This paper deals with quantifying the uncertainty associated with settlement predictions that are based on elastic theory. Because of spatial soil variability, deterministic solutions cannot provide information about the likely error of the prediction. Accordingly, it is only through luck that a particular deterministic prediction coincides with the observed settlement.William S. Kaggwa, Mei T. Cheong, and Mark B. Jaks

Cheong, M.

Jaksa, M.

Kaggwa, G.

Adelaide Research &amp; Scholarship

79ASSESSMENT OF THE LUCK ASSOCIATED WITH SETTLEMENTPREDICTIONS THAT ARE BASED ON ELASTIC THEORYWilliam S. Kaggwa, Senior Lecturer, Mei T. Cheong, Student, and Mark B. Jaksa, SeniorLecturer, Department of Civil and Environmental Engineering, The University of Adelaide, North Terrace, Adelaide, SA 5005, AustraliaT: 61-8-8303-5471; F: 61-8-8303-4359; E: wkaggwa@civeng.adelaide.edu.auABSTRACTThis paper deals with quantifying the uncertainty associated with settlement predictions thatare based on elastic theory.  Because of spatial soil variability, deterministic solutions cannotprovide information about the likely error of the prediction.  Accordingly, it is only throughluck that a particular deterministic prediction coincides with the observed settlement. INTRODUCTIONSettlement prediction is undertaken as part of either design of new structures or assessmentof the performance of existing structures.  Although both design and assessment deal withperformance, the underlying concepts of the two approaches to settlement prediction aredifferent.  The design process is concerned with ensuring that excessive settlements do notoccur and as such, the use of conservative soil parameters is common.  If the predictedsettlement is greater than the observed settlement, the design is deemed to be satisfactorysince larger settlements were incorporated in the design, less settlement should not causeany problems. On the other hand, in the assessment of performance the aim is to ensure that the predictionis as close as possible to the actual field observation.  The use of conservative values of soilparameters is no longer acceptable because the measure of success is how close aprediction is to the observed performance.  Accurate settlement prediction is very importantin staged construction.  It is also important in the assessment of existing defective structures,or where additional loads are proposed.Settlement prediction, within a few percentages of the observed settlement is consideredimpossible.  The difficulties faced by engineers revolve around the uncertainties associatedwith the loading conditions, the soil response or site conditions, and the prediction processadopted. The impact of variable loading conditions is more readily appreciated compared tothe other two issues.  It is instructive to evaluate the impact of the idealisations andsimplifications that are part and parcel of the prediction process.  Unfortunately this is notalways done with the rigour that it deserves. Vanmarcke (1) enumerated three major sources of uncertainties concerning soil responsecharacterisation. These are: 1. Natural heterogeneity of the soil, which arises from variations in mineral compositionand stress history of the site.KAGGWA, CHEONG & JAKSA802. Measurement errors, where measured values differ from actual field values due tosample disturbance, test imperfections and human factors.  These errors can bereduced by careful experimentation.3. Uncertainty attributed to the limited availability of information from site investigations.Soil profile characteristics therefore have to be inferred from limited field testing orlaboratory investigation of a limited number of soil samples.The problems of soil heterogeneity can only be minimised by thorough testing and modellingof the spatial variability.  However, not enough attention is paid to the effects of theseuncertainties in settlement prediction.  The shortcomings of using a deterministic approachled to the development of probabilistic methods in the late sixties most notably by Wu andKraft (2).  