Probabilistic approach to the design of anchored sheet pile walls

Computers and Geotechnics 26 (2000) 309±330 www.elsevier.com/locate/compgeo

Probabilistic approach to the design of anchored sheet pile walls C. Cherubini * Institute of Engineering Geology and Geotechnics, Technical University, Via Orabona 4, 70125 Bari, Italy Received 19 October 1998; received in revised form 22 March 1999; accepted 30 March 1999

Abstract Following a brief review of the physico-mechanical properties of soils, this work analyzes and comments upon some of the most frequently used approaches in anchored sheet pile wall design. The analysis highlights the conceptual di€erences between the various approaches, often leading, inevitably, to markedly diverging results. Although the probabilistic approach cannot be applied extensively, mainly because it is dicult to obtain statistical modelling of the soil mass, it nevertheless enables designers to avoid certain ambiguities that are present in the commonly used approach based on the Safety Factor. In addition, the probabilistic approach also permits handling of the calibrations required for the approaches to partial coecients to be e€ective and applicable to di€erent local conditions associated with the diversity of soils, di€erent modes of construction, etc. Numerical results obtained for a simple probabilistic model lead to conclusions which are certainly not exhaustive but may contribute signi®cant elements for re¯ection. # 2000 Elsevier Science Ltd. All rights reserved.

1. Introduction Up to now, the approach to the geotechnical design of retaining structures has always been essentially deterministic, i.e. with single values assigned to the mechanical properties of soils and with a single factor estimating the safety of the design, regardless of both the uncertainties pertaining to the calculation procedures and/or variability of the soil properties. Eurocode 7 incorporates design procedures implementing partial coecients to reduce strength parameters and to amplify actions, the same as other European

* Corresponding author. Tel.: +39-0805963363; fax: +39-0805963675. E-mail address: [email protected] 0266-352X/00/$ - see front matter # 2000 Elsevier Science Ltd. All rights reserved. PII: S0266-352X(99)00044-0

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National Codes. In addition, the possibility of de®ning the intervening properties on a statistical basis accounts for soil variability in the design approach. These socalled, ``semiprobabilistic'' methods can be calibrated either by referring to the working stress design (WSD) or, more rationally, on the basis of probabilistic methods taking into account the intrinsic variability of soils. The limitation of these methods lies in the lack of knowledge of the probability density distribution (often assumed to be normal) and the ¯uctuation scale (generally assumed from information given in the literature), associated with soil variability. Therefore, researchers should concentrate on developing probabilistic methods based upon a statistical characterisation of soil properties. In the following section, the current knowledge about the variability of soils is outlined. Thereafter, the paper gives a detailed account of classical methods used to calculate retaining structures by the limit state design approach and a probabilistic approach to the evaluation of safety is proposed. 2. The variability of soils Geotechnical parameters for direct use in evaluating stability or calculating deformation can be obtained by two major methodological approaches: . by laboratory testing, allowing ``direct'' measurements of the investigated parameters; . by in situ testing which requires ``transformation models'' to derive values of the parameters from measurements. In either case, however, the investigated soil volumes represent only a minor part of the volume subjected to stress variation in situ. The results of laboratory tests are a€ected essentially by uncertainties resulting from sampling and specimen preparation. Thus, the standard deviation of a soil property measured in the laboratory will relate to the soil's intrinsic variability, the e€ects of sampling disturbance, and of testing procedures. Let  be the ``lumped'' standard deviation, rp that is due to ``repeatability'', and h the standard deviation relating to the intrinsic variability of the investigated geotechnical parameter [1], it is, therefore, possible to write the following relationship:  r  2  2 ÿ rp …1† h ˆ where rp is only approximately known and appears to vary by some 20±30% of . Table 1 shows the most important statistical data relating to the coecients of variation of some geotechnical soil properties [2±6]. Unit weight variability is rather limited (between 1.0 and 27.9%); the friction angle ' is generally characterized by somewhat higher coecients of variations (between 1.0 and 87.2%). However, it appears that the highest coecients of variation

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(C.V.) relate to mean values of ' between 15 and 23 . For ' > 23 , the C.V. are contained between 3 and 15% (Fig. 1). High C.V. values are found for the compression index Cc (between 8.7 and 60%), whereas the lowest value observed for undrained shearing strength cu is 12% and the highest is 145%. However, two considerations can be made about the variability of the latter parameter: (a) C.V. values also diminish for undrained strength cu as the mean increases (See Fig. 2); (b) the very high variability observed with lower cu values can be explained by considering that these are usually measured for normally consolidated soils characterized by cu increasing with depth. It may well be then, that in some Table 1 Coecients of variation of some geotechnical properties Coecients of variation (%) of

No. of samples Mean Min. value Max. value Lower quartile Upper quartile

cu

Cc

0

c0

56 6.5 1.0 27.9 3.0 8.5

62 43.4 12.0 145.0 30.0 52.0

20 31.4 8.7 60.0 19.5 46.5

75 17.0 1.0 87.2 8.6 20.0

10 32.9 12.8 70.0 19.8 46.0

Fig. 1. Measured coecients of variation of friction angle plotted against mean values.

