Neural network based stochastic design charts for settlement prediction M.A. Shahin, M.B. Jaksa, and H.R. Maier
Abstract: Traditional methods of settlement prediction of shallow foundations on granular soils are far from accurate and consistent. This can be attributed to the fact that the problem of estimating the settlement of shallow foundations on granular soils is very complex and not yet entirely understood. Recently, artificial neural networks (ANNs) have been shown to outperform the most commonly used traditional methods for predicting the settlement of shallow foundations on granular soils. However, despite the relative advantage of the ANN based approach, it does not take into account the uncertainty that may affect the magnitude of the predicted settlement. Artificial neural networks, like more traditional methods of settlement prediction, are based on deterministic approaches that ignore this uncertainty and thus provide single values of settlement with no indication of the level of risk associated with these values. An alternative stochastic approach is essential to provide more rational estimation of settlement. In this paper, the likely distribution of predicted settlements, given the uncertainties associated with settlement prediction, is obtained by combining Monte Carlo simulation with a deterministic ANN model. A set of stochastic design charts, which incorporate the uncertainty associated with the ANN method, is developed. The charts are considered to be useful in the sense that they enable the designer to make informed decisions regarding the level of risk associated with predicted settlements and consequently provide a more realistic indication of what the actual settlement might be. Key words: settlement prediction, shallow foundations, neural networks, Monte Carlo, stochastic simulation. Résumé : Les méthodes traditionnelles de prédiction des tassements de fondations superficielles sur des sols pulvérulents sont loin d’être précises et consistantes. Ceci peut être attribué au fait que le problème d’évaluer le tassement des fondations superficielles sur les sols pulvérulents est très complexe et pas encore tout à fait compris. Récemment, on a montré que les réseaux artificiels de neurones (ANN) donnaient de meilleurs résultats que les méthodes traditionnelles pour prédire le tassement des fondations superficielles sur les sols pulvérulents. Cependant, en dépit des avantages relatifs de l’approche basée sur l’ANN, elle ne prend pas en compte l’incertitude qui peut affecter la grandeur du tassement prédit. Les ANN, comme la plupart des méthodes traditionnelles de prédiction des tassements, sont basées sur des approches déterministiques qui ignorent l’incertitude et fournissent ainsi des valeurs uniques de tassement sans indication du niveau de risque associé à ces valeurs. Une approche stochastique alternative est essentielle pour fournir une estimation plus rationnelle du tassement. Dans cet article, la distribution probable des tassements prédits, considérant les incertitudes associées à la prédiction du tassement, est obtenue en combinant la simulation de Monte Carlo avec un modèle déterministique d’ANN. On a développé un ensemble de diagrammes stochastiques pour la conception qui incorporent l’incertitude associée à la méthode d’ANN. On considère que les graphiques sont utiles, en ce sens qu’ils permettent au concepteur de prendre des décisions éclairées concernant le niveau de risque associé aux tassements prédits, et conséquemment fournissent une indication plus réaliste de ce que le tassement réel pourrait être. Mots clés : prédiction de tassement, fondations superficielles, réseaux de neurones, Monte Carlo, simulation stochastique. [Traduit par la Rédaction]
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Introduction Settlement prediction, as in many geotechnical engineering problems, is often affected by a considerable level of uncertainty. Such uncertainty may produce an unreliable estimation of the magnitude of settlement, while reliable set-
tlement prediction is essential for design purposes. Uncertainty affecting settlement prediction is generally caused by one or more of the following (Cherubini and Greco 1991; Krizek et al. 1977): (i) parameter (input variables) uncertainty and (ii) uncertainty associated with the model used for settlement prediction.
