Computers and Geotechnics 37 (2010) 359–373
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Computers and Geotechnics j o u r n a l h o m e p a g e : : w w w . e l s e v i e r . c o m / l o c a t e / c o m p g e o
Probabilistic analysis of retaining walls Ioannis E. Zevgolis a, , Philippe L. Bourdeau b *
a b
Edafomichaniki S.A., Athens, Greece School of Civil Engineering, Purdue University, USA
a r t i c l e
i n f o
Article history: Received 12 October 2008 Received in revised form 7 November 2009 Accepted 8 December 2009 Available online 11 February 2010 Keywords: Retaining structures Reliability analysis Stochastic modeling System reliability Monte Carlo Optimization
a b s t r a c t
A methodology for the probabilistic analysis of reinforced concrete cantilever walls is developed and described in this paper. The wall’s external stability under static conditions is addressed and modeled as a series system with correlated failure modes. Computations of reliability are performed using Monte Carlo simulations for assumed probability distributions of the backfill and foundation material engineering properties. A case example is analyzed based on the described methodology. The results indicated that risk, measured by the system probability of failure, is not a linear function of safety ratios. All three safety ratios in question were positively correlated, with bearing capacity being subjected to higher degree of uncertainty. The degree of correlation was found to have an important effect on the system probability of failure. Considering the width of the base as varying design parameter, the study also showed that first-order reliability bounds, which are often applied in practice, may lead to a noteworthy over- or under-estimation of the design. 2009 Elsevier Ltd. All rights reserved.
1. Introduction
Reinforced concrete cantilever walls are among the most common earth-retaining structures. Their design must satisfy two ma jor requirem requirement ents: s: internal internal stabilit stability, y, which is ensured ensured by sufficien sufficientt resistan resistance ce against against bending bending moments moments and shear shear forces, forces, and externa externall stabilit stability, y, which which means means that, that, except except for small small movem movements ents necessar necessary y to mobilize the earth pressures, the wall must be in equilibrium with with respe respect ct to exter external nal force forcess [51] [51].. Traditionally, Traditionally, conventional conventional design of external stability has been based on deterministic methods and on the concept of Factors of Safety (FS). Stability of the wall is examin examined ed as a system system consistin consisting g of different different mode modess of failure. failure. TypTypically, ically, these these modes modes are due to overturn overturning ing of the wall wall about about its toe, toe, due to sliding along its base, and due to bearing capacity inadequacy quacy of the found foundati ation on soil soil [62,15] [62,15].. Analy Analysis sis consi consists sts of exami examinin ning g separately every mode and modifying the design until the respective tive FS is larg larger er than than or equa equall to a pred predefi efine ned d valu value. e. In addi additi tion on,, the the eccentricity eccentricity of the loads’ resultant resultant with respect respect to the centerline centerline of the base shall intersect the base within its middle third. When the wall is constructed on slopped ground or in presence of complex stratification, deep-seated stability shall be addressed as well. Finally, an appropriate design must ensure that the anticipated total and differential settlements will not be excessive. Correspond Corresponding ing author. author. Address Address:: 6 Gavriilid Gavriilidou ou Street, Street, Athens Athens 11141, 11141, Greece. Greece. Tel.: Tel.: +30 210 2236467, mobile: +30 6944 926540; fax: +30 210 3303625. E-mail addresses:
[email protected] (I.E. Zevgolis), Zevgolis), bourdeau@pur due.edu (P.L. Bourdeau). *
0266-352X/$ - see front matter 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.compgeo.2009.12.003 doi:10.1016/j.compgeo.2009.12.003
Besides the questions in principles, for example the accuracy of the earth earth pressur pressure e theorie theories, s, the above above approa approach ch present presentss two inconveniences. The first one is that the uncertainty on the design load loadss and the soil soil resis resista tance nce is not not consi conside dered red explic explicitl itly, y, but implicit implicitly ly by using a FS based based on experience experience.. This approach approach is known known as allowab allowable le or working working stress design method method (ASD or WSD, respectively). For years, there has been a lot of effort to develop new reliability-based design methods that will be founded on rigor rigorous ous analys analysis is and statis statistic tical al treatm treatment ent of all includ included ed unceruncertainties tainties.. The greatest greatest challenge challenge in this effort is to identify identify and quantify the uncertainty on the soil resistance, i.e., the uncertainty of design design soil parameter parameterss (see for example example [42,70,64,35,37,52]). [42,70,64,35,37,52]). Nowadays there is a trend towards design methods that explicitly take the geotechnical variability into account. These methods have become known as Load and Resistance Factor Design (LRFD) in the USA, USA, a term borrowed borrowed from structural structural design design codes, codes, and Limit Limit State Statess Desig Design n (LSD) (LSD) in Canad Canada a and Euro Europe pe (see (see for for exam example ple [17,49,16,53,26]). [17,49,16,53,26] ). The second inconvenience of the conventional approach is that the analysis is not based on an overall estimate of the wall’s safety, but only on partial estimates of the safety relative ative to the individ individual ual modes modes of failure failure [4,14] [4,14].. So, the analysis analysis does does not provide an overall integrated measure of safety, but several measures (as many as the failure modes under consideration). Probab Probabilis ilistic tic analysis analysis offers offers the framewo framework rk to encounte encounterr the above above shortco shortcomin mings. gs. In this type of analys analysis, is, source sourcess of uncerta uncertainty inty are characterized and explicitly accounted for in the computation of the reliability. Each mode of failure can be analyzed separately and then corresponding modal reliabilities can be computed. But
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Nomenclature
ASD B c 1 C 1 c 2 c 2 CFS C 0 Cov COV C T D e E FS H0 K A K P LH LRFD LSD N nFi P A P [. . .] P F P P
allowable stress design footing width cohesive intercept of backfill material unit cost of consequences of a failure cohesive intercept of foundation material cohesive intercept along the interface of the wall–soil system Central Factor of Safety unit price of construction covariance coefficient of variation total expected cost of the wall embedment depth eccentricity of the loads’ resultant with respect to the centerline of the base event safety factor(s) total height of the wall (base to top) coefficient of active earth pressure coefficient of passive earth pressure length of heel (design variable) Load and Resistance Factor Design Limit States Design total number of trials in Monte Carlo simulations number of times that the safety ratio of the ith mode is less than one active (lateral) earth pressure probability of occurrence of the event within the brackets probability of failure passive (lateral) earth pressure