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Fig. 2. Measured coecients of variation of undrained cohesion cu plotted against mean values.

instances variability has been evaluated without ®rst eliminating this trend from the computations. It could be more correct, however, to study the variability of the ratio cu = 0 v0 . A few words should be spent on the probability density functions to which sets of data for certain geotechnical parameters are ®tted. Based on Pearson's 1 and 2 values and on signi®cance tests [7], one can often ®nd a symmetry in the distribution of frequencies which agrees well with a normal distribution [1]. Occasionally, the lognormal distribution also appears to ®t adequately some statistical distributions of geotechnical data [8]. To provide a full picture of the variability of a soil's geotechnical parameters, we should point out that the available data on e€ective cohesion are too few. Table 1 indicates that the e€ective cohesion coecient of variation varies from 12.8 (min) to 70% (max). Harr [9] reports values of the correlation coecient between c0 and '0 obtained from several authors. They are always negative in drained tests, varying from ÿ0.24 to ÿ0.70. Cherubini et al. [10] found that the correlation coecient between c0 and '0 is ÿ0.61 in the Matera Blue Clays. The ¯uctuation scale is a fundamental statistical indicator of spatial variability measuring the distance within which a soil property shows strong correlation [11]. ˆ2

…1 0

…z† dz

…2†

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where …z† is the autocorrelation coecient for a lag z;  would seem to be an attribute of a soil mass: the values of  are very similar for di€erent properties of the same soil [12]. Several forms of autocorrelation functions have been suggested by Vanmarcke [13] and are reported by Keaveny et al. [14]. Usually, a distinction is made between a ¯uctuation scale in the vertical and in the horizontal direction. The values of the vertical ¯uctuation scale v are generally between 0.5 and 2 m, while for the horizontal ¯uctuation scale h the corresponding values are generally of the order of a dozen meters. Some values of v and h are given by Cherubini [5]. The 0i s can be evaluated in di€erent ways as described by Wickremesinghe and Campanella [15], Vanmarcke [16] and Keaveny et al. [14]. The results of in situ tests appear to be easier to handle to evaluate the ¯uctuation scale. In particular, tests yielding almost continuous readings, or otherwise readings at short intervals (just a few cm) ®t the purpose best. Doubtless, it is important to have a good understanding of this parameter because it can signi®cantly reduce the working variance in relation to the kind of problem being investigated. Mostyn and Li [17] propose that probabilistic models not accounting for the spatial correlation of data should be discarded. The ¯uctuation scale  not being considered in the ``probabilized'' geotechnical model (slope stability, bearing capacity, etc) might engender failure probabilities that are often very high, and quite inconsistent with the frequencies observed in practice: the consequence is a lack of con®dence in the probabilistic model. Lastly, geotechnical data often show a clearcut trend with respect to depth. Data reported by Cherubini [5] show that both the overconsolidated Matera Blue Clay and London Clay, have mean values and standard deviations of undrained shear strength that are constant with depth. Conversely, other data, such as the undrained cohesion of the Nong Nhoo Hao Clay [18], show that there are some rather peculiar trends with a tendency of the investigated geotechnical property to rise both in its mean values and standard deviation. Unpublished data by Rethati on the Dunajuvaros Loess and reported in the paper by Cherubini [5] indicate a remarkable variability of C.V. values, for some geotechnical properties, in the upper 4±5 m of depth and a lower variability between 5 and 15 m. According to Li and White [19], three di€erent depth-related trends (Fig. 3), typical of most geotechnical properties, can be described statistically in brief as follows: . mean and standard deviation constant with depth; . linearly varying mean and constant s.d. with depth; . linearly varying mean and s.d. varying with depth. 3. Generalities about sheet pile design The following objectives should be achieved in a geotechnical study of sheet piles [2]:

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Fig. 3. The three main statistical models expressing the variation of a geotechnical property with depth.

a. Analysis of collapse conditions. b. Estimation of stresses and displacements of structural elements under working conditions. c. Estimation of soil displacements adjacent to the site of excavation under working conditions. For the collapse conditions, the study should attempt to test: 1. 2. 3. 4.

The stability of walls and any other retaining systems. The stability of excavations with respect to bottom heave. Stability with respect to piping. The structure's overall stability with respect to potential deep-seated failure, below the wall.