Received 24 March 2003. Accepted 13 September 2004. Published on the NRC Research Press Web site at http://cgj.nrc.ca on 16 February 2005. M.A. Shahin. Department of Civil, Mining and Environmental Engineering, University of Wollongong, NSW 2522, Australia. M.B. Jaksa1 and H.R. Maier. School of Civil and Environmental Engineering, The University of Adelaide, South Australia 5005, Australia. 1
Corresponding author (e-mail: [email protected]
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There are a number of major and minor factors that contribute to parameter uncertainty. The major factors include (i) poor knowledge of soil properties and (ii) uncertainty in forecasting the magnitude of the imposed loads. Uncertainty associated with poor knowledge of the soil properties is due to the natural spatial variability of soil, which is caused by variations in the mineral composition and characteristics of soil strata during soil formation. It is also due to insufficient description of soil characteristics as a result of limited spatial sampling. Uncertainty associated with this source can also be due to errors associated with the technique used to measure the actual soil properties. Theoretically, the uncertainty associated with the loads acting on foundations can be determined with an acceptable degree of accuracy (Greco and Cherubini 1993). However, realistically, it is unlikely that an accurate estimation of the magnitude of design loads can be made and thus loads should be treated as random variables (Corotis 1972; Peir and Cornell 1973; Melchers 1987). The minor factors that contribute to parameter uncertainty include (i) footing dimensions and (ii) footing embedment depth. These sources of parameter uncertainty are due to discrepancies between footing dimensions or the footing embedment depth implemented on site and those that appear in construction drawings as a result of human error. The second source of uncertainty is caused by the inherent error associated with the modeling technique used to characterize settlement prediction and is usually called model uncertainty (Frey 1998). This type of uncertainty is due to the simplified nature of models that are usually used to describe soil behavior, which are generally based on a number of assumptions that provide uncertainty. Unfortunately, model uncertainty is difficult to measure physically (Juang et al. 1991) and in most instances, the model used to describe a certain phenomenon is assumed to be a perfect predictor (i.e., has minimal or no prediction error). However, if sufficient measured and predicted data are available and assuming that the measured data are error free, then the uncertainty associated with the prediction method used can be quantified and assumed to correspond to model uncertainty. Most deterministic modeling methods for settlement prediction of shallow foundations on granular soils disregard the above uncertainties in their analysis and simulation. One way to include such uncertainties is to use stochastic simulation. Stochastic simulation has a significant benefit over deterministic methods in the sense that the degree of risk (i.e., uncertainty) associated with the model output can be quantified (Jaksa 1995; Whitman 2000). In recent times, artificial neural networks (ANNs) have been used successfully for settlement prediction of shallow foundations on granular soils and have been found to outperform the most commonly used traditional methods (Shahin et al. 2002a). However, thus far, ANNs have only been used for deterministic settlement prediction. In this paper, stochastic analysis is applied to an ANN model to obtain stochastic models of ANN settlement prediction of shallow foundations on granular soils that incorporate either parameter or model uncertainty. Applications of stochastic analysis to deterministic ANN models have been used in many situations in civil engineering. For example, Goh and Kulhawy (2003) used neural network
and reliability analysis to model the limit state surface. Papadrakakis et al. (1996) implemented neural network and Monte Carlo simulation for the reliability analysis of elastic–plastic structures. Papadrakakis and Lagaros (2002) also used neural networks and Monte Carlo simulation for reliability-based structural optimization. Tran-Canh and Tran-Cong (2002) used neural network and stochastic simulation for the computation of viscoelastic flow. Zou et al. (2002) used a neural network embedded Monte Carlo approach for the incorporation of input information uncertainty in water quality modeling. This paper has three objectives. The first objective is to present practical stochastic approaches that incorporate either parameter uncertainty or prediction method (model) uncertainty in the analysis of ANN settlement prediction of shallow foundations on granular soils. The second objective is to examine the effect of varying the parameter uncertainty on the magnitude of the predicted settlement. Finally, and most significantly, the third objective is to develop and provide a set of stochastic design charts that are based on the ANN method. These charts will be a valuable tool for the design of shallow foundations on granular soils. To demonstrate the first two objectives set out above, a numerical example is provided.
