good drainage, less compressibility and easy compactability. Also, the cohesive strength of the backfill influences the stability of the sheet pile wall more than that of frictional strength of cohesionless soil. D K Baidya and A GuhaRay

Conference on Numerical Modeling in Geomechanics CoNMiG-2017 However, for large variation of φ2, the depth of embedment required for a cohesionless backfill becomes excessively large. As COV of c2 increases, Rf for c2also increases. Rf for φ1 for cohesionless backfill and that of c2 for cohesive backfill decreases with increase in COV of c1. A key observation from the study is that the fluctuation of water table is highly influential in determining the design depth of embedment D of a sheet pile wall, both for cohesionless and cohesive backfill, if c2 shows a high COV value. The minimum depth of embedment is required when the water table is at the ground surface, while this depth increases as the water table drops down from ground surface and it becomes the maximum when water table reaches at the dredge level. The anchor force Ta is found to be highly influential in determining the design penetration depth D and the safety against sliding mode of failure in anchored sheet pile wall. On the other hand, the position of the anchor rod does not significantly contribute to the stability of the sheet pile wall. For anchored sheet pile walls embedded in cohesionless soil, the wall-pile friction angle δ is also found to be important for the stability of the wall. For anchored sheet pile wall embedded into cohesive soil, role of δ is found to be insignificant. 7.4. Reinforced Earth Walls This study describes a reliability-based analysis for internal (pull-out and tensile failure of reinforcements) and external (overturning, sliding and bearing capacity failure) stability of MSE walls under static loading conditions that explicitly accounts for uncertainties in design parameters. The study shows that for the external modes of failure i.e. overturning and bearing, no failure is expected with the variation of φ1 (within expected range). However, sliding mode of failure shows a small Pf for COV of φ1greater than approximately 10%. A sensitivity study was initially conducted to identify the random variables which have maximum influence on Pf. Results of the sensitivity study show that the tensile strength, vertical spacing and length of the reinforcement and internal friction angle of the backfill soil have a significant effect on the probability of internal failure of MSE walls. Taking into account the variations of random variables, Probabilistic Risk Factors are proposed for variation of φ1.It is seen that as COV of φ1 increases, Rf for φ1 increases, while that for the other four variables remain almost constant. It is also observed that as the value of the ultimate tensile strength of the reinforcement Tall increases, the spacing between the layers also increase and accordingly, the length of the reinforcement layers also change. Finally design guidelines for spacing and lengths of reinforcements are provided for different tensile strengths and different variations of φ1. 7.5. Finite Slopes A reliability based design method is applied to finite slopes and probabilistic risk factors for different geotechnical random variables are proposed based on the risk factor based approach. Analysis of a finite slope identifies that probability of failure changes considerably by varying both the friction angle and cohesion of the soil. However, from sensitivity analysis, cohesion is found to be the most important parameter affecting the stability of the soil slope. It may be seen that if variation of c remains within 10%, more economic section can be adopted compared to that adopted based on deterministic approach. But if the variation is more than 10%, modification in the slope section is needed for the safety of the slope.


Conference on Numerical Modeling in Geomechanics CoNMiG-2017 Two case studies are carried out to study the effectiveness of the proposed method. In the first case, the failure in undrained condition of a finite slope in soft, sensitive clay in Porsmossen, Stockholm, Sweden is considered and the effect of variability of soil parameters is investigated. It has been observed that the variation in the measured value of cohesion of the soil is one of the major factors for instigating the failure of the slope and Rf value for cohesion has been proposed to account for these variations. In the second case, the failure of a cut slope in saturated glacial clay in Congress Street, Chicago is analyzed. From analysis, it has been found that the cohesion value of the 4th soil layer is most sensitive towards slope stability. Although it has been observed that the values of cohesion of different layers show considerable variation from their mean values, the analysis by risk factor based approach highlights that these variations are not the major factors for the cause of slope failure. 8. CONCLUSIONS In view of the uncertainties at various stages of geotechnical design process, the deterministic methods of analysis based on factor of safety, without due consideration of the effects of various sources of uncertainty, may not produce realistic results. Researchers have recognized the importance of uncertainties and their dominant role in the design of geotechnical structures, and researchers and engineers have been working together for introducing reliability concepts in current practice as well as in codes. An algorithm to formulate probabilistic risk factors (Rf) for each geotechnical random variable based on their quantity and nature of variations has been presented in the present study. This probabilistic risk factor may be envisioned as analogous to partial safety factors for different geotechnical random variables, based on a probabilistic framework, which will take into account the variations of the random variables. A gravity retaining wall has been analyzed by the probabilistic risk factor based approach under static loading conditions. The probabilistic risk factors (Rf) have been determined for each random variable and introduced in analysis to account for the uncertainty in the design variables as well as to identify the potential failure modes of the structure. The major findings of the study on gravity retaining wall may be highlighted as follows:  For the chosen parameters and variability associated with geotechnical properties of the retaining wall, the internal angle of friction (φ1) of the backfill soil and the cohesion of the foundation soil (c2) have been found to be the most sensitive parameters for static analysis.  Probabilistic Risk Factors (Rf) have been proposed to take into account for the effects of both severity and occurrence. This factor comes out to be 1.7 for φ1with COV=20% in static loading case. The proposed reliability based design approach exhibits promising features and wide applicability for the analysis and design of geotechnical earth structures. It may help to develop confidence to the practicing engineers in handling different types of uncertainties and the proposed methodology may be adopted in practice for safe and economic design of different earth structures.