more importantly, and provided some modeling simplifications are accepted, an overall measure of reliability of the retaining wall can be assessed. The key advantage of the systemmodeling approach is that it provides a single index for quantifying the structure’s overall reliability, instead of partial and unrelated modal reliabilities. This greatly facilitates the use of reliability as a criterion of design optimization and decision support. For gravity or semi-gravity earth-retaining structures, the model in question is a series system where failure of at least one component results in the system’s ruin. Among important characteristics of the series systemin the case of a retaining wall is that its components (i.e., the resistances to failure modes) are not independent; they all depend to some extent on the same design parameters or material and soil properties. However, a commonly accepted simplification consists in ignoringthis dependency[31,4,7,60]. Another simplification consists in computing system reliability bounds corresponding to complete independence and/or perfect correlation. For example, Ang and Tang [2] and Tang [61] evaluated the limit values of the overall reliability of a reinforced concrete cantilever wall for two extreme hypotheses of correlation between the failure modes: non-correlated and perfectly correlated. For a bi-modal system (overturning and sliding) they showed that the domain of variation of the reliability in function of the degree of modal correlation is quite narrow when there is a dominant mode of failure. Same conclusion was drawn by Blazquez and Der Kiureghian [11] for three modes of failure of a gravity wall subjected to seismic loading. Bourdeau and Gutierrez [14], also considering thetwo limiting cases of perfect correlation and non-correlation, studied the ranges of wall width that influence these correlations. In the absence of a dominant failure mode, their study showed that the ef-
qb,L qmax qmin R S SR V W CONC W SOIL WSD
c1 c2 el ka/2 k(1a/2)
l hli1a q r RF R RF SL RM OT RM R RV u1 u2
u2
ultimate bearing capacity pressure maximum soil pressure minimum soil pressure probability of success (e.g., reliability) shear resistance along the interface between base and foundation soil safety ratio unit volume of the concrete weight of the concrete components of the wall (footing and stem) weight of the backfill and the soil above the toe of the wall working stress design unit weight of backfill unit weight of foundation material mean standard error lower critical value of a standard normal PDF upper critical value of a standard normal PDF mean value confidence interval coefficient of linear correlation standard deviation summation of resisting forces summation of sliding forces summation of overturning moments of the forces summation of resisting moments of the forces sum of the vertical forces acting on the wall friction angle of backfill material friction angle of foundation material interface friction angle along the wall’s base and the soil
fect of correlations on the system reliability decreases as the width of thefootingincreases. Usinga simplified cost function, they also indicated that the correlations may have an impact on the geometry of the wall, and consequently on the total expected cost of construction. Unless the reliability bounds are close to each other, none of the above simplified solutions is very satisfactory. This is so, because in the case of non-correlation of the failure modes, the reliability is underestimated, while in the case of perfect correlation of the failure modes the reliability is overestimated. The degree of under- or over-estimation is not always clear, especially in the absence of a dominant failure mode. A pioneering work presenting a methodology of system reliability analysis that avoids the above shortcomings is the work of Biernatowski and Puła [10]. Their study provides a probabilistic procedure for analyzing the stability of massive bridge abutments, based on simulation techniques. Recently, Low [40] presented an approach for calculating the system reliability of a bi-modal system (overturning and sliding) based on the method of Low et al. [41]. One of the objectives of this paper is to revisit the issue of dependency between failure modes that had been observed by Biernatowski and Puła [10]. A methodology is presented for the reliability analysis of retaining walls that takes into account the dependency between different failure modes. This dependency is investigated and quantified through their correlation, and its influence on the system reliability is addressed. Computations are performed using a Monte Carlo simulation algorithm for assumed probability distributions of the random variables. The impact of failure mode correlation on the system reliability and, in the light of these data, the relevance of common simplifications is discussed. Finally, considering the width of the base as varying design
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Fig. 1. Geometry of the analyzed wall.
parameter, the study investigates the significance of the modal correlations in terms of optimal design of the retaining wall. The methodology presented in this study is envisioned as a supplementary tool to recently developed LRFD and Eurocode.
2. System reliability 2.1. Stability considerations
The type of retaining wall being considered herein for reliability analysis is shown in Fig. 1. It is representative of backfilled reinforced concrete walls frequently used in practice. Only the wall’s external stability under static conditions is considered through analysis of overturning around the toe, base sliding, foundation bearing capacity, and location of the resultant of applied loads (eccentricity). These modes of failure or instability are illustrated in Fig. 2. Note that in cases of special structures, such as retaining walls that serve as bridge abutments, walls constructed on slopped ground and/or walls constructed in the presence of complex stratification, deep-seated stability analysis (also called rotational or overall stability) would be necessary and should be included in the model. However, in the present work, assuming the wall is built on leveled ground without a complex stratification, this is
not necessary. Moreover, settlement analysis is not included in the model in its current formulation. An overall reliability analysis of a retaining wall shall include analysis of the structural failure modes. However, these are not addressed in the present study, because the authors concentrate on geotechnical sources of uncertainty and consider, as a reasonable simplification, that the internal probability of failure would be much smaller than the external modes. This is so because the resistance of the reinforced concrete has lesser variability than soil shear strength. In other words, in geotechnical modes, both load and resistance are affected by soil parameters randomness, whereas in structural reliability only the load is affected by soil variability. So, the approximation that is made here is that the structural mode reliability is close to one (relatively to geotechnical modes), and this component is neutral in the system reliability. Of course, a comprehensive model should include it specifically, but the goal here is to investigate the geotechnical design process. Besides, a complete analysis of the structural failure modes would have to include at least control against bending moment and shear at stem, at toe, and at heel (i.e., six more safety factors). In addition, in order for the study to be accurate, elements of reinforced concrete design would have to be included. Such an analysis falls beyond the scope of the present study, whose focus is the geotechnical design process.
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in which K A is the coefficient of active earth pressure, c 1 the unit weight of the backfill, H 0 the total height of the wall (base to top), and c 1 is the cohesive intercept of the backfilled material. For horizontal backfill and vertical wall stem, K A is given by: u ¼ 11 þ sin sin u
1
K A
ð2Þ
1
in which u 1 is the friction angle of the backfilled material. – The lateral earth pressure acting on the front of the wall–soil unit. This is usually taken as the passive earth pressure P P that, assuming zero friction at the interface, is given by:
¼ 12 K c D þ 2c
P P
P 2
2
2
p ffiffi ffi
K P D
ð3Þ
in which K P is the coefficient of passive earth pressure, c2 the unit weight of the foundation material, D the embedment depth, and c 2 is the cohesive intercept of the foundation material. In this study the material below, in front, and above the toe of the wall is assumed to be the same. The coefficient of passive earth pressure K P is given by: u ¼ 11 þ sin sin u
2
K P
ð4Þ
2
in which u 2 is the friction angle of the foundation material. – The soil pressure q that acts vertically on the base of the wall. Its maximum and minimum values are given by:
V 1 B
6e B
P ¼
qmax;min
ð5Þ
P in which V is the summation of the vertical forces acting on the wall, B the width of the base of the wall, and e is the eccentricity of the loads’ resultant with respect to the centerline of the base given by:
e
Fig. 2. Modes of instability: (a) overturning, (b) sliding, (c) bearing capacity, and (d) excessive eccentricity.