However, the probabilistic approach is not widely used in practice because of thestatistical terminology and concepts that are not familiar to most geotechnical engineers.Moreover, it is perceived that adopting probabilistic analyses would require an inordinateamount of data, time and effort. VARIABLE SOIL CONDITIONS MEANS VARIABLE OUTCOMESMinimal site investigations, due to perceived time and cost constraints often result in limitedsubsurface data, leading to problems during the design stage.  The resulting time delays andbudget blowouts invariably exceed any savings on site investigation.  There is no doubt thatsettlement prediction based on inadequate knowledge is risky and fundamentally unreliable.Soil parameters vary spatially and hence a single prediction is as useful as a singlemeasurement of a soil parameter for the entire soil profile.  As soil is naturally variable, thereought to be a range of likely predictions.  As an example, consider a circular footing 2 m indiameter supporting 100 kPa average pressure, at a site underlain by soft clay.  Thesettlement of the footing can be evaluated based on elastic theory (Equation 1) to assess theperformance of the structure. ( )21 ν−=ρ ρsEqdI(1)where ρ = vertical displacement; q = applied pressure; d = diameter of footing; E = Young’smodulus; and ν = Poisson’s ratio.Consider two values of Young’s modulus Es = 2 MPa and 6 MPa, obtained from results oftwo boreholes drilled close to the location and from laboratory testing.  Based on a value ofinfluence factor, Iρ = 1.05, based on Poulos and Davis (3),and Poisson’s ratio ν = 0.2, thepredicted settlements are summarised in Table 1. Table 1: Possible settlement predictions for the exampleIt can be seen that there can be three possible outcomes of predicted settlement.  It is notreadily apparent which value will be close to the observed value, and as such the reliabilityof the estimated settlement cannot be quantified. Basis Young’s Modulus(MPa)Predicted settlement (mm)Borehole 1 2 50Borehole 2 6 17Average value 4 25ASSESSMENT OF THE LUCK ASSOCIATED WITH SETTLEMENT PREDICTION81DESCRIPTION OF SOIL VARIABILITYInadequacy of the Deterministic ApproachIn the conventional method, analyses are based on the idealisation of the soil profile as aseries of uniform soil layers, each soil layer being represented by a single characteristicvalue of the soil parameter.  This characteristic value may be determined from laboratorytesting, in-situ measurements or a combination, plus experience.  It is basically engineer-dependent. Figure 1 illustrates the idealisation of the soil profile based on cone tip resistancedata from a CPT test.  The soil profile is subdivided into 5 uniform soil layers, with each layerbeing represented by a single value of cone tip resistance.  It should be noted that thecharacteristic values are engineer-dependent where conservative engineers would use lowervalues while less conservative engineers may use the average values. Figure 1: Idealisation of soil profile as a series of uniform layers based on CPT testdataApplication of Statistical ConceptsThe spatial variation of any soil parameter, vz can be represented by a mean or trendfunction tz , which corresponds to the average value, and a random variation from the trend,ζz , as shown in Equation 2.zzz tv ζ+= (2)According to Vanmarcke (1), the spatial randomness ζz, is described by the standarddeviation σ, which measures the intensity of fluctuation, and the scale of fluctuation δ, whichmeasures the distance within which a soil property shows relatively strong correlation, asshown in Figure 2.C P T  fo r C D 1001000200030004000500060000 100 200 300 400 500C o n e  T ip  R e s is tan ce , q c (M P a)Depth, mm300 MPa40 MPa230 MPa110 MPa190 MPaKAGGWA, CHEONG & JAKSA82It is wrongly assumed that a lot of testing is required in order to derive the mean andstandard deviation of a soil parameter.  This is not always necessary, however, becausethere is extensive published data on soil parameters.  