Deterministic ANN model for settlement prediction The present study uses an ANN model to obtain deterministic settlement predictions of shallow foundations on granular soils. The ANN model was developed by Shahin et al. (2002a) and uses feedforward multilayer perceptrons (MLPs) that are trained with the back-propagation algorithm (Rumelhart et al. 1986). Details of the ANN model development are beyond the scope of this paper and are given by Shahin et al. (2002a). The model has five inputs representing the footing width, B; the net applied footing load, q; the average blow count obtained using a standard penetration test (SPT) over the depth of influence of the foundation, N, as a measure of soil compressibility; the footing geometry (length to width of footing), L/B; and the footing embedment ratio (embedment depth to footing width), Df /B. The single model output is foundation settlement, Sm. The database used for model development comprises a total of 189 individual cases (Shahin et al. 2002a), which is amongst the largest data sets used to develop such models. The data were obtained from the literature and span a wide range of the input and output data records, as summarized in Table 1. The available data were divided randomly into three sets: training, testing, and validation; in such a way that they are statistically consistent and thus represent the same statistical population (Masters 1993; Shahin et al. 2004). The training set was used to adjust the model free parameters (i.e., connection weights), the testing set was used to decide when to stop training to avoid overfitting, and the validation set was used to test the predictive ability of the model in real-world situations. In total, 80% of the data were used for training and 20% were used for validation. The training data were further divided into 70% for the training set and 30% for the testing set. The optimum model geometry was determined using a trial-and-error approach in which ANN models were © 2005 NRC Canada
Can. Geotech. J. Vol. 42, 2005 Table 1. Data ranges used for developing the ANN model. Model variables
Footing width, B (m) Footing net applied load, q (kPa) Average SPT blow count, N Footing geometry, L/B Footing embedment ratio, Df /B Measured settlement, Sm (mm)
0.8 18.3 4.0 1.0 0.0 0.6
60.0 697.0 60.0 10.5 3.4 121.0
trained with one hidden layer and 1, 2, 3, 5, 7, 9, and 11 hidden layer nodes, respectively. The optimal network parameters were obtained by training the ANN model with different combinations of learning rates and momentum terms. A model with two hidden layer nodes, a learning rate of 0.2, a momentum term of 0.8, tanh transfer function for the hidden layer nodes, and sigmoidal transfer function for the output layer node was found to perform best. Details of the above parameters are given by Shahin et al. (2002a). The performance of the ANN model is summarized in Table 2. It can be seen that the model performs well, as it has high correlation coefficients, r, low root mean squared errors (RMSE), and mean absolute errors (MAE) between the measured and predicted settlements for all three data sets (i.e., training, testing, and validation). A comparison carried out by Shahin et al. (2002a) on the validation set utilizing the ANN model and three of the most commonly used traditional methods (i.e., Meyerhof 1965; Schmertmann et al. 1978; Schultze and Sherif 1973), indicated that the ANN method provides more accurate settlement predictions than the traditional methods (Table 3). To facilitate the ANN technique for deterministic settlement prediction of shallow foundations on cohesionless soils, the information obtained from the ANN model was translated into a relatively simple handcalculation formula (Shahin et al. 2002b) and also into a set of design charts (Shahin 2003) suitable for practical use. An illustrative set of the deterministic design charts based on the ANN is shown in Fig. 1.
Overview of stochastic analysis of settlement prediction Over the last decade or so, interest in applying more rational stochastic analyses in the field of geotechnical engineering, rather than the less accurate traditional deterministic solutions, has increased rapidly (Tang 1993). In particular, for example, in the area of settlement prediction of shallow foundations, Padilla and Vanmarcke (1974) developed a stochastic approach for settlement prediction of a onedimensional model based on a first-order probabilistic description of loads and soil properties. Fraser and Wardle (1975) used a first-order probabilistic analysis to develop a model for the determination of total and differential settlement of raft foundations resting on layered cross-anisotropic elastic soils, taking into account the uncertainty associated with the imposed loads and supporting soil modulus. Cherubini and Greco (1991) presented a probabilistic approach for settlements predicted using the method proposed by Arnold (1980) to estimate the settlement of spread footings on sand, taking into account the uncertainty associated
Table 2. Performance of the ANN model. Data set
Training Testing Validation
0.930 0.929 0.905
10.01 10.12 11.04
6.87 6.43 8.78
Note: r, correlation coefficient; RMSE, root mean squared errors; MAE, mean absolute errors.