B 2
M R
P P P
M OT
6 V P P M R and M OT are the summations of overturning where resisting and driving moments, with respect to the toe of the base. Typically, the resultant is required to intersect the base of the wall within the middle third; hence the entire area beneath the base is theoretically subjected to compression [51]. Numerically this means that the eccentricity must be smaller than or equal to one sixth of the base length. If the resultant falls on the right side of the base centerline, then based on Eq. (6) the eccentricity becomes negative. So the condition that must be met is: B e 6 7 6
¼
jj
ðÞ
ðÞ
According to conventional geotechnical design procedures for this type of wall, the mass of backfill overlaying the stem is assumed to form a resisting block attached to the structure [51]. Active earth thrust on the fictive vertical interface between this resisting mass and the retained backfill is computed according to Rankine’s theory. For dry backfill conditions, and in addition to the self-weight of the wall (W CONC) and the weight of the soil above the base (W SOIL ), the following pressures are applied on the wall:
– The shear resistance S acting along the interface between the base of the wall and the foundation soil. Typically, this is given by:
– The lateral (active) earth pressure PA acting on the back of the wall–soil unit:
In principle, the wall is safe when the loads that tend to activate a mechanism of instability, are smaller than or equal to the loads that tend to resist to this mechanism (capacity–demand model). In a traditional deterministic analysis following ASD, a safety factor would be computed for each of the postulated modes of failure
¼ 12 K c H0 2c
P A
A 1
2
1
p ffiffi ffi ffi
K A H0
ð1Þ
S Bc 2
¼ þ
X
V tan u2
ð8Þ
in which c 2 is the cohesive intercept along the interface, and u2 is the interface friction angle along the wall and the soil at the base.
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using nominal (characteristic) values of the controlling parameters. Each of the resulting safety factors would then be required to be greater than a recommended empirical value, specific to the particular failure mode. Required safety factors are typically in the order of 1.5–2 for overturning, 1.25–2 for base sliding, and 2–3 for bearing capacity [62,15]. The fact that these required values are larger than one is an acknowledgement of important uncertainty being present in the design process, though this is not explicitly analyzed. Additionally, the eccentricity is required to be less than B / 6, according to Eq. (7). In fact, some references recommend the eccentricity criterion as the criterion against overturning [63]. This approach was considered more appropriate by the authors and it is followed in the present study. Further reasoning for this decision is provided in the Appendix A. It is acknowledged that design analyses are simplifications of reality and these may produce ‘‘biased” prediction of actual performance. Model uncertainty, in the context of reliability analysis, is still a domain largely unexplored. Its assessment would require statistical databases of prediction vs. performance be assembled and analyzed. In absence of such assessment, it is important to recognize that computed reliability or probability of failure have only a relative meaning. This can be used in comparative analyses, such as optimization studies, provided these comparisons are based on the same models. 2.2. Modal probabilities of failure
In contrast to a deterministic approach, in a probabilistic analysis of the wall stability, sources of uncertainty are characterized and explicitly accounted for in the computation of the 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). Safety ratios with respect to sliding and bearing capacity inadequacy, SR SL and SR BC, respectively, are expressed by:
SRSL
¼
PP
¼ qq
SRBC
FR P P S FSL P A
¼ þ
b;L
max
P
ð9Þ ð10Þ
P
where F R and F SL are the summations of the base sliding resisting and driving forces, and qb,L and qmax are the foundation ultimate bearing pressure and maximal applied pressure. Note that the above mathematical expressions are similar to those of safety factors but, in contrast with the latter, the safety ratios are functions of soil and material parameters affected by uncertainty. As these parameters are modeled as random variables, SR SL and SR BC are functions of randomvariables and therefore, are randomquantities, too. In this study a performance function, SR e, is also defined for the eccentricity. Based on Eq. (7), this can be defined as:
¼ B=6 jej
SRe
ð11Þ
For all three postulated mechanisms of instability, failure is defined as the case where the corresponding SR is less than one. It shall be emphasized that the term failure herein does not necessarily imply a collapse of the wall or a catastrophic failure, but it refers to any unacceptable difference between expected and observed performance [39]. Another option, instead of using safety ratios, would have been to use safety margins (expressed as the difference between resistance and load). The probability P Fi that [SR i]<1 for any given mechanism i is given by the following expression:
P Fi
¼ P ½SR < 1
ð12Þ
i
while the reliability is given by:
Ri
¼ 1 P ¼ P ½SR P 1 Fi
i
ð13Þ
Fig. 3. Schematic representation of a system in series.
2.3. Assessment of system reliability
The global stability of the wall is modeled as a system in series with zero redundancy. This means that if at least one of the postulated modes of instability occurs, then the system fails. On a probabilistic context, the global failure is the event in which any of [SR e < 1], [SR SL < 1], [SR BC < 1] occurs, and its probability of occurrence, P F, is given by the union of these events:
P F
¼ P ðSR < 1Þ [ ðSR e
SL < 1
Þ [ ðSR
BC < 1
Þ
ð14Þ
ð15Þ
where P[–] denotes the probability of the event indicated within the brackets. The global stability is the event in which [SR e P 1], [SR SL P 1], and [SR BC P 1] occur simultaneously. The probability of occurrence of this event is the reliability of the event given by:
R
¼ P ðSR P 1Þ \ ðSR e
SL P
1
Þ \ ðSR
BC P
1
Þ
An illustration of the series system concept is given in Fig. 3. The global stability of the wall is shown at the end of the chain whose links (components) are the three modes of instability. If any of the links breaks, then the global stability is lost. In general, computation of P F or R by exact integration of the involved multivariate functions is impractical. This difficulty can be overcome by using approximate methods that have been developed specifically for system reliability analysis. Among these are the Hasofer and Lind [30] procedure or the system reliability bound methods [1,2]. Other methods include the first- and
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second-order reliability methods, either on their classical format or using concepts of asymptotic analysis [32]. The simplest in the present case would consist in determining the first-order reliability bounds of the system. If the safety ratios are all positively correlated, then the system probability of failure and the system reliability can be, respectively, written as: k
max P Fi
f g
6 P F 6 1
1
P Fi
i 1
¼
k
Y
Y ð
ð16Þ
Þ
Fig. 4. First-order reliability bounds for positive correlations.