There are numerous empiricalcorrelations based on world-wide studies, as well as data derived from local experience, tohelp estimate the mean value of a particular parameter.  The standard deviation can beestimated from local data by approximating the standard deviation as one-sixth of the rangeof available data.Figure 2: Characterisation of spatial variability of a soil parameter within a soil layer(Phoon and Kulhawy 4)IMPLICATIONS OF SOIL VARIABILITY ON PREDICTED SETTLEMENTSThe computer program FLEA5 developed by Small and Booker (5), based on finite layeranalysis of elastic materials, was used to investigate the effects of spatial variation ofYoung’s modulus on the vertical displacement at the centre of a flexible circular footing.  Theflexible circular footing was supported by a clayey soil 8 m thick.  A pressure of 100 kPa wasapplied to the footing.  The soil layer was subdivided into 49 sub-layers (Figure 3), with eachlayer assigned a random value of Young’s modulus, determined from Monte Carloformulation to simulate a normal distribution function with specified mean and standarddeviation. Figure 3: Schematic of model problem analysed using program FLEA5The spatial variation of Young’s modulus is described by the coefficient of variation COV (E),the ratio of standard deviation σ, to mean value µ.  Monte Carlo simulation was adopted inthe analyses and 200 realisations, or runs, were performed using FLEA5 for each coefficient8 mdSub-layerFootingASSESSMENT OF THE LUCK ASSOCIATED WITH SETTLEMENT PREDICTION83of variation of Young’s modulus.  The Young’s modulus profiles were generated using aproprietary random number generator. Each profile contains 49 independent random datapoints that are uniformly distributed and generated based on a mean of 0.5 for COV (E) =5%, 10%, 15%, 20%, 25%, and 30%.  Only Case I and Case III of Figure 1 were used tostudy the influence of variation of Young’s modulus on the vertical displacement of shallowfoundation.  The effect of variations in Poisson’s ratio on settlement prediction was notexamined as Fraser and Wardle (6), Cambou (7), Baecher and Ingra (8), and Fenton andGriffith (9) showed that uncertainty in vertical displacement is relatively insensitive toPoisson’s ratio.  The analyses also investigated the effect of varying the width to depth ratioon the settlement.  In the analyses values of depth to radius ratio, (h/a) = 2, 4, 8 and 20,were used. In order to avoid introducing effects due to variations of scale of fluctuation, thedepth of each sub-layer was kept constant while the radius of the foundation was varied.It should be noted that the use of 200 runs implies errors of estimates of the order of 7% forthe mean, and 10% for the standard deviation, of the predicted settlement.  Bearing theseerrors in mind, there are two effects of spatial variability on settlement.  The first effect is onthe expected (mean) settlement, expressed by the settlement influence factor in Equation 1.This effect is shown in Figure 4. It can be seen that the settlement influence factor increasesas the variability of the soil increases.  Another way of representing this effect is to define avariability factor, Iv, shown in Figure 5, defined by:ρ=ρ .Vm I (3)where, ρm is the mean settlement for a specific value of COV (E) and ρ is the settlement of auniform layer with constant Young’s modulus.1.01.41.82.22.63.00 5 10 15 20n o rm alis e d  d e p th  o f s o il,  h /asettlement influence factor,  I ρE la s tic s o lu tio n , ν= 0 .2E la s tic s o lu tio n , ν = 0COV(E)=30%COV(E)=5%aPh zFigure 4: Settlement influence factor for COV (E) = 5% and 30%, compared withsolutions of Meyerhof (10) for a uniform soil profileThe second effect of variable Young’s modulus is on the variability of the predictedsettlement, expressed in terms of the coefficient of variation of normalised settlement, COV(ρi/ρm).  