Table 3. Performance of ANN and traditional methods for the validation set. Performance measure
Schultze and Sherif (1973)
Schmertmann et al. (1978)
r RMSE (mm) MAE (mm)
0.905 11.04 8.78
0.440 25.72 16.59
0.729 23.55 11.81
0.798 23.67 15.69
with the reliability of the technique used for settlement prediction. Brzakala and Pula (1996) also combined finite element analysis with stochastic simulation to provide a probabilistic solution for the estimation of settlement of shallow foundations, considering three basic sources of input parameter uncertainty: random shape of the subsoil (location of an interface between two strata), random material parameters, and random loads. Fenton et al. (1996) estimated probabilistic measures of total and differential settlement of spread footings on elastic soils using a two-dimensional finite element model combined with Monte Carlo simulation, taking into account the variability of the soil modulus of elasticity. Sivakugan and Johnson (2002) applied probabilistic analysis to settlement prediction of four deterministic traditional methods including the methods proposed by Terzaghi and Peck (1967), Schmertmann et al. (1978), Burland and Burbidge (1985), and Berardi and Lancellotta (1994), considering the prediction method uncertainty.
Stochastic settlement prediction To determine the impact of the parameter uncertainty or model uncertainty on predicted settlements obtained from the ANN technique, Monte Carlo simulation is applied to the deterministic ANN model described previously. Monte Carlo simulation attempts to generate a random set of values from known or assumed probability distributions of some variables involved in a certain problem. Full details of the Monte Carlo technique are given by many authors (e.g., Hammersley and Handscomb 1964; Rubinstein 1981). The approach used to obtain the uncertainty in settlement prediction associated with each of the two types of uncertainty © 2005 NRC Canada
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Fig. 1. Illustrative set of deterministic design charts based on the ANN model (L /B = 1.0 and Df /B = 0.0).
considered is given below, and an illustrative numerical example follows. It should be noted that, in the section below, when parameter uncertainty is considered, the ANN model is assumed to be a perfect predictor (i.e., it has no prediction
error or model uncertainty), as mentioned previously. On the other hand, when the model uncertainty is considered, the model input variables are assumed to be error free (i.e., they have no parameter uncertainty). © 2005 NRC Canada
Inclusion of parameter uncertainty For an individual case of settlement prediction, the procedure for obtaining the stochastic solution that incorporates parameter uncertainty is as follows: (1) The values of each input variable (i.e., B, q, N, L /B, Df /B) are generated randomly by knowing or assuming their probability density functions (PDF) and any correlation that exists among them; (2) Assuming that the ANN model is a perfect predictor, the deterministic predicted settlement is obtained from the ANN model developed by Shahin et al. (2002a) using either the model itself, the hand-calculation formula derived by Shahin et al. (2002b), or the set of design charts developed by Shahin (2003); (3) The above two steps are repeated thousands of times as part of Monte Carlo simulation; and (4) Finally, the settlements obtained are used to determine the cumulative distribution function (CDF) or to plot the cumulative probability distribution from which the probability of nonexceedance (PN/E), or level of risk, associated with a certain prediction can be estimated. Among the five inputs to the ANN model, two, the footing net applied pressure, q, and the average SPT blow count, N, are likely to include more than marginal parameter uncertainty, and thus, in this work, are assumed to be random variables. As mentioned earlier, footing dimensions contribute to parameter uncertainty to a lesser degree and are thus assumed to be deterministic for practical purposes. In addition, the input variable of footing embedment depth, Df, is also assumed to be deterministic. A number of studies have attempted to characterize the uncertainty associated with q and N, as discussed below. According to Melchers (1987), loads acting on structures can be divided into two broad groups: natural loads (e.g., wind and earthquake) and human-imposed loads (e.g., dead loads and live loads), and the magnitude of each varies with time and location. Consequently, estimation of total loads imposes uncertainty. As a guide, Auvinet and Rossa (1991) showed that the coefficient of variation, COV (i.e., standard deviation/mean), of permanent loads for