Obviously, a study of this kind is quite complex. The work by Pane and Tamagnini [20] deals with these problems and points out the results that can be provided by the FDM (®nite di€erences) and FEM (®nite elements) calculation codes in comparison with the more commonly used limit equilibrium (LE) analysis. The latter method can be regarded as a useful means for evaluating collapse conditions in professional practice(ULS 1 according to Eurocode 7 [58]) whereas, when dealing with working conditions, we can resort to the so-called simpli®ed ``spring-like'' approaches (Subgrade Reaction Method Ð SRM). In the analysis developed by Pane and Tamagnini, the LE approach seems to be satisfactory both for cantilever and for anchored sheet piles as against other more sophisticated methods, provided that pressure relates to reliable ka and kp coecients, accounting for the soil-structure friction e€ect. However, based upon calculations with the FDM method, the same authors declare that, because of substantial di€erences in the distribution of the vertical stress, the real pressure coecient on the active side appears to be higher than the

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active coecient, whereas the ``resisting'' pressure coecient is lower than the passive coecient. Pane and Tamagnini conclude that these two contrasting e€ects combine in such a way that the same distribution of horizontal pressures is produced in the two approaches, so that the predictions given by the limit states methods about the conditions of failure must be regarded as a lucky coincidence. Nevertheless, here a simple model is proposed that allows the analysis of reliability to be made and is based on the limit state design approach. As explained later in this paper, this kind of analysis requires knowledge of certain statistical parameters of the soil, which is at present an unusual and costly procedure and one of the reasons why probabilistic approaches are seen with some suspicion [21]. It is quite obvious that, with advancing research, more sophisticated calculation methods based on the probability approach will yield more complete and undoubtedly better results and will allow comparisons to be made with real conditions (strains and pressures during construction and operation). Such monitoring appears to be a necessary tool for an adequate calibration of calculation methods. 4. Usual methods of limit state design analysis Some limit state design methods for anchored sheel pile walls are brie¯y reviewed in the following paragraphs (note that stresses in the tie rod and sheet pile wall, taken as structural elements, are not considered here); (a) The working stress design (WSD) is widely used in North America [22]. The depth of embedment below the dredge line is determined by considering the equilibrium of moments about the anchor point (Fig. 4a): P0A LA ‡ PWA LWA ˆ

P0P LP ‡ PWP LWP F

…3†

where P0A and P0P are the resultant e€ective active and passive forces; PWA and PWP are the resultant water pressures acting on either side of the wall; L0s are the distances with respect to the anchor point. The factor of safety F (applied to passive pressure alone), is normally assumed to be equal to 2, or in any case greater than 1.5. Obviously the water pressures assumed in Fig. 4 have been simpli®ed by not considering seepage through the soil mass. Stress in the tie rod, supposed to be horizontal, simply results from the horizontal equilibrium: T ˆ P0A ‡ PWA ÿ

P0p ÿ PWP F

…4†

Shriver and Valsangkar [63] report that such stress is usually increased (for example, by 30%) to account for the inevitable uncertainties in the calculation of thrusts as estimated according to Rankine-Coulomb.

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Fig. 4. a±d, Schematic representations of main deterministic calculus approaches at limit equilibrium.

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(b) In the ``Strength Method'' (SM), a partial coecient of safety, fs , is applied to the friction angle. The aim of this method is to adequately reduce the parameters characterized by the greatest uncertainty. Here again the depth of penetration is obtained by writing an equation of moment equilibrium around the anchor point (Fig. 4b), i.e. ÿ
ÿ
P0A 0m LA ‡ PWA LWA ˆ P0P 0m LP ‡ PWP LWP

…5†

where P0A …0m † and P0P …0m † are the resultant active and passive forces whose factored friction angle is derived from the expression: 0m ˆ tanÿ1



tan 0 fs

 …6†

0 is the soil's friction angle; fs is allowed to range between 1.25 and 1.5 for granular soils. A similar procedure has been adopted in Eurocode 7, which also requires a characteristic value to be de®ned (the value of  in this case) that, if assumed on a statistical basis, implies that a sucient number of data must be processed, whereas, if selected according to one's own experience then it will necessarily be conditioned by a subjective interpretation. (c) In the ``Limit States Design'' (LSD) method, the depth of penetration (see Fig. 4c) is obtained from the expression P0AF LA ‡ PWAF LWA ˆ P0PF LP ‡ PWPF LWP

…7†

where P0AF and P0PF are the factored resultant active and passive forces and; PWAF and PWPF are the factored water pressures. The factors mentioned above are de®ned as follows: ÿ
P0AF ˆ fq
P0A 0f ÿ
P0PF ˆ ft
P0P 0f P0WAF ˆ fWA PWA

P0WPF ˆ fWP PWP   tan 0 0f ˆ tanÿ1 f

where 0f is the diminished friction angle; fq is the load amplifying factor; ft is the strength diminishing factor; fWA and fWP are water pressure modi®cation factors. The Canadian Geotechnical Society [23] suggests the following values:

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fq ˆ 1:25 ft ˆ 0:8 fWA ˆ 1:25 fWP ˆ 0:8 f ˆ 0:85 The Ontario Highway Bridge Design Code [24] suggests the same values for the di€erent factors, though with two exceptions: fWA ˆ 1:10 fWP ˆ 0:90 (d) Potts and Burland's method [62], also known as the Revised Method (RM), proposes a global factor of safety taken as the ratio of the moments of the net resisting pressure divided by the moment of the net activating pressure (Fig. 4d). Hence, the safety factor is here expressed by Fr ˆ

P0

P0PN0 LP AN LA ‡ PWN LWN

…8†

According to Potts and Burland, the recommended value for this factor is at least equal to 2.0. Two more methods are reported by Burland et al. [25] and are brie¯y described as follows (Fig. 5). (e) One method uses values that are cleared of pressures as shown in Fig. 5a and consequently de®nes the Safety Factor [26] as follows:

Fig. 5. a,b Further schematic representations of limit equilibrium approaches.