Ri 6 R 6 min Ri
ð17Þ
f g
i 1
¼
where P Fi and R i are the modal probabilities of failure and modal reliabilities, respectively, relative to each of the three individual failure modes and k = 3 (since three failure modes are examined in the present study). If the safety ratios are negatively correlated, then: k
06R6
Y
R i
ð18Þ
i 1
¼
Qk Qk The terms 1 i¼1 ð1 P Fi Þ and i¼1 Ri are the exact system probability of failure and system reliability in the case of modal independence. The terms max {P Fi} and min{Ri} correspond to the exact system probability of failure and system reliability when the safety ratios are perfectly and positively correlated. Because, as shown in Fig. 4, the interval defined by Eq. (18) is usually very large, the first-order reliability bounds solution is useful mainly in the case of positive correlations between safety ratios (i.e., safety ratios are either simultaneously increasing or decreasing functions of the engineering properties of the soils). This is usually, but not always, the case in practice. For instance, increases in the friction angle values have generally positive impact on Eqs. (9)–(11), while increases in the unit weights may, in some cases, have negative impact. So, due to the complexity of the above equations, a safe conclusion regarding their behavior upon a simultaneous variation of these variables cannot be drawn. Therefore being free from the constraining assumption of positive correlations is an advantage [72]. The first-order bounds described above may be improved by taking into account the correlation between pairs of the potential modes of failure. The resulting improved bounds necessarily require the probabilities of joint events and thus may be called bi-modal or second order bounds [2]. For k failure modes (in this study k = 3), the second order lower and upper bound failure probabilities are given by the following correlation [36,33]:
" ( X þ X X k
P F 1
i 1
max 0; P Fi
i 2
¼
k
k
P Fi
6
i 1
¼
i 2
¼
Xð j 1
¼
max P E i E j j
P E i E j
½ ð Þ
)#
Þ
chastic technique that generates a great number of repeated simulation processes (realizations). Each simulation is based on the generation of a series of values of one or more random variables. The procedure requires complete definition of the random variable probability distributions, but, through simple computations, it provides empirical outcomes of numerically simulated random realizations of the safety ratios. Then, statistical estimates can be obtained for the system reliability, safety ratios, their coefficients of correlations, or other quantities of interest. The Monte Carlo technique is extensively documented in geotechnical engineering literature. Among other applications, it has been used for probabilistic analysis on earth pressure problems, retaining structures, deep foundations, slope stability problems, and other (see for example [59,4,10,68,27,43]). The recent evolution of personal computers has lead to more accurate and sophisticated use of the technique than three decades ago, and often the method is combined with finite elements analyses [23,25]. It is noted here that due to the complexity of the studied problem and the correlations between the failure modes, analytical solution through first or second-order reliability methods (both based on the Taylor Series expansion) may be cumbersome and difficult to achieve. However, first or second-order reliability methods improved by concepts of asymptotic analysis, such as the ones presented by Hohenbichler et al. [32], could be alternatively used in order to solve the problem without significant difficulties. On the other hand, the Point Estimate Method would be adequate for computing the reliability, provided that assumptions are made on the distributions of the safety ratios, but it would not provide a direct assessment of their correlation. In the case of Monte Carlo simulations, and assuming that the modes of instability are adequately represented by Eqs. (9)– (11), the dependency between these modes can be evaluated by means of coefficients of linear correlation ( q). For any pair of i– j modes, q is given by [58]: qSR SR
6
i
P F
ð19Þ
where E i and E j are the individual failure events (with E 1 being the largest set and P F1 being its probability of occurence). A similar version of the above second order bounds has been proposed by Ditlevsen [22]. The bi-modal bounds depend on the ordering of the individual failure modes. This means that different orderings of the individual failure modes may yield different values in the above equation, so the bounds corresponding to different orderings may have to be evaluated to determine the narrowest bounds [2]. In addition, in cases of correlated failure modes (such as in the examined study), the calculations of the joint probabilities P (E iE j) remain difficult. The approach used in the present study is a still approximate, but more direct computation of the system reliability using a Monte Carlo simulation algorithm. Monte Carlo simulation is a sto-
j
¼
Cov SR i ; SR j rSR i rSR j
ð20Þ
in which r SR i ; rSR j are the standard deviations of the variables SR i, SR j, and Cov is their covariance. The covariance is defined as the expected value E of the products of ð SR i SR i Þ and ð SR j SR j Þ:
Cov SRi ; SR j
½
h
¼ E
SR i
SR
i
SR j
SR
where SR i ¼ E ½SR i and SR j ¼ E ½SR j .
j
i
ð21Þ
2.4. Design optimization
Reliability analysis offers the framework for optimization of the design process. In simple risk-decision problems, an objective function, often expressed in monetary units, is defined in terms of one or more decision variables. The optimal design is determined by the values of decision variables that maximize (in case of benefit) or minimize (in case of cost) the objective function [2]. Previous literature includes optimization analyses for retaining walls based on the first-order reliability bounds [31,14,7].
I.E. Zevgolis, P.L. Bourdeau/ Computers and Geotechnics 37 (2010) 359–373
Optimization in the current study is based on the actual system reliability given by Eq. (15). Qualitatively, the objective function is expressed in terms of cost by the following expression:
Total Expected Cost
¼ Initial Costðof ConstructionÞ þ Risk ð22Þ
In engineering context, risk is often defined as the product of probability of occurrence of an event by the consequence of this event. Based on this, the objective function is expressed by the following equation [14]:
C T
¼ C V þ P ðC V þ C V Þ 0
F
0
1
C1 C 0
þ þ
¼ V 1
P F 1
ð24Þ
So, the normalized total expected cost is expressed as a function of the unit volume of the concrete, i.e., as a function of the geometry of the wall. The ratio C 1/C 0 can be viewed as a risk factor, assigned on a case-by-case basis. Based on Table 1 and Fig. 1, the volume of interest is given by:
¼ ðH L þ H L þ H 0 L Þ 1 linear meter
V
T T
T H
S
Table 1
Geometric characteristics of the analyzed wall.