Mexico City buildings is 8%. Melchers (1987) stated that dead loads are commonly assumed to be closely approximated by a normal distribution with a COV of 5%–10%. Rao (1992) stated that dead loads are usually described by a normal distribution with a COV of 10%. Krizek et al. (1977) demonstrated that the uncertainty associated with the estimation of the total imposed loads follows a normal distribution, and Fraser and Wardle (1975) illustrated that the COV of the total imposed loads that can be encountered in practice is equal to 14%. Padilla and Vanmarcke (1974) also showed that, if dead and live loads were assumed to be stochastically independent, the resulting variability of their sum would have a COV of 10%. Uncertainty associated with the SPT blow count, N, as a measure of soil compressibility is considerable because of the many factors that affect SPT results (Orchant et al. 1987). Fletcher (1965) identified 13 factors that affect the SPT, which can be categorized into the following two major groups: (i) equipment effects (e.g., hammer, hammer drop system, drill rods, and sampler) and (ii) procedural–operator effects (e.g., height of hammer drop, seating of the sampler,
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errors in counting blows, and cleaning of borehole). Lee et al. (1983) reported that the uncertainty associated with the average SPT blow count can be assumed to follow a normal distribution with the COV ranging from 27% to 85% and they recommended a value equal to 30%. Inclusion of model uncertainty The stochastic solution that incorporates the model uncertainty is based on the assumption that previous measured settlements of foundations may be employed to predict the settlements of other foundations in similar conditions (Cherubini and Greco 1991). The uncertainty of the prediction method (model uncertainty) can be examined by calculating the settlement ratio, k (Cherubini and Greco 1991; Sivakugan and Johnson 2002), which is defined as the ratio of the predicted settlement to the actual measured settlement, assuming that the measured settlement is error free. If a set of predicted and measured settlements is available, the settlement ratios can be calculated and used to obtain the probability density function (PDF) of k. A number of studies have attempted to characterize the PDF of k. For example, Greco and Cherubini (1993) demonstrated that the distribution of k could be approximated by a lognormal distribution for settlement predictions obtained from the methods proposed by Arnold (1980) and Papadopoulos (1992). Sivakugan and Johnson (2002) showed that k can be represented by a beta distribution for settlements predicted using four traditional methods, (i.e., Terzaghi and Peck 1967; Schmertmann et al. 1978; Burland and Burbidge 1985; Berardi and Lancellotta 1994). In the present work, the distribution of k is obtained using the 189 case records used by Shahin et al. (2002a) to develop the deterministic ANN model, as shown later. A Monte Carlo simulation can then be conducted to estimate the uncertainty associated with the predicted settlements. The detailed procedure is as follows: (1) The PDF of k is estimated using a set of predicted and measured settlements; (2) For an individual case of settlement prediction, the deterministic settlement is calculated using: (i) the ANN model developed by Shahin et al. (2002a), (ii) the handcalculation formula (Shahin et al. 2002b), or (iii) deterministic design charts (Shahin 2003); (3) A random value of k is generated from the PDF of k obtained in Step 1; (4) From the definition of k, the deterministic predicted settlement in Step 2 is divided by the generated random value of k from Step 3, and the corresponding actual settlement is calculated; (5) Steps 3 and 4 are repeated for many iterations (Monte Carlo simulation); and (6) The settlements obtained as part of the Monte Carlo simulation are used to estimate the CDF or to plot the cumulative probability distribution from which the probability of nonexceedance (PN/E), or level of risk, associated with a certain settlement prediction, can be estimated.
Numerical example The following case study is examined. A rectangular footing, the dimensions of which are 2.5 m × 4.0 m, is founded © 2005 NRC Canada
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115 Table 4. Statistics for parameter uncertainty used in the numerical example. Settlement parameter
Footing width, B (m) Net applied footing load, q (kPa) Average SPT blow count, N Footing geometry, L /B Footing embedment ratio, Df /B
2.5 350 16 1.6 0.6
Deterministic 35 4.8 Deterministic Deterministic
deterministic 10 30 deterministic deterministic
— Normal Normal — —
Note: —, not applicable.