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Fˆ

LPN SPN LaN SaN

319

…9†

(f) According to the other one, developed by Tschebotario€ [27], the value dmin of penetration depth must be so calculated that La1 Sa1 ˆ LP1 SP1

…10†

By assuming a depth d > dmin , the safety factor is de®ned as F ˆ d=dmin

…11†

Burland et al. [25] compare the results that can be obtained with some of the above methods and conclude that the most e€ective de®nition is the one that requires the safety factor to be applied to the friction angle. However, because this method is not easily applicable to heterogeneous soils, these authors suggest their own solution which they de®ne as logical and not contradictory, whatever the situation. (See also the papers by Evangelista and Viggiani [28] and Cortellazzo and Mazzuccato [29] for more details on comparisons between the above methods.) However, we are proposing here a few more comments. The estimation of the safety factor by means of the ratio between e€ective and minimal depth of penetration is conceptually di€erent to other safety evaluations. The other methods for estimating depth of penetration can be separated into two groups: 1. Methods that identify the safety factor through an equilibrium of moments between acting and resisting thrusts. 2. Methods that diminish the strength parameters (the friction angle, in our case) with or without ampli®cation of actions. Certainly, each of the described methods was developed according to the various researchers' personal experience in conjunction with construction conditions and the soil's geotechnical properties. Currently, researchers are endeavouring to identify methodological procedures that have, as far as possible, a large acceptance and leave (as does Eurocode 7) to individual local realities the selection of partial coecients [30,31]. Just by way of example, Fig. 6 (from a study by Smoltczik [32]) highlights the important di€erences in the depth of penetration and in the stress of the tie rod, due to the di€erent codes of some European countries, for de®nite geometry, watertable level and physicomechanical properties of the soil. Di€erences are very strong, especially with respect to depth of penetration and, above all, of tie rod stress: there is no need for further comments, except that it is imperative to develop methodological approaches that are as homogeneous as possible.

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Fig. 6. Comparative analysis of sheet pile wall depth and anchor thrust according to various existing National Codes in Europe.

5. The estimation of safety and probabilistic approaches As mentioned above, estimating safety from a single coecient that must be compared with a ratio (of forces, moments or pressures) may introduce a number of di€erent approaches many of which imply a possible intrinsic ambiguity of that same ratio. Li et al. [33] have pointed out that in certain cases (slopes and retaining walls), action (demand) and resistance (capacity) can be de®ned in di€erent ways resulting in di€erent values of the deterministic ratio (Safety Factor). On the other hand, Greco and Varone [34] point out that conventional deterministic analysis, is in fact a statistical (Bayesian)type of approach accounting for the historical frequencies of collapse associated with the various values of the safety factor and, thereby, with the various uncertainties inherent in the analysis. Furthermore, according to Nguyen [35] ``a design with a factor of safety of 2.5 does not mean that the structure is twice as safe as that with a factor of safety of 1.25, and likewise, a factor of 1.5 for one potential mode of failure cannot warrant a

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safer design than that based on a calculated factor of safety of 1.4 but for a di€erent potential failure mode''. Ditlevsen and Madsen [36] state clearly that probabilistic (structural) analysis is the art of formulating a mathematical model within which one can ask and get an answer to the question ``What is the probability that a structure behaves in a speci®c way given that one or more of its material properties or geometrical dimensions and properties are of a random or incompletely known nature and/or that the actions of the structure in some respects have random or incompletely known properties?'' According to these authors, probabilistic analysis can then be seen as an extension of the deterministic one, and therefore it appears that it is not true to say that the deterministic methods (WSD) have not carried out, or do not continue to carry out properly, the function of providing ``safety'' in geotechnical design. A de®nite process of continuous adjustment has made it possible to develop deterministic factors of safety such that engineering works and, in the speci®c case discussed here, also geotechnical design can bene®t from good overall reliability. However, it is quite true that growing knowledge about the variability of materials and the need to quantify safety in a more rational manner in any case leads to greater detail in imposing safety factors (consider partial coecients) and also requires a more speci®c understanding of the probability that a limit situation (of collapse or strain) may occur. Comparisons can be made between the safety (or reliability) of each separate collapse (or strain) mechanism. It is, however, essential to point out that this change in perspective is no easy matter, at least from a professional viewpoint, as it requires replacing the estimation of a safety factor that appears to be reassuring with the estimation of a hazard that implies the presence of a probability, that the unfavourable event may occur. Let us now brie¯y review the probabilistic approaches, although we refer the reader to the works by Madsen et al. [37], Ditlevsen and Madsen [36], Casciati and Faravelli [38], Harr [9] and, for geotechnical problems in particular, to Rethati [1] and Smith [61]. Instead, those concerned speci®cally with sheet pile wall design are referred to the papers published by Cortellazzo and Mazzuccato [29], Scarpelli et al. [40], Cherubini and Garrasi [41,42], Cherubini et al. [43], Smith [39], Ramachadran [44], De Quelerij [45]. The Safety Margin (SM), de®ned as the di€erence between capacity and demand, resolves the ambiguity inherent to the capacity/demand ratio but cannot represent (deterministically) a, so to speak, ``universal'' indicator of safety since it is not dimensionless. It is well known that Cornell [46] proposed an index for the evaluation of safety, called the Reliability Index'' ( ),which is given by the ratio of the safety margin expected value to its standard deviation, i.e. ˆ

