Stochastic Modeling of Redundancy in Mechanically Stabilized Earth (MSE) Walls

Stochastic Modeling of Redundancy in Mechanically Stabilized Earth (MSE) Walls

Ioannis E. Zevgolis1, S.M. ASCE and Philippe L. Bourdeau2, M. ASCE

1 Geotechnical Engineer, PhD; 7-9 Gavriilidou Str., Athens, 11141, Greece (formerly: PhD Student, Purdue University, School of Civil Engineering); [email protected] 2 Assoc. Professor; Purdue University, School of Civil Engineering, 550 Stadium Mall Dr., West Lafayette, IN, 47907-2051, USA; [email protected] ABSTRACT: A probabilistic model is developed in order to assess the reliability of the internal stability of MSE walls. Geotechnical uncertainty is explicitly considered by modeling shear strength properties of reinforced, retained, and foundation soil as random variables following beta distributions. The model is developed in two steps. First, an in-series configuration addresses the reliability per reinforcement layer and provides a profile of reliability with depth. Second, an r-out-of-m configuration is used to model the inherent redundancy of MSE structures. Reliability analyses are performed using Monte Carlo simulations and propagation of failure is modeled using transition probabilities and Markov stochastic processes. As an illustration, a case example of an MSE wall used as direct bridge abutment is analyzed in the context of the developed model. INTRODUCTION

Reliability analysis of mechanically stabilized earth (MSE) walls requires consideration of several failure mechanisms and how these are affected by various sources of uncertainty. In conventional design methods, the reinforced soil mass is assumed to behave as a rigid body in terms of external stability and safety assessment is performed with respect to sliding, bearing capacity, and excessive eccentricity. In terms of internal stability, MSE walls are analyzed with respect to tensile and pull out resistance of the reinforcement elements (Elias et al. 2001; AASHTO 2002). Particularly with respect to internal stability, MSE walls are inherently redundant systems. This means that if a reinforcement element fails, the remaining elements are expected to assume additional responsibility so that the system is still in a position to continue its operation, even if not as initially intended. So, the reliability of the internal stability of an MSE wall is a function of its redundancy, although commonly accepted simplifications ignore this aspect. Provided some modeling simplifications are accepted, stochastic models offer the framework to determine not only the

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original reliability per layer of reinforcement of an MSE wall, but also the updated reliability given that one or more layers have already failed. In this paper, the development of such a model is described. Internal stability is approached in two steps using Monte Carlo simulations. The framework for consideration of redundancy and propagation of failure is formulated based on transition probabilities and Markov stochastic processes. As an illustration, an example case of an MSE wall used as direct bridge abutment is analyzed in the context of the developed model. FORMULATION OF THE PROBABILISTIC MODEL Performance functions In contrast to deterministic approaches, in probabilistic analysis of MSE walls, sources of uncertainty are characterized and explicitly accounted for in the computation of reliability. In order to represent limit states of equilibrium, it is convenient to define performance functions by analogy with safety factors, as Safety Ratios (SR). So, safety ratios with respect to tensile and pull out failure of each reinforcement layer i, [SRTi] and [SRPOi] respectively, are expressed by:

max,

⎡ ⎤ =⎣ ⎦i

YT

i

FSRT

for i = 1, 2, … , m (1)

,

max,

⎡ ⎤ =⎣ ⎦i

R iPO

i

PSR

T for i = 1, 2, … , m (2)

where m is the number of reinforcement layers, FY is the tensile strength of the reinforcement elements, Tmax,i is the maximum tensile force applied on each reinforcement layer i, and PR,i is the pull out resistance of each reinforcement layer i. Definitions of FY, Tmax,i and PR,i are provided in specifications and guidelines (Elias et al. 2001; AASHTO 2002). Failure is defined as the case where the corresponding SR is less than one. Then, the probability of failure (PF) for any mechanism is given:

[ ] 1FP P SR= < for any mechanism (3) Assessment of reliability Each reinforcement layer is modeled as an in-series system. This means that if the layer fails either in tension or in pull out, then that specific layer does not contribute anymore to the internal stability of the structure. So, failure of an individual layer i is the event in which and/or 1

iTSR⎡ <⎣ ⎤⎦ 1iPOSR⎡ ⎤<⎣ ⎦ occurs. The probability of

occurrence of the failure ( ) is given by the union of these two events: , iF INTP