at a depth of 1.5 m below the ground surface. The soil beneath the footing is sand that extends to a depth in excess of two times its width. The net applied footing load is 350 kPa and the average SPT blow count is 16. Estimation of parameter uncertainty To apply the stochastic solution that incorporates parameter uncertainty, the steps described previously for the inclusion of parameter uncertainty are followed. The statistical data representing the uncertainty associated with the settlement parameters are taken to be equal to those commonly encountered in practice and recommended in the literature, as described earlier, and are shown in Table 4. In addition, the database used by Shahin et al. (2002a) for the development of the ANN model is utilized to determine the coefficient of correlation between the footing net applied pressure, q, and the average SPT blow count, N, which was found to equal 0.4. The statistical data in Table 4 are used to generate sample values for q and N (Step 1) and the corresponding deterministic settlement is calculated using the ANN model developed by Shahin et al. (2002a) (Step 2). As mentioned previously, when considering the parameter uncertainty, the ANN model is assumed to be a perfect predictor and consequently, the uncertainty associated with the parameters q and N is the only source of uncertainty. It should be noted that the values generated for q and N are obtained so as to be within (i) the range of data that can be expected in practical applications and (ii) the ranges of the input data used for training of the ANN model. The PC-based software @Risk (Palisade 2000) is used for this purpose. Steps 1 and 2 are repeated until a convergence criterion is achieved (Step 3). To determine whether convergence has been achieved, the statistics describing the distribution of the predicted settlements are calculated at fixed numbers of simulations and compared with the same statistics at previous simulations. Convergence is deemed to have occurred if the change in the statistics describing the distribution of predicted settlement is 1% or less. It was found that 1300 simulations are sufficient to achieve convergence. The predicted settlements obtained for the 1300 simulations are used to plot the cumulative probability distribution curve from which different probabilities of nonexceedance are obtained (Step 4). The results are shown in Fig. 2, which also includes the predicted single value deterministic settlement associated with the numerical example, and are summarized in Table 5 (columns 1 and 3). The results shown in columns 2 and 4 of Table 5 will be explained later in this section. The predicted deterministic settlement for the numerical example that is shown in Fig. 2
is obtained from the ANN model developed by Shahin et al. (2002a) and is found to be 13.3 mm. It can be seen from Fig. 2 that there is an approximately 50% probability that the settlement could be higher than the deterministic single estimation of 13.3 mm. This result indicates that the uncertainty associated with q and N can considerably affect settlement and thus, should not be neglected in the analysis and simulation of settlement prediction. In addition, there are probabilities of 75%, 80%, 85%, 90%, and 95% (i.e., probability levels that may be needed for design purposes) that the settlement will not exceed 18.6, 20.4, 22.4, 25.4, and 31.3 mm, respectively (Table 5). As discussed earlier, uncertainty estimation of q and N varies considerably (i.e., the COV varies from 5% to 14% for q and from 27% to 85% for N). Consequently, it is worthwhile to carry out a parametric study to examine the effect of changing the COV for q and N on the magnitude of settlement prediction for the numerical example. Using the ANN-based stochastic approach that incorporates parameter uncertainty, two different combinations of the values of the COV for q and N are examined. The minimum values recommended in the literature for the COV of q and N (i.e., 5% for q and 27% for N) are used for one trial and the maximum values (i.e., 14% for q and 85% for N) are used for the other. The PN/E for the predicted settlement using the three different combinations of the COV for q and N are shown in Table 5. It can be seen from Table 5 that the predicted settlement becomes more conservative as the COV for q and N increases. For example, if the PN/E required in the design of a footing under consideration is 95%, the predicted settlement will not exceed 29.2 mm when the COVs for q and N are equal to 5% and 27%, respectively. In other words, there is a 5% chance that the predicted settlement will exceed 29.2 mm when the COVs for q and N are equal to 5% and 27%, respectively. However, to achieve the same level of risk, this settlement will exceed 31.3 mm when the COVs for q and N are increased to 10% and 30%, respectively. Moreover, for a level of risk of 5%, the predicted settlement will exceed 42.6 mm when the COVs for q and N are increased further to be equal to 14% and 85%, respectively. The results in Table 5 also illustrate that to achieve a level of risk of only 5%, the predicted settlement obtained using the maximum combination of the COV for q and N (i.e., 14% for q and 85% for N) is approximately 45% more than the predicted settlement obtained when the minimum combination of the COV for q and N (i.e., 5% for q and 27% for N) is used. This suggests that estimating the correct values of the uncertainty associated with q and N is very important © 2005 NRC Canada
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Fig. 2. Cumulative probability distribution incorporating parameter uncertainty for the numerical example.
Table 5. Predicted settlements accounting for parameter uncertainty of different q and N values for the numerical example.
Fig. 3. Gamma distribution of k.