SM SM

…12†

This de®nition re¯ects a number of problems related to the expression of the safety margin, so much so that Hasofer and Lind [47] identi®ed as the distance between the origin and limit state boundary in a transformed uncorrelated parameter space.

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Recently, starting from the matrix formulation of the Hasofer-Lind reliability index, namely q …13† ˆ min …x ÿ m†T
Cÿ1 …x ÿ m† x2F

where x is a vector representing the set of random variables; m their mean values; C the covariance matrix; F the failure region. Low and Tang [48] proposed a more intuitive interpretation of , noting that the previous expression suggests that the Hasofer-Lind index can be calculated by minimizing the quadratic form subject to the constant that the ellipse (for 2D cases) or ellipsoid or hyperellipsoid just touches the surface or hypersurface of the failure region F. To calculate by this method, a spreadsheet can be used successfully as shown by Low and Tang; when dealing with two random variables, this method appears to be highly descriptive, also from a graphical point of view. In conclusion, for a probabilistic type of calculation approach in geotechnics, one has to: . de®ne one or more performance functions that will ®t the problem to be investigated; . know the mean (unbiased) values of the basic variables to be introduced into the model; . know the (unbiased) s.d. of the same variables and, possibly, the probability density function; . know the value of the ¯uctuation scale. Each listed step has its own weak points. In particular, the estimation of the ¯uctuation scale often appears to be dicult because of the large amount of data required for its determination. This and other similar reasons for ``uncertainty'' are not valid reasons for avoiding the de®nition of the uncertainties (and variabilities). On the contrary, the greater the uncertainties, ``the more urgent is the need for reliability analyses'' [49]. 6. The probabilistic approach The schematic representations of actions and resistances and the geometrical data describing a sheetpile wall are shown in Fig. 7. The equilibrium relative to the anchor point that is required for evaluating penetration depth de®nes the ®rst expression of the safety margin SM1 ˆ Mp ÿ Ma

…14†

where Mp is the moment of passive thrust relative to the anchor point; Ma is the moment of active thrust relative to the anchor point.

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Fig. 7. Schematic representation for probabilistic calculus of embedment depth and anchor thrust.

A second expression of the safety margin is needed to evaluate stress in the tie rod, after the depth of penetration has been established, namely SM2 ˆ Sp ‡ T ÿ Sa

…15†

where Sp and Sa are the passive and active thrusts; T is the anchor strength. A sheetpile wall with a free height H of 10 m, and with the anchor placed at 0.2 H from the top was considered. A cohesionless soil was considered with a deterministic unit weight equal to 19 kN/m3 while the only variable geotechnical parameter was taken to be the friction angle  with the following values for the mean and coecient of variation:  ˆ 24 ÿ 28 ÿ 32 C:V: ˆ 10ÿ20% The vertical ¯uctuation scale v of the friction angle was taken to be equal to 0.5 and 1.0 m, values which are compatible with those found in the literature.

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The reduction of variance was developed separately for active and passive pressures according to the equation: ÿ2L

   v v L 1 1ÿ for ˆ > L 4L v 2

…16†

Concerning the passive pressure coecients kp , we chose to use the values reported by Caquot and Kerisel [50] for an earth-wall friction  equal to =2 and to . Some analytical laws have been derived from tabulated values of the passive coecient kp : kp ˆ

a ‡ cx ‡ ex2 1 ‡ bx ‡ dx2

…17†

where for  ˆ =2 the constants assume the following values: a ˆ 0:99527 b ˆ 0:03111 c ˆ 0:01041 d ˆ 0:00024 e ˆ 7:5959 E ÿ 5 and for  ˆ  a ˆ 1:10359 b ˆ ÿ0:03418 c ˆ 0:00849 d ˆ 0:000295 e ˆ 0:000308 The case where the earth-wall friction angle is equal to 0 was not considered in this work. Some calculations on this point are reported in the article by Cherubini and Colella [51]. Comrel's calculation code [52] was used to develop the analyses, assuming gaussian distribution of the random variable . 7. Comment on results Fig. 8 shows the depth D of penetration obtained for the di€erent mean values of  and for the two indicated values of C.V., with a ¯uctuation scale taken to be 0.5 m. As mentioned above, the earth-wall friction angle was assumed to be equal to =2 and . A value of ˆ 3 was imposed [53], roughly corresponding to a collapse