This is shown in Figure 6. It can be seen that the coefficient of variation of settlementincreases with increasing variation of Young’s modulus.KAGGWA, CHEONG & JAKSA84R2 = 0.99511.001.051.101.150.0 0.1 0.2 0.3coefficient of variation, COV(E)variability factor, Iv =( ρv / ρ)Figure 5: Variation of variability factor of settlement with variability of Young’smodulus using (h/a) = 2 to 200.00.10.20.30 0.1 0.2 0.3 0.4COV ( E )COV ( ρI / ρm ) h/a = 20h/a = 2Figure 6: Relation between the coefficients of variation of settlement and Young’smodulus for a circular footing on a uniform soil profileEVALUATION OF PROBABILITY OR ‘LUCK’ IN SETTLEMENT PREDICTION The results shown in Figures 5 to 7 can be used to evaluate the likelihood that the observedsettlement will lie within a specified range of predicted values.  By incorporating probabilityinto the prediction process, a clearer picture emerges about the likely range of settlements.This is very helpful in decision-making, thereby quantifying the risks inherent in theprediction. The method is illustrated using the following example.A flexible circular footing is to be constructed over a site, with 30-metre deep variable soillayer overlying bedrock.  The width of the footing is fixed at 3 m.  The footing is to carry apressure of 50 kPa.  Based on past experience, the Young’s modulus of the alluvium isvariable but the average value is 5 MPa.  Laboratory testing of soil samples from the siteshows that the Young’s modulus of the soil at site can be as high as 9.5 MPa.  At a nearbylocation, the lowest value of Young’s modulus obtained from laboratory testing is 3.5 MPa.ASSESSMENT OF THE LUCK ASSOCIATED WITH SETTLEMENT PREDICTION85Determination of Statistical ParametersBased on experience and soil testing, the variation of Young’s modulus of the soil can bedetermined by computing the mean and standard deviation, from which COV (E) isdetermined.  The standard deviation can be estimated from the difference between thehighest value obtained or expected, and the mean value based on testing and experience.  Ifthe two sites are treated independently, then we have two cases.  For Case 1 (based on thedifference between maximum and average values), σ = 1.5 and COV (E) = 0.3, and for Case2 (based on difference between average and minimum values), σ = 0.5 and COV (E) = 0.1.Incorporation of the Effects of Spatial Variability of Young's ModulusIn the example, the width to depth ratio, (B/h) = 0.10 (diameter of 3 m and soil layerthickness of 30 m).  The mean settlement ρm can be determined using the expected (mean)value of Young’s modulus and displacement influence factor, I (e.g. Poulos and Davis, 3).The variability factor Iv, for a COV (E) of 0.1 and 0.3 can be obtained from Figure 6. Iv = 1.02and 1.11 for COV (E) = 0.10 and 0.30, respectively.  Accordingly, the mean settlement,ρm = 31 mm for COV (E) = 0.1 and ρm = 33 mm for COV (E) = 0.3.The standard deviation of settlement can be determined from the radius to depth ratio, (a/h)= 0.05 (footing radius of 1.5 m and soil layer thickness of 30 m) and the use of Figure 7.  Thecoefficient of variation COV (ρi/ρm) = 0.19 for a COV (E) =0.3 and COV (ρi/ρm) = 0.04 for aCOV (E) =0.1.  It can be shown that ρσ = 1.2 mm for COV (E) = 0.1 and ρσ = 6.3 mm forCOV (E) = 0.3.  Thus the predicted settlement is 31 +1.2 mm for COV (E) = 0.1 and33 +6.3 mm for COV (E) = 0.3. Evaluation of Probability Associated with PredictionIn order to determine the probability of occurrence of a particular range of values ofsettlement, the standard normal distribution chart can be used.  The mean and standarddeviation of the estimated settlement are standardised using Z = (x–µ)/σ, where x is themagnitude of settlement of interest.Suppose we are interested in evaluating the likelihood that the observed settlement will liebetween 28 and 30 mm or 25 and 35 mm.  