COV for q and N (%) PN/E (%)
q = 5, N = 27
q = 10, N = 30
q = 14, N = 85
75 80 85 90 95
18.0 19.6 21.3 24.2 29.2
18.6 20.4 22.4 25.4 31.3
19.5 23.6 28.1 34.2 42.6
because uneconomical design of footings results from increasing values of the COVs for q and N. Estimation of model uncertainty The steps described previously for the inclusion of model uncertainty are used as follows. The PDF of k for the ANN method is obtained from the 189 data records that are used by Shahin et al. (2002a) for the development of the ANN model (Step 1). As mentioned previously, when considering model uncertainty, the input variables included in the 189 data records are assumed to be error free and consequently, the difference between the measured and predicted settlements (i.e., model uncertainty) is the only source of uncertainty. The PC-based software @Risk (Palisade 2000) is used to determine the PDF that provides the best fit to the 189 data points. For a given set of data values, @Risk can identify the probability distribution that best fits these values from 38 candidate distributions, and it provides statistical properties that describe the distribution. The theoretical distribution that is found to best match the actual distribution of k is the Gamma distribution (Fig. 3). The statistical properties of the Gamma distribution obtained are given in Table 6. The deterministic single solution of settlement prediction was obtained previously from the ANN model given by Shahin et al. (2002a) and was found to be 13.3 mm (Step 2). From the statistical properties of the Gamma distribution obtained, random values of k are generated (Step 3). The nu-
Table 6. Gamma distribution parameters of k. Statistical parameter
Minimum Maximum Mean Std. dev. Shape parameter (α) Scale parameter (β)
0.30 10.30 1.40 1.10 1.22 0.97
merical example is recalculated by dividing the settlement predicted in Step 2 by the generated value of k obtained from Step 3, and a corresponding actual settlement is calculated (Step 4). The above procedure (Steps 3 and 4) is re© 2005 NRC Canada
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Fig. 4. Cumulative probability distribution incorporating model uncertainty for the numerical example.
peated 700 times (Monte Carlo simulation) until a convergence criterion equal to 1% is achieved (Step 5). The predicted settlements obtained for the 700 simulations are used to plot the cumulative probability distribution curve from which different probabilities of nonexceedance are obtained (Step 6). The results are shown in Fig. 4, which also includes the deterministic single settlement value of 13.3 mm, and are summarized in Table 7. It can be seen from Fig. 4 that there is a probability of approximately 72% that the settlement will not exceed the deterministic single estimate of 13.3 mm, which means that there is 28% probability that the settlement could be higher than the deterministic estimate of 13.3 mm. This result indicates that the uncertainty associated with the prediction method can affect settlement and thus, should not be neglected in the analysis and simulation of settlement prediction. In addition, there are probabilities of 75%, 80%, 85%, 90%, and 95% that the settlement will not exceed 13.7, 15.2, 16.8, 19.1, and 20.8 mm, respectively, (Table 7). The results of the preceding numerical example indicate that, for an individual case of settlement prediction, to account for model or parameter uncertainty, there is a range of predicted settlements that should be considered for design purposes. For instance, if 90% PN/E (i.e., 10% risk) is required, the predicted settlements that should be considered for design will range from 19.1 mm, to account for model uncertainty (Table 7), to 34.2 mm, to account for parameter uncertainty (Table 5). It should be noted that the maximum settlement value of this range (i.e., 34.2 mm) could be lower depending on the adopted values of the COV for q and N, as shown in Table 5.
Stochastic settlement prediction design charts The stochastic simulation that incorporates the prediction method (model) uncertainty is used to develop a generic set of stochastic design charts based on the ANN model. The charts are expected to be a useful tool for practitioners from
Table 7. Predicted settlements accounting for model uncertainty for the numerical example. PN/E (%)
Predicted settlement (mm)
75 80 85 90 95
13.7 15.2 16.8 19.1 20.8
which the level of risk associated with predicted settlement can be readily obtained. It should be noted that, as mentioned earlier, parameter uncertainty (i.e., uncertainty estimation associated with q and N) varies considerably. Consequently, similar charts that incorporate parameter uncertainty are not developed in this work and, for a certain case of settlement prediction, the steps described earlier for the inclusion of parameter uncertainty need to be carried out individually to account for a specific combination of the uncertainty estimation of q and N. Since the ANN model predicts the most accurate settlement estimates to date, the subsequent stochastic design charts are considered to be the most reliable of those currently available. The procedure that is used to develop the charts is as follows: (1) A random synthetic value of predicted settlement, which accounts for an individual case of settlement prediction, is generated between the ranges given in Table 1; (2) The approach, outlined previously, which incorporates model uncertainty for obtaining a stochastic settlement prediction is applied to the settlement predicted in the previous step and the corresponding CDF is obtained; (3) From the above CDF, the 75%, 80%, 85%, 90%, and 95% probabilities of nonexceedance are determined; (4) Another random synthetic value of predicted settlement is generated by increasing the value generated in Step 1 by 5% of the total range between the minimum and maximum values given in Table 1; © 2005 NRC Canada
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Fig. 5. Stochastic ANN-based design charts for settlement prediction.PN/E, probability of nonexceedance.