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Fig. 8. Results in terms of embedment depth (the ¯uctuation scale assumed is equal to 0.5 m).

probability of approximately 10ÿ3. The ®gure also shows that the higher the value of the earth-wall friction the lower the depth of penetration. The latter is remarkably increased if the coecient of variation is higher, all other conditions being equal. The mean value of  plays a fundamental role. All other conditions being unchanged, passing from a 10 to 20% coecient of variation generally implies an increase in length of more than half a meter. A clearly de®ned value of , namely 28 , is now submitted for the reader's special attention. The following ®gures show, for  ˆ 28 , how and the Safety Factor F (according to the WSD method) vary with the variations of the dimensionless ratio (H+D)/H (see Figs. 9a±d). Obviously no statistical parameter appears in the WSD method, while the probabilistic method introduces a comparison not only between di€erent coecients of variation but also between the two values of the ¯uctuation scale v . It is quite clear then that passing from a smaller to a larger ¯uctuation scale results in a much longer depth of penetration while the other parameters (both statistical and other) are unchanged. It is equally clear how the coecient of variation a€ects the value of the Reliability Index. The smaller the C:V:, the higher the Reliability Index, the depth of penetration being equal. Note however that, with the parameters considered here, the depth of penetration, with the WSD method and with F ˆ 2, is always greater than that obtained with the probabilistic approach with ˆ 3. This is especially true in the case of v ˆ 0:5 with C:V: ˆ 20% whereas with v =1.0 m the di€erences are smaller.

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Fig. 9. Variation of safety factor and reliability index for  ˆ 28 and the di€erent values shown of coecient of variation, v ; . a:  ˆ =2 v ˆ 0:5 m, b:  ˆ  v ˆ 0:5 m, c:  ˆ =2 v ˆ 1:0 m, d:  ˆ  v ˆ 1:0 m.

If a 10% coecient of variation is assumed, then the depth of penetration for a Reliability index ˆ 3 are in any case much lower then the depths obtainable with the, so to speak, deterministic method. Table 2 sums up the values of stress in the tie rod obtained by means of the horizontal equilibrium of forces by imposing ˆ 3 in the probabilistic model (depth of penetration obtained probabilistically, again for ˆ 3). The ®rst value in the table

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Fig. 9. (continued).

was obtained with a ¯uctuation scale of 0.5 m and the second is for v =1.0 m. Note that the values are little in¯uenced either by the earth-wall friction angle or by the value taken up by the ¯uctuation scale. For the same , a stronger in¯uence is observed when the value of the coecient of variation is subject to changes. Instead the results on stress in the tie rod are less strongly a€ected by the ¯uctuation scale and by the wall-earth friction angle.

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Table 2 Values of anchor thrust (assuming ˆ 3 for the second performance function) T (kN)

 ˆ 24  ˆ 28  ˆ 32

 ˆ =2 ˆ  ˆ =2 ˆ  ˆ =2 ˆ

C:V: ˆ 10%

C:V: ˆ 20%

286±294 276±285 225±231 218±225 182±189 175±183

311±336 301±325 250±270 243±263 202±222 195±218

8. Conclusions Probabilistic methods are currently being applied only within geotechnical engineering research due to a number of reasons: among them, the problems involved in, and the cost of, developing a soil model that will account for soil variability. Add to that the problem of calibrating the calculation model which, at any rate, is also shared by the deterministic approaches. Still, it is worth insisting on the investigation at the probabilistic approach so that the approaches to partial coecients may be tested and calibrated as they are being increasingly used in Europe and worldwide. Concerning the probabilistic approach proposed as a means of estimating penetration depth for anchored sheet pile walls, some computer outputs, appropriately developed in the form of diagrams, have quite clearly shown how strongly the possibility of arriving at a given value of the reliability index for the investigated problem is in¯uenced by the ¯uctuation scale and by the coecient of variation of the geotechnical parameter involved (here ). For the value of CV and the assumed value of v , the value of penetration depth is less than the corresponding value calculated by the classical method (WSD) where the safety factor F is assumed to be equal to 2. Obviously these are only partial results that cannot be generalized as this would require more complete analyses and an indispensable comparison with the reality of soil behaviour. References [1] Rethati L. Probabilistic solutions in geotechnics. Amsterdam: Elsevier, 1989. [2] Kulhawy H, Roth NJS, Grigoriu NB. Some statistical evaluations of geotechnical properties. Mexico City: VI ICASP, 1991 (pp. 705±712). [3] Cherubini C, Giasi CI, Rethati L. The coecients of variation of some geotechnical parameters. In: Li, Lo, editors. Proceedings of the Conference on Probabilistic Methods in Geotechnical Engineering. Canberra, 1993. pp. 179±83. [4] Cherubini C. The variability of geotecnical parameters. In: Breysse D, editor. Probabilities and materials-tests, models and applications. Dordrecht: Kluwer Academic Publishers, 1993. p. 69±80.