Table 2 is a summary of the computations.Table 2: Summary computation table for probability of settlement falling in givenrange for the example problemSettlementrange(ρl – ρu)(mm)(1)COV(E)(2)Settlement,mean andstandarddeviation (mm)(3)Range of valuesof  Z(Zl – Zu)(4)Range ofprobabilities(pl – pu)(5)Probabilityofsettlementfalling inrange(6)Comment onprobability ofoccurrence (or luck)(7)28 to 30 0.1 31 and 1.2 –2.500 to –0.833 0.006 to 0.202 0.196 1 in 5 chance28 to 30 0.3 33 and 6.3 –0.794 to –0.476 0.214 to 0.317 0.103 1 in 10 chance25 to 35 0.1 31 and 1.2 –5.000 to 3.333 0.00 to 0.9996 0.9996 Extremely likely25 to 35 0.3 33 and 6.3 –1.270 to 0.317 0.102 to 0.625 0.5225 1 in 2 chanceIt can be seen that for the profile with COV (E) = 0.1, there is a 20% likelihood that theobserved settlement will be between 28 and 30 mm compared to 10% for COV (E) = 0.3.  Inother words, if the designer predicted settlements to be between 28 and 30 mm, theKAGGWA, CHEONG & JAKSA86likelihood of success (or luck) would be 10 to 20%.  On the other hand, if the designer wasinterested in a less precise estimate, and adopted a wide range for the settlement, say 25 to35 mm, then for COV (E) = 0.1, this range essentially brackets the likely observations(99.96% likelihood). In the case of COV (E) = 0.3, there is only 52% likelihood that theobserved settlement will be within that range.CONCLUSIONSIt has been shown that the effects of variation of Young’s modulus within a soil profile, on thepredicted elastic settlements, can be readily quantified and incorporated in routine analyses.Increasing spatial variability increases the expected (mean) settlement and the variationabout this mean value.  A simple procedure has been presented to enable prediction of thestatistical moments of the predicted settlement, and how these can be used to evaluate thelikelihood that a particular settlement prediction will eventuate.NOTATIONCOV coefficient of variation (ratio of standard deviation to mean value)Iρ settlement influence factorIv variability factor for settlement to account for randomly variable soilREFERENCES1. Vanmarcke, E.H (1977). “Probabilistic modeling of soil profiles.” Journal of theGeotechnical Engineering Division, ASCE, Vol. 103, No GT11, pp. 1227-1245.2. Wu, T.H. & Kraft, L. M. (1967). “The probability of foundation safety.” Journal SoilMechanics and Foundation Division. ASCE, Vol.93, No. SM5, pp. 213-231.3. Poulos, H.G., and Davis, E. (1974). Elastic Solutions for Soil and Rock Mechanics.Wiley, New York.4. Phoon, K-K and Kulhawy, F.H. (1999). Characterisation of geotechnical variability.Canadian Geotechnical Journal, 36: pp. 612-624.5. Small, J.C. and Booker (1999). Program Finite Layer Elastic Analysis Version 5. Centrefor Geotechnical Research, The University of Sydney, Australia.6. Fraser, R.A. & Wardle, L.J., (1975). “Raft foundation – case study and sensitivityanalysis.” Proceedings-2nd International Conference on Applications of Statistical &Probability in Soil & Structural Engineering, 15th –18th September, Aachen, F.RGermany, Deutsche Gesellschaft fur Erd-und Grundbau e.v 43 Essen 1,KronprinzenstraBe 35a, pp. 89-99.7. Cambou, B. (1975). "Application of first order uncertainty analysis in the finite elementmethod in linear elasticity." Proceedings 2nd International Conference. Application ofStatistics and Probability in Soils and Structural Engineering, Aachen, pp. 67-88.8. Baecher, G.B. & Ingra, T.S. (1981). “Stochastic FEM in settlement predictions.” Journalof the Geotechnical Engineering Division, ASCE, Vol.107, No GT4, pp. 451-463.9. Fenton, G.A. and Griffiths, D.V. (2002). "Probabilistic foundation settlement on spatiallyrandom soil. Journal of Geotechnical and Geoenvironmental Engineering, ASCE,Vol.128, No 5, pp. 381-390.10. Meyerhof, G.G. (1965). "Shallow foundations." Journal of Soil Mechanics and FoundationDesign, ASCE, Vol. 91, SM2, pp. 21-31.

Assessment of the luck associated with settlement predictions that are based on elastic theory

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Assessment of the luck associated with settlement predictions that are based on elastic theory

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Assessment of the luck associated with settlement predictions that are based on elastic theory

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Adelaide Research & Scholarship