( ) (, 1 i i iF INT T POP P SR SR⎡= < ∪ <⎣ )1 ⎤

⎦ (4)

GEOCONGRESS 2008: GEOSUSTAINABILITY AND GEOHAZARD MITIGATION 1180


where [ ]P − denotes the probability of the event indicated within the brackets. Since

the events and 1iTSR⎡ ⎤<⎣ ⎦ 1

iPOSR⎡ ⎤<⎣ ⎦ are not mutually exclusive (nor independent), the above can be written as following:

( ) ( ), 1 1 - 1 1i i i i iF INT T PO T POP P SR P SR P SR SR⎡ ⎤⎡ ⎤ ⎡ ⎤= < + < < ∩ <⎣ ⎦ ⎣ ⎦ ⎣ ⎦ (5)

The components of the above equation are given by:

,1 i

i

F TT

nP SR

N⎡ ⎤< =⎣ ⎦ (6)

,1 i

i

F POPO

nP SR

N⎡ ⎤< =⎣ ⎦ (7)

( ) ( ) , -1 1 i i

i i

F T POT PO

nP SR SR

N⎡ ⎤< ∩ < =⎣ ⎦ (8)

where and are the number of Monte Carlo realizations in which the safety ratio of the respective mode is less than one, is the number of times in which the two safety ratios are simultaneously less than one, and N is the total number of Monte Carlo realizations. Once the probabilities of failure per layer ( ) have been found for all layers, then a profile of probability of failure with depth can be obtained. Such a profile would indicate which reinforcement layers are subjected to higher risk of failure.

, iF Tn , iF POn

, iF T POn − i

, iF INTP

CONSIDERATION OF REDUNDANCY As already explained in the introduction, MSE walls are characterized by redundancy. Therefore, it is of interest to determine not only the original reliability per layer of reinforcement, but also the reliability of a layer given that another layer has already failed. An appropriate model for doing so would be an r-out-of-m system. This model refers to a system of m components, r of which must be operable for the system to survive (Ang and Tang 1984; Harr 1987). In the case of an MSE wall, m is the total number of reinforcement layers and r is the number of reinforcement layers that shall not fail in order for the structure to remain in operation. The updated profile of reliability with depth will be assessed using transition probabilities and Markov stochastic models. Markov processes are stochastic processes that can be used in probabilistic modeling of systems that satisfy the following criterion: the transition of the system from one state to another depends only on the current state of the system and not on the previous states. In Markovian models, the probability of transition from one state i to another state j is called transition probability, and will be denoted herein as pi→j.



For a system that has (m+1) possible states, the array of all transition probabilities can be written in the form of an (m+1)×(m+1) matrix, called transition probability matrix Π. Details about the theory of Markov models can be found in classical textbooks (Benjamin and Cornell 1970; Ang and Tang 1984; Harr 1987). So, a fundamental step in Markovian models is to clearly define all possible states that the system can move from, and towards to, during its lifetime. Considering an MSE structure with m reinforcement layers, the following states (herein called “states of failure”) are defined:

State of failure 0 All layers are intact State of failure 1 One layer has already failed State of failure 2 Two layers have already failed … … State of failure i i layers have already failed … … State of failure m-1 m-1 layers have already failed State of failure m All layers have failed

where states of failure 0 and m are the original and final states, respectively. Taking into consideration the nature of an MSE structure, the following statements can be made for the model: · The model does not allow for reverse of failure. In other words, having reached a

certain state i, the system cannot return to i-1, i-2, … states. · When failure occurs at a certain state i, it may or may not propagate to a

following state j. If the system remains in its present state, then this is called an absorbing state.

· The model is free from restrictions of continuous propagation of failure. This means that when failure propagates, it may do so to any of the remaining states.

The transition probability matrix takes the following form (Zevgolis 2007):

0 0 0 1 0 0 0

1 1 1 1 1

... ... ...0 ... ... ...

0 0 ... ... ...