Characteristic
Notation
Value (m)
Free height Depth of embedment Base thickness Toe length Stem thickness Heel length
H D H T LT LS LH
6 1.5 0.8 0.8 0.8 1.5–3.5
ð23Þ
in which C T is the total expected cost of the wall, C 0 the unit price of construction, V the unit volume of concrete, P F the probability of failure of the wall, and C 1 is the unit cost of consequences of a failure. The terms V and P F are the decision variables of Eq. (23). In the second part of the equation, the first term, C 0V , represents the initial cost of construction of the wall and the second term, P F(C 0V + C 1V ), represents the cost of failure. For simplicity, the cost of reconstruction of the wall after a potential failure is assumed to be the same with the initial cost of construction. Moreover, the term V refers only to the volume of concrete, i.e., it does not include the volume of excavated material. Finally, the consequences of failures by the three modes of instability are assumed to be the same and all together expressed by the term C 1. A more detailed analysis would have considered the consequences of each mode separately. However, this would require modeling deformation (such as foundation settlement and tilting), which in any case is not an option with limit equilibrium models. So, such an analysis falls beyond the scope of this study. Besides, the cost of consequences of a failure is difficult to estimate, because intangible as well as tangible factors must be considered [31]. By normalizing with C 0, Eq. (23) can be written as following:
C T C 0
365
ð25Þ
Optimal design is the one for which C T/C 0 becomes minimum. The length of heel for which this situation happens is called the optimal length of heel, L H(OPT). 2.4.1. Case example In the case example being analyzed, geometric characteristics of the wall are considered deterministic quantities. These are indicated in Table 1, with reference to notations in Fig. 1. One of these, the heel of the wall, L H, is used as variable design parameter. 2.5. Random variables 2.5.1. Spatial variability When extensive site investigations are performed and soil properties are measured at a large number of points of a particular pro ject site, it may be possible to analyze these data statistically and to describe their spatial variability, within homogeneous sub-regions, as a continuous random field [65]. The main elements of such a descriptive random field are the expected value, variance and autocovariance function (or the autocorrelation function) of the analyzed soil property. A number of examples have been published where the random fields were numerically simulated. Sensitivity analyses have shown, for instance, that short autocorrelation distances for strength properties tend to reduce the variance of the
soil resistance in limit equilibrium problems, and may result in smaller probabilities of failure, as compared to long autocorrelation distances (e.g., [24,25,28,46,54–56]). However, even when large arrays of data are available for a site, empirical autocorrelation functions are inaccurate and difficult to estimate with confidence [12,13]. It should also be noted that two-dimensional modeling of a geotechnical problem is incompatible with a correct random field description of the site because, considering a twodimensional cross-section as representative of the site implicitly assumes all other parallel cross-sections are identical and their properties perfectly autocorrelated in the transverse direction [6,5]. In the present case example, no such detailed geostatistical description of the site is considered. Subgrade soil and backfill properties are assumed to be estimated based on a limited number of data or expert opinion. This situation is likely to be the case in practice when rather simple retaining earth systems are to be designed. These properties are modeled here as an ‘‘estimated” field using random variables that are assumed homogeneous (or perfectly autocorrelated) over the volume of interest. This is similar to the approach used for instance by Harr [29] or Low [40]. It is acknowledged that this approach likely leads to conservative estimates of failure probabilities, but this should not affect the generality of the discussion on system modal correlations and design optimization. 2.5.2. Cross-correlation Regarding the cross-correlation between the involved random variables, and particularly between friction angle and cohesive intercept, literature has not been clarifying [23]. Values varying in both negative and positive range have been quoted by several authors (see for example [67,18]). Fenton and Griffiths [23], investigating a bearing capacity problem, have found that even correlation extremes have only a minor influence on the stochastic behavior of the bearing capacity. Zevgolis [71], examining the external stability of a reinforced soil wall, has shown that positive cross-correlations overestimate the probability of failure, while negative cross-correlations underestimate it. Therefore, in the absence of reliable data that would indicate the true value of cross-correlations, one should be cautious with using any particular value. Arbitrary use of positive values as a safety net is too conservative, taking into account that real values are usually expected to be negative. So, the model in its current formulation does not take into account cross-correlations between variables, i.e., the random variables are considered independent to each other. In addition, the paper assumes statistical independence between the foundation and backfill soil properties. This is so because the backfill would be borrowed from the site (thus its properties be similar or strongly correlated to those of the foundation soil), only if it is adequate with respect to drainage, compaction and frost sensitivity, etc. In this study it was assumed this was not the case and therefore these two materials are unrelated. Overall, the above simplifying assumptions about the correlation structure of the data set should slightly affect the system reliability analysis numerical results but without loss of general validity.
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2.5.3. Probability distribution function Each one of the random variables is represented by its first two order moments (mean value l and standard deviation r, or coefficient of variation COV), and its minimum and maximum value. Therefore, based on the principle of maximum entropy [34], the random variables are modeled using Beta (i.e., Pearson type I) distributions, with characteristics listed in Table 2. For the use of beta distribution in civil engineering applications see Oboni and Bourdeau [47] and Harr [29]. It is noted here that other type of distributions, such as normal, lognormal, or gamma, could have been another option for modeling the random variables. Nevertheless, at least one bound of these distributions goes to infinity (i.e., the upper bound in the case of lognormal and gamma distributions, and both lower and upper bounds in the case of normal distribution), which is not physically consistent with the physical parameters examined in this study. Practically, the unbounded extreme implies that the variate may take on values many times greater than its expected value. Although theoretically this is possible, the probability of such an occurrence is fairly small [29]. The mean values are selected as representative of typical soils encountered on site (i.e., coarse-grained backfill material, more fine-grained foundation material). The coefficients of variation, COV, are consistent with representative data reported in literature (e.g., [29,37,52]). In the absence of extensive site-specific data base, minimal and maximal values are selected based on an arbitrary rule of ±4 standard deviations from the mean. From a statistical point of view, such a range guarantees a high probability of occurrence: 93.75% using Chebyshev’s inequality, and 97.22% using Gauss’ inequality for symmetrical cases. At the same time, the physical meaning and the likely range of fluctuation is consistent with these values. Note that for l 4r, the minimum value of c 2 becomes negative, and therefore in this example the minimum value was taken equal to zero. The interface properties along the base of the wall, u 2 and c 2 , are factored down with respect to the random variables u 2 and c 2, respectively, by 0.7, which is a value consistent with common practice [69]. As a result, these properties are also random variables, modeled as linear functions of u2 and c 2, following a beta distribution. The passive pressure, P P, is neglected in the calculations of all safety ratios, which is often the case in design [51,62], and all bearing capacity factors, including depth and inclination factors, are calculated based on Meyerhof’s formulae. 2.6. Monte Carlo realizations
Analyses are performed for a wide range of heel’s length, from LH = 1.5m to L H = 3.5 m, in increments of 0.1 m (21 cases in total). As a deterministic point of reference, Central Factors of Safety (CFS) is also computed. The CFS is defined as the Factor of Safety computed with the mean values of the parameters [29], in contrast with actual design Factors of Safety that would result from using conservative estimates of the parameters expected values. For the probabilistic simulations thirty thousand realizations (trials) are performed for each examined case. This was a reasonable number for keeping errors in the estimated probabilities within tolerable limits. Analyses were also performed for LH > 3.5 m, however the mean standard errors in these cases were high. Another ap Table 2
Probabilistic parameters (input) of soil properties.