C. Cherubini / Computers and Geotechnics 26 (2000) 309±330

329

[5] Cherubini C. Data and considerations on the variability of geotechnical properties of soils. Proc. of the ESREL`97 Lisbon 1997; 2: 1583±91. [6] Becker DE. Eighteenth Canadian geotechnical colloquium: limit states design for foundations Part II. Development for the national building code of Canada. Canad Geotec Journal 1996;33:984±1003. [7] Kottegoda NT, Rosso R. Statistics, probability and reliability for civil and enviromental engineers. New York: Mc-Graw Hill, 1997. [8] Chowdhury RN. Recent developements in landslide studies: probabilistic methods. In: State of the Art Session VII International Symposium on Landslides. Toronto, 1984. p. 209±28. [9] Harr ME. Reliability based design in civil engineering. New York: McGraw Hill Book Company, 1987. [10] Cherubini C, Cucchiararo L, Germinario S, Giasi CI. Elaborazione statistica preliminare sulle principali caratteristiche geotecniche delle Argille Azzurre della CittaÁ di Matera. In: Proceedings of 2nd Workshop ``Informatica e Scienze della Terra'', GIAST, Sarnano, 1990. p. 63±70. [11] Campanella RG, Wickremesinghe DJ, Robertson PK. Statistical treatment of cone penetrometer test data.Proceedings of ICASP, 5 Vancouver, 1987; 2; 1011±9. [12] Bao CG, Huang WF, Chang QN. Reliability analysis of bearing capacity for a gravity wharf foundation in random ®eld theory. In: Proceedings of the Conference on Probabilistic Methods in Geotechnical Engineering, Canberra, 1993. p. 291±4 [13] Vanmarcke EH. Probabilistic characterization of soil pro®les. In: Specialty workshop on site characterization and exploration. Northwestern University, Illinois, USA, 1978. p. 199±219. [14] Keaveny JM, Nadim F, Lacasse G. Autocorrelation functions for o€shore geotechnical data. In: 5th Int. Conf. on Structural Safety and Reliability ICOSSAR, 1989. p. 263±70. [15] Wickremesinghe D, Campanella RG. Scale of ¯uctuation as a description of soil variability. In: Proceedings of the Conference on Probabilistic Methods in Geotechnical Engineering. Canberra, 1993. p. 233±9. [16] Vanmarcke EH. Probabilistic modelling of soil pro®les. Journ of Geot Eng Div ASCE 1977;103(H):1227±46. [17] Mostyn GR, Li KS. Probabilistic slope analysis. State of play. In: Proceedings of the Conference on Probabilistic Methods in Geotechnical Engineering. Canberra, 1993. p. 89±109. [18] Bergado DT, Chang JC. Application of on interactive computer aided probabilistic slope stability assessment for embankments os soft Bangkok Clay. In: Proceedings of The International Symposium on Computer and Physical Modelling in Geotechnical Engineering, Bangkok, 1989. p. 25±52. [19] Li KS, White W. Probabilistic characterization of soil pro®le. Reasearch studies from the Department of Civil Engineering. University College. Australian Defence Force Academy UNSW Report No 20, 1987. [20] Pane V, Tamagnini C. Problemi generali delle analisi delle opere di sostegno. IV Convegno Nazionale Ricercatori d'Ingegneria Geotecnica. Perugia, 1997. Vol. II, p. 1±120. [21] Gudehus G. Numerical methods versus statistical safety in geomechanics. Numerical Methods in Geomechanics 1988. 85±86. [22] US Department of the Navy. Foundations and earth structures. Washington (DC): NAVFAC Design Manual 7.2, 1986. [23] Canadian Foundation Engineering Manual. MontreÂal: Canadian Geotechnical Society, 1985. [24] Ontario Highway Bridge Design Code. Ontario Ministry of Transportation and Comunications. Highway Engineering Division. Toronto, 1983. [25] Burland JB, Potts DM, Walsh NM. The overall stability of free and propped embedded cantilever retaining walls. Ground Engineering 1981;14(5):28±38. [26] Piling Handbook. British Steel Corporation, 1979. [27] Tschebotario€ GR. Foundations, retaining and earth structures. New York: Mc Graw-Hill, 1973. [28] Evangelista A, Viggiani C. Opere di sostegno. Atti delle Conferenze di Geotecnica. XIII Ciclo, Torino, 1987. [29] Cortellazzo G, Mazzucato OO. Coeciente di sicurezza e probabilitaÁ di rottura delle paratie ancorate. Rivista Italiana di Geotecnica 1995;XXIX(2):95±108. [30] Jappelli R. Premises for geotechnical instructions. Seminario Internazionale. Raccomandazioni per il restauro strutturale dei beni architettonici. CNR, Ravello, Italy, 1995. p. 15±31.