0 0 ... 0 ... ...

0 0 ... 0 ... 0 ... 1

i j

i j

i i i j i m

j j j m

p p p p pp p p p

p p p

p p

→ → → → →

→ → →

→ →

→ →

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥Π = ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

m

m→

→ (9)

The above is an upper triangular matrix, with ( )( )2 1m m+ + 2 non-zero elements. The following equations hold true for this matrix: 0 1i jp →≤ ≤ for any i and j (10)



1

1m

i jj

p →=

=∑ for any i (11)

0i jp → = if i > j (12)

1m mp → = (13)

In order to determine the transition probabilities that compose the matrix Π, an iterative process is followed (Zevgolis 2007). In the first iteration, Monte Carlo simulation is performed considering that all reinforcement layers are in place. This is the state of failure 0. is calculated based on equations 5 to 8, and the corresponding profile of probability of failure is obtained. In addition, the number of realizations where simultaneous failures of different layers occur, is counted. The number of realizations of simultaneous failure of 0 out of m layers, corresponds to state of failure 0, the number of realizations of simultaneous failure of 1 out of m layers, corresponds to state of failure 1, and so on. Finally, the number of realizations of simultaneous failure of m out of m layers, corresponds to state of failure m. Then the probability of transition from state of failure 0 to state of failure i (p

, iF INTP

0→i) is given by:

,00 F i

i

np

N→

→ = for i = 0, 1, 2, …, m (14)

where nF,0→i is the number of realizations of simultaneous failures of i out of m layers, and N is the total number of realizations in the Monte Carlo simulation. The array of transition probabilities of equation 14 can now be written in the form of a 1×(m+1) row matrix Π0 as following:

[ ]0 0 0 0 1 0 0 ... ...i mp p p p→ → → →Π = (15) The subscript 0 means that refers to the state of failure 0 as the initial state for this simulation. In the second iteration, Monte Carlo simulation is performed again, this time considering that one reinforcement layer has already failed (so, is obtained) . The iterative process is repeated m+1 times in total. Every time the failing layer is assumed to be the most critical one. Assembling the row matrices Π

0Π

1Π

0, Π1, …, Πm into one matrix, gives us the transition probability matrix of equation 9. CASE EXAMPLE A case example of an MSE wall used as direct bridge abutment is analyzed as an illustration of the developed model (Figure 1).



2.3m

6m

8.3m

2m

7m

γF = 17 kN/m3

265 kN/m

10 kN/m2

5.25 kN/m

0.6mγREINF = 20 kN/m3

γRET = 20 kN/m3

Figure 1. Schematic representation of the analyzed case example The present model includes four random variables (Table 1). Each one is represented by the first two order moments (mean value μ and coefficient of variation C.O.V.) and the minimum and maximum value. Therefore, based on the principle of maximum entropy, the random variables are modeled using beta distributions (Oboni and Bourdeau 1985; Harr 1987). All other properties (such as unit weights, reinforcement characteristics, and loading conditions) are considered deterministic variables with values shown in Figure 1. Note that perfect autocorrelation is assumed over the volume of interest for all four random variables. This simplifying assumption should slightly affect the analysis numerical results but without loss of general validity.

Table 1. Probabilistic parameters of random variables

Soil property Notation μ C.O.V. Min MaxFriction angle of reinforced backfill φREINF (o) 34 0.10 20.4 47.6Friction angle of retained backfill φRET (o) 30 0.15 12 48 Friction angle of foundation soil φF (o) 20 0.20 4 36

Cohesion of foundation soil cF (kN/m2) 40 0.30 0 88 Figure 2 illustrates the variation of the computed standard deviations, σ[SRT] and σ[SRPO], with depth. As shown in the Figure, while the variation of σ[SRT] is relatively constant, this is not the case for σ[SRPO], which demonstrates a continuous increase with depth. This implies that SRPO is subjected to higher degree of uncertainty compared to SRT. In addition, it is shown that the uncertainty of SRPO increases with depth.