Soil property
c1
u1
c 1
c2
3
kN/m
u2
c 2
Unit
kN/m2 kN/m3
kN/m2
l
COV
Min
Max
20 32 0 18 25 30
0.07 0.12 0 0.07 0.20 0.30
14.4 16.6 0 13 5 0
25.6 47.4 0 23 45 66
proach would have been to prescribe the targeted accuracy and let the number of simulations be selected in order to achieve this accuracy. For each realization, a different group of the five soil parameters (random variables) is generated following a beta distribution with characteristics listed in Table 2. As a result, each group corresponds to a different outcome regarding the three modes of instability. The probability of failure is then given by:
P Fi
¼ nN
Fi
ð26Þ
in which n Fi is the number of times that the safety ratio of the i th mode is less than one and N is the total number of trials that are performed. The generation of the values of the random variables is performed using the random number generator (RNG) of the risk analysis software Crystal Ball v.7.2, which operates as an add-into Microsoft Excel [20]. Crystal Ball allows for the definition of design variables as randomvariables that follow a probability mass or density function (pmf or pdf, respectively). Using the defined random variables, the program performs Monte Carlo simulations and provide outcome in terms of a wide range of elements,such as probability distribution functions (PDFs), cumulative distribution functions (CDFs), statistical parameters, and other. A great advantage of the program is that allows for full extraction of data in Microsoft Excel, including both input and output based on which the simulation is performed. This gives the user the opportunity not only to perform his/her own statistical analysis, but also to examine carefully and validate the results that are provided by the program. Crystal Ball has been increasingly used in the recent past for risk analysis in civil and geotechnical engineering applications [44,66,9,50]. Based on comparative evaluation of six available software programs that offer Monte Carlo capabilities, Metzger et al. [45] recommended Crystal Ball as the best one, for risk assessment problems that can be implemented on a spreadsheet. As far as the (pseudo-)random number generator is concerned, Crystal Ball uses a multiplicative linear congruential generator (LCG). The recursive formula that is being used has the following format [20,21]:
Z nþ1
31
¼ ð62; 089; 911 Z Þmodð2 1Þ n
ð27Þ
The above generator has a full period of length, equal to (232 1) 1 = 2147,483,646, i.e., the cycle of random numbers repeats after more than two billion trials. Several researchers have tested and evaluated the above recursive formula and have classified it as one of the best available for regular risk assessment applications [48,57,8,38]. 3. Results and discussion 3.1. Reference to deterministic analysis
Fig. 5 is provided as a reference to deterministic analysis, and it shows the variation of Central Factor of Safety (CFS i) computed for each instability mode in function of heel’s length (LH). As shown in the Figure, the rates of increase of the CFS are quite different from one mode to another: CFSSL and CFSBC increase linearly with increasing L H, while CFSe increases non-linearly. The type of information shown in Fig. 5 presents three inconveniences. First, it does not provide any output on the probability of occurrence of any instability mode and it does not scale safety. This means that for instance, a design decision for increasing LH does not reflect the increase in the reliability of the structure. Second, it does not provide any information related to the dependency of the instability modes with each other, and as a result it does not provide any information on the global stability of the wall. Third, this information in its current format cannot be easily used on an optimization analysis. These inconveniences can be eliminated when, in conjunction to conventional design, a probabilistic analysis is performed.
I.E. Zevgolis, P.L. Bourdeau/ Computers and Geotechnics 37 (2010) 359–373
Fig. 5. Central Factor of Safety vs. length of heel.
367
Fig. 7. Typical histogram for CFSBC.
3.2. Modal probabilities of failure
Typical obtained histograms for Central Factor of Safety with respect to sliding, bearing capacity and eccentricity are provided in Figs. 6–8, respectively. Table 3 tabulates the basic statistical parameters obtained from the analysis. Statistical errors associated with these parameters can be estimated using the mean standard error (el) and the corresponding confidence intervals (hli1a). These two are expressed as [3]:
¼ p rN
el
ffiffi ffi
hli ¼ 1 a
l
ð28Þ r
þ k p N l þ kð a=2
ffiffi
;
ffiffi ffi p r
1 a=2
Þ
N
ð29Þ
Fig. 8. Typical histogram for CFS e.
Table 3
Statistical parameters of the resulting safety ratios (for L H = 3 m).
Statistics
Fig. 6. Typical histogram for CFS SL .
Realizations Mean Median Variance Standard deviation Coefficient of variation Coefficient of skewness Coefficient of kurtosis Minimum Maximum Range width
Notation
SRSL
SR BC
SR e
n
30,000 1.74 1.70 0.18 0.42 0.24263 0.56365 3.47 0.61 4.12 3.51
30,000 3.99 3.35 6.17 2.48 0.62325 2.18 11.51 0.54 32.39 31.85
30,000 1.89 1.79 0.28 0.53 0.28104 1.24 5.64 0.90 6.93 6.03
l lm Var
r COV b1 b2
min max –
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368 Table 4
Error estimates (for L H = 3 m).
Statistics
Notation SRSL
SR BC 3
SR e 3
el Mean standard error 2.434 10 14.344 10 3.061 103 3 3 Conf. interval – 95% hli0.95 4.771 10 28.115 10 5.999 103 Conf. interval – 99% hli0.99 6.280 103 37.008 103 7.897 103
in which (1 a) is a specified probability and k a/2 and k (1a/2) are the lower and upper critical values of a standard normal PDF (see [3] for more details). The values of the above are provided in Table 4 for L H = 3 m. Based on the above, it is indicated that bearing capacity computations are subjected to higher degree of uncertainty ðCOVCFSBC ¼ 0:6232Þ, compared to sliding and eccentricity (COVCFSSL ¼ 0:2426 and COVCFSe ¼ 0:2810, respectively). This is consistent with the conclusions drawn from the study of Biernatowski and Puła [10], as well. This is so because bearing capacity primarily depends on the cohesion and friction angle of the foundation soil, which are both expected to demonstrate higher dispersion compared to the friction angle of the backfilled material, and because the bearing capacity coefficients depends in non-linear way on friction angle (particularly for higher values). This has an impact on the corresponding errors and confidence intervals, too (Table 4). For instance, the sampling error of the mean of CFSBC is approximately six times as big as the equivalent error of CFS SL and almost five times as big as that of CFS e. As far as the symmetry of the distributions is concerned, all three appear to be skewed with the long tail of the distribution being on the right side of the mean (b1 > 0). This is well illustrated in the histograms, especially in the case of CFSBC, which shows the higher degree of skewness. Moreover, CFSBC shows the higher degree of kurtosis. Figs. 9 and 10 graphically present the modal probabilities of failure (P Fi) plotted in logarithmic scale, with respect to LH and CFSi, respectively. It is shown that the modal probabilities of failure are not linear functions of L H and CFS. For instance, CFSBC = 2.4 corresponds to P FBC ¼ 4:95%, CFSBC = 3.0 to P FBC ¼ 1:48% and CFSBC = 4 to P FBC ¼ 0:29% (Fig. 10). So, CFSBC alone does not provide any information as to the likelihood of a failure occurrence, nor as to the achieved improvement of stability when design is modified.
Similar conclusions are drawn for sliding and eccentricity. On the other hand, as expected, the same level of safety for the three postulated mechanisms of instability is achieved for different values of LH and CFS. Another observation from the above Figures is the sensitivity of P Fe with varying L H. Specifically, the probability drastically changes within a short range of LH. It is worth mentioning that for CFSe = 1, which is the requirement in conventional design, the probability that e will exceed B /6 is approximately 50%. Note that for the example analyzed here, CFSe = 1 corresponded to CFSOT = 2.4 (CFS against overturning), which means that even a safe design with respect to overturning does not necessarily result in safety with respect to the location of the resultant of the forces. Opposite to eccentricity, P FSL and P FBC demonstrate a smoother variation with varying length of heel.