Conference on Numerical Modeling in Geomechanics CoNMiG-2017 REFERENCES Baecher, G.B. and Christian, J.T. (2003), Reliability and Statistics in Geotechnical Engineering, Wiley, New York.Duncan, J.M. (2000), Factors of Safety and Reliability in Geotechnical Engineering, J. Geotech. Geoenviron. Eng., ASCE, Vol. 126, No. 4, pp. 307-316. GuhaRay, A., Ghosh, S. and Baidya, D.K. (2014) “Risk Factor based Design of Cantilever Retaining Walls”, Geotechnical and Geological Engineering, An International Journal (ISSN: 0960 -3182), Springer, Netherlands; Vol. 32(1): 179-189; DOI 10.1007/s10706-013-9702-y. GuhaRay, A. and Baidya, D.K. (2014) “Partial Safety Factors for Retaining Walls and Slopes under Static Loading: A Reliability based Approach”, Geomechanics and Engineering, An International Journal (ISSN: 2005 -307X), Techno Press, Korea; Vol. 6 (2): 99-115; DOI 10.12989/gae.2014.6.2.099. GuhaRay, A. and Baidya, D.K. (2015) “Reliability based Analysis of Cantilever Sheet Pile Walls backfilled with different soil types using Finite Element Approach”, International Journal of Geomechanics, ASCE, doi 10.1061/(ASCE)GM.1943-5622.0000475, 06015001, pp 1-11. GuhaRay, A. and Baidya, D.K. (2012) “Reliability coupled Sensitivity based Design Approach for Gravity Retaining Walls”, Journal of the Institution of Engineer (India): Series A, (ISSN: 2250 -2149) Springer, Vol. 93(3): 193-201; DOI 10.1007/s40030-013-0023-1. GuhaRay, A. and Baidya, D.K. (2016) “Reliability coupled Sensitivity based Seismic Analysis of Gravity Retaining Wall using Pseudo-Static Approach”, InternationalJournal of Geotechnical and Geoenvironmental Engineering, ASCE doi: 10.1061/(ASCE)GT.1943-5606.0001467, Vol. 142 (6), pp. 04016010 – 1-13. Harr, M. E. (1984), Reliability-based design in civil engineering, 1984 Henry M. Shaw Lecture, Dept. of Civil Engineering, North Carolina State University, Raleigh, N.C. Kulhawy, F.H. (1992), On the evaluation of soil properties, ASCE Geotech. Spec. Publ. No. 31, pp. 95–115. Lacasse, S. and Nadim, F. (1997), Uncertainties in characterizing soil properties, Publ. No. 201, Norwegian Geotechnical Institute, Oslo, Norway, pp. 49–75. Rausand, M. and Hoylan, A. (2004), System Reliability Theorie, Models, Statistical Methods, and Applications, 2nd edn.Wiley Series in Probability and Statistics. Saltelli, A., Tarantola, S., and Chan, K.P.S. (1999), A quantitative model independent method for global sensitivity analysis of model output, Technometrics, Vol. 41, pp. 39–56. Singh, A. (1971), How reliable is the factor of safety in foundation engineering?, First International Conference on Application of Statistics and Probability in Soil and Structural Engineering, Hong Kong, Hong Kong University Press, pp. 389-424. Stamatis, D.H. (1995), Failure Mode and Effect Analysis: FMEA from Theory to Execution, American Society for Quality (ASQ), Milwaukee, Wisconsin. USACE (1997), Engineering and design introduction to probability and reliability methods for use in geotechnical engineering, Engr. Tech. Letter No. 1110-2-547, Department of the Army, Washington, D.C.