0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Standard deviation of safety ratios, σ[−]

16

5

4

3

2

1

0

Dep

th, z

(m)

σ[SRT]

σ[SRPO]

Figure 2. Variation of σ[SRT] and σ[SRPO] with depth Based on the computations, and for the analyzed case example, the top layer of reinforcement was the most critical. This was reasonable, considering that this layer is located below the bridge seat. So, failure was assumed to initiate from that one. This may not always be the case, particularly in conventional (non-abutments) MSE walls. Following the iterative procedure that was described earlier, different profiles of probabilities of failure corresponding to different states of failure, were obtained. These are shown below, in terms of the transition probability matrix, for the top two layers:

0.92935 0.06423 0.00560 0.00028 0.00033 0.00007 0.00008 0.00002 0.00003 0 00 0.83458 0.12525 0.03195 0.00625 0.00030 0.00013 0.00145 0.00008 0 00 0 0.54998 0.29860 0.11537 0.01928 0.00815 0.00827 0.00035 0 00 0 0 ... ... ... ... ... ... ... ...0 0 0 0 ... ... ... ... ... ... ...0 0 0 0 0 ... ... ... ... ... ...0 0 0 0 0 0 ... ... ... ... ...0 0 0 0 0 0 0 ... ... ... ...0 0 0 0 0 0 0 0 ... ... ...0 0 0 0 0 0 0 0 0 ... ...0 0 0 0 0 0 0 0 0 0

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦1

In this example, the number of reinforcement layers is m = 10, so the transition matrix is an (m+1)×(m+1) = 11×11 matrix. So for instance, the element

corresponds to the following transition probability: given that the top 2 layers have already failed (state of failure 2), the probability that there will not be any further propagation of failure is 54.998 %. The probability that there will be

3 3 0.54998→ =p



propagation to the third from the top layer is 29.860 % ( 3 4 0.29860→ =p ), to the fourth from the top is 11.537 % ( 3 5 0.11537→ =p ), and so on. CONCLUSIONS

MSE walls are often characterized by redundancy. This means that in the event of failure of one or more layers of reinforcement, the wall does not necessarily collapse, because the remaining layers assume additional responsibility in terms of loads. This is typically an aspect that is ignored by previous models. In this paper, a stochastic model was developed in order to assess the reliability of MSE walls with respect to their internal stability, taking into account the redundant nature of this type of structures. So, using an r-out-of-m configuration the proposed model is able to determine not only the original reliability per layer of reinforcement, but also the updated reliability given that one or more layers have already failed. Propagation from one state of failure to another were modeled using transition probabilities and Markov stochastic models. To illustrate the developed methodology, a case example of an MSE wall used as direct bridge abutment was analyzed. For this example, the results indicate that the mechanism subjected to higher risk of failure is the one in pull out. In addition, the transition probability matrix provides the probabilities of failure propagation for three different states of failure. ACKNOWLEDGEMENTS Financial support provided by the Empirikion Foundation to the first author during his doctoral studies is greatly appreciated. REFERENCES AASHTO. (2002). "Standard Specifications for Highway Bridges." American

Association of State Highway and Transportation Officials, 17th edition, Washington D.C., USA.

Ang, A. H.-S., and Tang, W. H. (1984). Probability Concepts in Engineering Planning and Design. Volume II: Decision, Risk and Reliability. John Wiley & Sons.

Benjamin, J. R., and Cornell, C. A. (1970). Probability, Statistics, and Decision for Civil Engineers. McGraw-Hill.

Elias, V., Christopher, B. R., and Berg, R. R. (2001). "Mechanically stabilized earth walls and reinforced soil slopes – Design & construction guidelines." FHWA-NHI-00-043, US Department of Transportation, Federal Highway Administration, Washington D.C., USA.

Harr, M. E. (1987). Reliability-Based Design in Civil Engineering. McGraw-Hill. Oboni, F., and Bourdeau, P. L. (1985). "Simplified use of the beta distribution and

sensitivity to the bound locations." Structural Safety, 3(1), 63-66. Zevgolis, I. E. (2007). "Numerical and Probabilistic Analysis of Reinforced Soil

Structures." PhD Dissertation, Purdue University, West Lafayette, IN, USA.



Stochastic Modeling of Redundancy in Mechanically Stabilized Earth (MSE) Walls

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