Fig. 9. Length of heel vs. modal probabilities of failure (log).
Fig. 11. Joint distribution of CFS e–CFSSL .
Fig. 10. Central Factor of Safety vs. modal probabilities of failure (log).
I.E. Zevgolis, P.L. Bourdeau/ Computers and Geotechnics 37 (2010) 359–373
Fig. 12. Joint distribution of CFS BC–CFSSL .
3.3. Dependency of modes of instability
Figs. 11–13 graphically present the joint distributions of CFS taken two by two for the case of LH = 3 m. It is noticed that CFSBC demonstrates some very high values (up to 33). Opposite to this, the ranges of CFSe and CFSSL are much lower, with their upper bounds at around seven and four, respectively. This is so because the equation providing CFSBC is very sensitive with respect to varying parameters and can result in high values if random variables’ values are close to their upper bound. Results in terms of coefficient of linear correlation with varying LH are shown in Fig. 14. The highest degree of correlation was found between CFSSL and CFSBC. In terms of Monte Carlo realizations, this meant that for the analyzed example most of the times where failure by sliding occurred, failure by bearing capacity was also taking place, and
Fig. 13. Joint distribution of CFS BC–CFSe.
369
Fig. 14. Coefficients of linear correlation vs. length of heel.
vice versa. High degree correlation was also found between CFSSL and CFSe, while CFSBC and CFSe were correlated on a lower degree. The trend of correlations appears to be gradually decreasing with increasing L H. Overall, it is clear that the three failure modes are correlated, to a higher or lower degree, to each other. 3.4. System reliability
Fig. 15 graphically presents the system probability of failure ðP FSYS Þ as calculated by Eq. (14), with varying length of heel. As expected, P FSYS falls within the range of the first-order bounds, which
are calculated according to Eq. (16). However, the question is whether or not these bounds are appropriate to use, especially in the absence of a dominant failure mode. In regards to the latter
Fig. 15. System probability of failure (log) vs. length of heel.
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370
Fig. 16. Computed system probability of failure and modal probabilities of failure.
comment, Fig. 16 illustrates P FSYS together with the three modal probabilities of failure. It is shown that the system is controlled by the eccentricity for L H up to 2.7 m, and by sliding for L H up to 3.5 m. In other words, for different values of LH different modes of instability control the design and therefore a safe conclusion regarding the appropriate of use of the first-order bounds cannot be made in advance. For instance, in this case example and for a target P FSYS ¼ 0:01ð1%Þ, LH shall be 3.3 and 3.5 m based on the lower and upper bounds, respectively (Fig. 15). Using Eq. (14), L H shall be 3.5 m. So, assuming a modal independence leads to over-design of the heel by 0.2 m. On an attempt to quantify the degree of overor under-estimation, Fig. 17 illustrates the first-order bounds probabilities of failure normalized by P FSYS (Eq. (16)). So, axis Y shows Q3 the ratios
1
i¼1
P F
ð1P Fi Þ
SYS
fP Fi g (complete independence) and max (perfect P F SYS
Fig. 17. Ratios of P FSYS with first order probabilities of failure.
Fig. 18. Optimum designs for different C 1/C 0.
correlation). For instance, for design value of LH = 3 m, the above ratios are 1.407 and 0.911, respectively. 3.5. Design optimization
In this study, optimization analysis was performed for a wide range of risk factors C 1/C 0 and representative results are provided in Fig. 18. As shown in the Figure, LH(OPT) increases as the risk factor increases. For instance, when the unit cost of consequences of a failure is equal to the unit cost of initial construction ( C 1/C 0 = 1), the optimum length of heel is 3 m. Another interesting observation is made when comparing optimization curves for first-order reliability bounds. In this context, Fig. 19 shows the curves obtained for a risk factor equal to 1. The lower and upper curves represent
Fig. 19. Optimum designs for P FSYS and first-order bounds.
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Optimum heel length. C 1/C 0
1 2 3 4 5
LH(OPT)
Perfect correlation
P FSYS (Eq. (17))
Complete independence
3 3.1 3.3 3.3 3.4
3.1 3.2 3.3 3.4 3.4
3.2 3.3 3.4 >3.5 >3.5
perfect correlation and complete independence, respectively, between instability modes. As shown in the Figure, LH(OPT) shows a tendency to increase from perfectly correlated to totally independent failure modes. Table 5 tabulates the different L H(OPT) for varying risk factors. So, the assumption of complete independence (which is mostly used in the literature) may overestimate L H(OPT) and therefore lead to a conservative design decision. On the other side, assuming perfect correlation between the different modes of instability modes usually results in slightly lower L H(OPT).
4. Summary and conclusions
In this study, the stability of reinforced concrete cantilever wall was analyzed following a probabilistic approach. Only the wall’s external stability under static conditions was considered through analysis of base sliding, foundation soil bearing capacity, and excessive eccentricity. In order to represent limit states of equilibrium, performance functions were defined as safety ratios. Overall stability was addressed as a system in series. Engineering properties of the backfilled and foundation soil were considered random variables following a beta distribution. As such, they were each represented by their mean, coefficient of variation, minimum, and maximum value. Spatial variability and cross-correlations among the random variables were not included in the model. Geometric characteristics were considered deterministic quantities, and the heel’s length was used as variable design parameter. Analyses were performed for a wide range of this variable, using a Monte Carlo simulation algorithm. A case example was analyzed in order to illustrate the above methodology, and the following conclusions (for this example) were drawn from the analysis: Computations of bearing capacity are subjected to higher degree of uncertainty, compared to sliding and eccentricity (a conclusions consistent with the observations made by Biernatowski and Puła [10]). The same level of safety for the three postulated mechanisms of instability is achieved for different values of the design variable L H. In addition, the modal probabilities of failure are not linear functions of the corresponding safety ratios (nor of the design variable LH). This is why safety ratios do not scale safety. Safety ratios are all positively correlated. Their correlation decreases as L H increases. Particularly high degree of correlation was found between CFS pairs of sliding – bearing capacity and sliding – eccentricity. The system probability of failure fell within the first-order bounds. However, depending on the target reliability and the presence or not of a dominant failure mode, these bounds may lead to a noteworthy over- or under-estimation of the system probability of failure. Obtaining a unique value for the system probability of failure allows for the optimization of the design process. The results of a simple optimization analysis, performed for illustrative purposes, indicated that the optimum design variable may significantly increase with increasing risk factors. In addition, it was shown that the assumption of complete independence or perfect correlation between the failure modes, may have an impact on the design variable.