330

C. Cherubini / Computers and Geotechnics 26 (2000) 309±330

[31] Jappelli R. Eurocodice 7: Progettazione geotecnica. Scopi, principi e compatibilitaÁ con le norme italiane Rivista Italiana di Geotecnica, 2 1996;3:26±35. [32] Smoltczyk U. Geotechnics-Research and Reality. In: Proceedings of the ninth Danube-European Conference on SMFE, Budapest, 1990. p. 31±7. [33] Li KS, Lee Ik, Lo CR. Limit state design in geotechnics. In: Li, Lo, editors. Probabilistic methods in geotechnical engineering. Balkema, Rotterdam, 1993. p. 29±47. [34] Greco VR, Varone R. Il metodo Monte Carlo nelle analisi probabilistiche di stabilitaÁ dei pendii. XVI Convegno Nazionale di Geotecnica, Bologna 1986;II:73±82. [35] Nguyen VU. Safety factors and limit states analysis in geotechnical engineering. Discussion, Canadian Geotechnical Journal 1985;22:144. [36] Ditlevsen O, Madsen HO. Structural reliability methods. New York: John Wiley & Sons, 1996. [37] Madsen HO, Krenk S, Lind NC. Methods of structural safety. London: Prentice Hall, 1986. [38] Casciati F, Faravelli L. Fragility analysis of complex structural systems. New York: Research Studies Press Ltd. J. Wiley & Sons, 1991. [39] Smith GN. The use of probability theory to assess the safety of propped embedded contilever retaining walls. Geotechnique 1987;35:451±60. [40] Scarpelli G, Mazzaferro P, Fruzzetti V. Progetto di opere di sostegno con metodi probabilistici. IV Convegno Nazionale dei Ricercatori Universitari in Geotecnica Perugia, 1997. p. 95±108. [41] Cherubini C, Garrasi A. A probabilistic approach to the study of an anchored bulkhead stability. In: Sixth International Conference on Applications of Statistic and Probability in Civil Engineering. Mexico City, 1991. p. 833±8. [42] Cherubini, C., Garrasi A. Designing anchored sheet pile walls by probabilistic methods. In: Li, Lo, editors. Probabilistic methods in geotechnical engineering. Canberra. Balkema, 1993. p. 319±24. [43] Cherubini C, Garrasi A, Petrolla C. The reliability of on anchored sheet-pile wall empedded in cohesionless soil. Canadian Geotechnical Journal 1992;29(3):426±35. [44] Ramachandran K. Reliability analysis of propped embedded cautilever walls. In: Proceedings 6th International Conference on Numerical Methods in Geomechanics. Innsbruck, 1988. Vol. 2, p. 1225±30. [45] De Quelerij, L. Probabilistic analysis of anchored sheet piling. In: Proceeding 12th International Conference on Soil Mechanics and Foundation Engineering. Rio de Janeiro, 1989. Vol.2, p. 869±72. [46] Cornell, C.A. First order uncertainties analysis of soils deformation and stability. In: Proc. ICASP1, 1971. p. 129±44. [47] Hasofer AM, Lind NC. Exact and invariant second moment code format. A.S.C.E. Journal Engineering Mechanics Division V 100 EM1 1974; 111±121. [48] Low BK, Tang WH. Ecient reliability evaluation using spreadsheet. Journal of Engineering Mechanics 1997;123(7):749±52. [49] Lacasse S. Reliability and probabilistic methods XIII ICSMFE, New Delhi, 1994. p. 225±7. [50] Clayton CRF, Milititsky S, Woods RI. Earth pressure and earth-retaining structures. 2nd ed. London: Blackie Academic & Professional, 1993. [51] Cherubini C, Colella F. Probabilistic analysis of anchored sheet pile walls stability in cohesionless soils. Third International Conference on Computational Mechanics. Santorini, 1998. p. 561±565. [52] Comrel. Reliability consulting programs. GMBH Munchen, 1997. [53] Meyerhof GG. Limit state designs in geotechnical engineering. Struct Safety 1982;1(1):67±71. [58] Eurocode 7 Geotechnical design. Part 1: general rules. CEN European Committee for Standardization. Bruxelles, 1997. [61] Smith GN. Probability and statistics in civil engineering. London: Collins, 1986. [62] Potts DM, Burland JB. A parametric study of the stability of embedded earth retaining structures. TRRL Report SR813. Transport and Road Research Laboratory, 1983, Crowthorne, UK. [63] Shriver AB, Valsangkar AJ. Anchor rod forces and maximum bending moments in sheet pile walls using the factorial strength approach. Com Geotech J 1996;33:815±21.