(5) Steps 2 to 4 are repeated until the maximum synthetic value of predicted settlement is reached; and (6) For each probability level of nonexceedance, the synthetic deterministic settlements are plotted against stochastic settlements and a set of design charts is obtained, as shown in Fig. 5. For any individual case of settlement prediction within the ranges of the data shown in Table 1, the deterministic single settlement prediction can be obtained from the ANN model given by Shahin et al. (2002a), the hand-calculation formula (Shahin et al. 2002b), or the deterministic design charts (Shahin 2003). The corresponding stochastic settlement can be readily obtained from Fig. 5, accounting for a certain desired PN/E. For example, if the deterministic ANN model predicts a settlement of 22 mm and reliability levels (i.e., probabilities of nonexceedance) of 90% and 95% are
required, the corresponding design stochastic settlements (Fig. 5) are 32 and 34 mm, respectively. It should be noted that the applicability of these charts is constrained by the range of the 189 data records used to characterize the uncertainty associated with the settlement ratio, k (Table 1). However, the range of applicability of the approach can be extended in the future, by retraining the ANN model and regenerating the design charts should additional data records become available. It should also be noted that, as mentioned earlier, the stochastic solution that incorporates prediction method (model) uncertainty relies on the estimation of the PDF of k and consequently, as many case records of settlement prediction as possible are needed to obtain a reliable estimate of the PDF of k. In this work, all the available data at hand (i.e., 189 cases) are used for the estimation of the PDF of k, as mentioned previously. For further verification © 2005 NRC Canada
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of the stochastic design charts, their results need to be compared with the actual measured settlements of some additional case records, once they become available.
Summary and conclusions Stochastic approaches that utilize the Monte Carlo technique were used to generate stochastic settlement prediction of shallow foundations on granular soils from an ANN model. The proposed stochastic approaches incorporate either parameter uncertainty or prediction method (model) uncertainty and enable the uncertainty associated with predicted settlements to be quantified in the form of a cumulative probability distribution function that provides the designer with the level of risk associated with exceeding a given predicted settlement. A parametric study was also carried out to examine the effect of varying the uncertainty associated with the factors affecting settlement (i.e., coefficient of variation, COV, for the imposed load, q, and soil property, N) on the uncertainty of the predicted settlements. The proposed stochastic approaches were applied to a numerical example for illustration. A series of ANN-based design charts that incorporate model uncertainty were developed for routine use in practice. The results of the numerical example that incorporates parameter uncertainty indicated that there was a probability of approximately 50% that the settlement could be higher than the deterministic single estimation with COVs of 10% and 30% for q and N, respectively. The results also indicated that over the range of COVs for q and N suggested in the literature, the design settlements ranged from 29.2 mm to 42.6 mm for a nonexceedance probability of 95%. These results indicated that the uncertainties associated with q and N can considerably affect settlement and thus, they should not be neglected in the analysis and simulation of settlement prediction. This also implied that it is important to collect sufficient data to characterize the uncertainty associated with q and N, as the results obtained were very sensitive to these variables. On the other hand, the results of the numerical example that incorporates the prediction method (model) uncertainty indicated that there was a probability of approximately 28% that the settlement could be higher than the deterministic single estimation. This result indicated that model uncertainty affects settlement and thus should not be neglected in the analysis and simulation of settlement prediction. It was shown in this work that the developed stochastic charts can be used to predict settlements for a certain desired reliability level given the deterministic settlement predicted from the ANN model developed by Shahin et al. (2002a), which is believed to be a useful tool in the design of shallow foundations on cohesionless soils.
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