371
Overall, reliability assessment is not a substitute to the conventional methods of design, but it is a complementary to them tool, necessary on a risk analysis context. Nowadays, the latter one becomes more and more an essential part of civil engineering pro jects. The present methodology can provide a framework for more complex systems analysis in geotechnical engineering and help optimizing designs in presence of uncertainty. Appendix A
Overturning rarely governs the design of retaining walls [62,15]. Some authors even suggest that this type of failure is indirectly accounted for, if the eccentricity is kept smaller than B/6 and the wall is safely designed with respect to sliding [19]. When overturning is addressed separately, through a safety factor dealing with overturning resisting and driving moments, it is usually evaluated based on the assumption that rotation takes place around the bottom point of the wall’s toe. However, this is not always a kinematically realistic assumption. The location of the actual point of rotation depends on the foundation soil characteristics. On a compressible material the wall would start plunging before the overturning mechanism is mobilized, so the point of rotation would ‘‘move” towards the right side of the base. In addition to these qualitative comments, it can be shown that a control against overturning is redundant in the design process, if the resultant of forces intersects the base of the wall within the middle third, i.e., if e < B/6. Following is the proof of this statement. Supposedly a safety factor (FS) applied in the ratio of overturnP P ing driving and resisting moments, M OT and M R , respectively, is used as the criterion against overturning of the wall about its toe. This is defined as:
FSOT
¼
M R M OT
PP
If FSOT < 1, then design is performed again until FSOT > 1. If FSOT > 1, then:
M R > 1 M OT
PP
X X P P P ) M R >
)
M R
M OT
V
M OT
)
X X M R
M OT > 0
> 0
In this case, based on Eq. (6), the eccentricity may be either smaller or larger than B /6. So, in addition to control with respect to FSOT, a separate check addressing the eccentricity shall be performed in order to determine its value. Supposedly, on the other hand, that control with respect to FSOT is not performed and one directly determines the eccentricity according to Eq. (6). Then, if e > B6, design is performed again until e < B6. If e < B6, then the following hold true. Assuming the resultant falls on the left side of the base centerline, the eccentricity is defined as:
B 2
MR
M OT
P P P ¼ P P P )
e
V
In this case we have:
B e < 6
B 2
M R
M OT
B < 6
B < 3
M R
P P P ) P X X ) ) P V
M OT
V
But B3 > 0 always holds true, therefore:
MR
P P P V
M OT
> 0 > 1
M R
M OT > 0
M R > 1 MOT
) FS
OT
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Assuming that the resultant falls on the right side of the base centerline, and in order to define the eccentricity as a positive number, the eccentricity is defined as:
e
¼
MR
MOT
P P P P P P ) V
B 2
In this case we have:
MR
e > 0
M OT
V
B > 0 2
MR
M OT
P P P ) P X X ) ) P V
>
B 2
But B2 > 0 always holds true, therefore:
MR
P P P
MOT
V
> 0 > 1
M R
MOT > 0
MR > 1 M OT
) FS
OT
So, whenever the eccentricity is smaller than B /6, the safety factor against overturning is always larger than one. Therefore, a separate control against overturning is not necessary. In summary, using a safety factor against overturning requires a separate calculation of the eccentricity. On the other hand, using the eccentricity as the criterion for overturning, makes the calculation of the safety factor a redundant step. These statements are true for safety ratios and eccentricity defined as in the present study. References [1] Ang AH-S, Amin M. Reliability of structures and structural systems. J Eng Mech Div 1968;94(2):671–91. [2] Ang AH-S, Tang WH. Probability concepts in engineering planning and design. Decision, risk and reliability, vol. II. John Wiley & Sons; 1984. [3] Ang AH-S, Tang WH. Probability concepts in engineering – emphasis on applications in civil and environmental engineering. 2nd ed. John Wiley & Sons; 2007. [4] Athanasiou-Grivas D. Reliability analysis of retaining structures. In: Ingles OG, editor. 3rd International conference on application of statistics and probability on soil and structural engineering (ICASP 3), Kensington NSW, 29 January–2 February 1979, Sydney, Australia; 1979. p. 632–41. [5] Auvinet G. Probabilistic approach in geotechnical engineering practice. In: Proceedings, 4eme conference nationale sur la fiabilite des materiaux et structures (JNF’05), Clermont-Ferrand, Universite Blaise Pascal, France, Keynote Lecture; 2005. 9 p [in French]. [6] Auvinet G, Gonzalez JL. Three-dimensional reliability analysis of earth slopes. Comput Geotech 2000;26(3–4):247–61. [7] Babu GLS, Mythily M, Rao DB. Probabilistic design of retaining walls. In: Lacasse S, Singh SK, editors. International conference on offshore and nearshore geotechnical engineering (GEOShore), 2–3 December 1999, Mumbai, India; 2000. p. 383–6. [8] BarryTM. Recommendations on the testing and use of pseudo-random number generators used in Monte Carlo analysis for risk assessment. Risk Anal 1996;16(1):93–105. [9] Benardos AG, Kaliampakos DC. A methodology for assessing geotechnical hazards for TBM tunnelling – illustrated by the Athens Metro, Greece. Int J Rock Mech Min Sci 2004;41(6):987–99. [10] Biernatowski K, Puła W. Probabilistic analysis of the stability of massive bridge abutments using simulation methods. Struct Safety 1988;5(1):1–15. [11] BlazquezR, Der Kiureghian A. Seismic reliability of retainingwalls. In:Lind NC, editor. 5th International conference on application of statistics and probability in soil and structural engineering (ICASP 5), 25–29 May 1987, BC (Canada): Vancouver; 1987. p. 1149–56. [12] Bolle A. Investigation and allowance for spatial variability. Rev. FranÇaise de Geotech 2000;93:55–66. [13] Bourdeau PL, Amundaray JI. Non-parametric simulation of geotechnical variability. Geotechnique 2005;55(2):95–108. [14] Bourdeau PL, Gutierrez A. Influence de la corrélation entre modes de défaillance sur le dimensionnement des soutènements massifs. In: Bourdeau PL, editor. Symposium on reliability-based design in civil engineering, 7–9 July 1988, EPFL, Lausanne, Switzerland; 1988. p. 343–50. [15] Bowles JE. Foundation analysis and design. 5th ed. McGraw-Hill; 1997. [16] Cardoso AS, Fernandes MM. Characteristic values of ground parameters and probability of failure in design according to Eurocode 7. Geotechnique 2001;51(6):519–31. [17] CEN. Eurocode 7: geotechnical design. Part 1: general rules, ENV 1997-1. Comité Européen de Normalisation – European Committee for Standardization, Brussels; 1994. [18] Cherubini C. Reliability evaluation of shallow foundation bearing capacity on c0 , u 0 soils. Can Geotech J 2000;37(1):264–9.
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