load test, pressuremeter test and screw plate load test. However, most of the available methods are restricted by simplifying the problem by incorporating several.
Predicting the Settlement of Shallow Foundations on Cohesionless Soils Using Back-Propagation Neural Networks
by M. A. Shahin M. B. Jaksa H. R. Maier
Department of Civil & Environmental Engineering University of Adelaide
Research Report No. R 167 February, 2000
ABSTRACT Artificial neural networks (ANNs) are a form of artificial intelligence (AI), which in their architecture attempt to simulate the biological structure of the human brain and nervous system. In this report, back-propagation neural networks are used to predict the settlement of shallow foundations on cohesionless soils. More than two hundred cases of actual measured settlements are used to develop and verify the ANN model. The predicted settlements found by utilising ANNs are compared with the values predicted by three commonly used deterministic methods. The results indicate that artificial neural networks are a promising method for predicting settlement of shallow foundations on cohesionless soils, as they outperform the conventional methods.
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CONTENTS ABSTRACT ...........................................................................................................i CONTENTS ..........................................................................................................ii 1. INTRODUCTION ...........................................................................................1 2. OVERNIEW OF NEURAL NETWORKS .....................................................2 2.1 Natural neural networks ............................................................................2 2.2 Artificial neural networks ..........................................................................3 2.3 Network type .............................................................................................4 3. NEURAL NETWORK MODEL FOR SETTLEMENT PREDICTION .........5 3.1 Computer program .....................................................................................5 3.2 Model database...........................................................................................5 3.3 Model development ...................................................................................5 3.3.1 Data division......................................................................................6 3.3.2 Model inputs......................................................................................7 3.3.3 Model architecture.............................................................................8 3.3.4 Model optimization ...........................................................................8 3.3.5 Stopping criteria ................................................................................8 3.3.6 Model validation................................................................................8 4. RESULTS ........................................................................................................9 5. COMPARISON WITH CONVENTIONAL METHODS .............................11 6. SUMMARY AND CONCLUSION ...............................................................12 7. REFERENCES ..............................................................................................14 APPENDIX – Notation ......................................................................................16
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1. INTRODUCTION It is generally understood that sand deposits are much more heterogeneous than their clay counterparts. As a result, differential settlements are likely to be higher in sand deposits than in clay profiles (Maugeri et al., 1998). Because cohesionless soils exhibit high void ratios and high degrees of permeability, settlement occurs in a short time; immediately after load application. Such quick settlement causes relatively rapid deformation of superstructures, which results in an inability to remedy damage and to avoid further deformation. The two major criteria that control the design of shallow foundations on cohesionless soils are the bearing capacity of the soil beneath the footing and settlement of the foundation. However, settlement usually controls the design process, rather than bearing capacity, especially when the breadth of footing exceeds 1 metre (3–4 ft) (Schmertmann, 1970). As a consequence, settlement prediction is a major concern and is an essential criterion in the design process of shallow foundations. The problem of estimating the settlement of shallow foundations on cohesionless soils is very complex due to the uncertainty associated with the factors that affect the magnitude of this settlement. Among these factors are the distribution of applied stresses, the stress-strain history of the soil, the effect of soil compressibility, and the difficulty in obtaining undisturbed samples of cohesionless soil. The geotechnical literature has included many methods, both theoretical and experimental, to predict settlement of shallow foundations on cohesionless soils. Many settlement prediction methods have focussed on correlations with in-situ tests, such as the cone penetration test (CPT), standard penetration test (SPT), dilatometer modulus test (DMT), plate load test, pressuremeter test and screw plate load test. However, most of the available methods are restricted by simplifying the problem by incorporating several assumptions for the factors that affect the settlement of shallow foundations. These factors are difficult to quantify and frequently uncertain. Comparative studies for the available methods (Jeyapalan and Boehm, 1986; Gifford et al., 1987; Wahls, 1997) indicate inconsistent prediction for the magnitude of settlement calculated by these methods. As a result, an alternative method is needed, which reduces the uncertainty involved in settlement prediction. In recent times, artificial neural networks (ANNs) have been applied to many geotechnical engineering tasks and have demonstrated some degree of success. For example, ANNs have been used in pile bearing capacity prediction (Lee and Lee, 1996; Abu Kiefa, 1998), stress-strain modelling of sands (Ellis et al., 1995), interpretation of site investigation (Zhou and Wu, 1994) and in seismic liquefaction assessment (Goh, 1994). However, to the authors’ best knowledge, no study has yet focussed on the prediction of the settlement of shallow foundations on cohesionless soils. 1
In this report, ANNs are used as an alternative technique to predict the settlement of shallow foundations on cohesionless soils. More than two hundred cases of actual field data are used for model development. The results obtained are compared with three conventional methods. 2. OVERVIEW OF NEURAL NETWORKS McCulloch and Pitts (1943) developed the first artificial neuron. However, it was not until the psychologists David Rumelhart, of University of California at San Diego, and James McClelland, of Carnegie-Mellon University, developed the backpropagation algorithm for training multi-layer perceptrons, that interest in ANNs flourished (Rumelhart et al., 1986 a, b; McClelland and Rumelhart, 1988). Recently, ANNs have been applied extensively to many prediction tasks. ANNs are able to determine the relationship between a set of input data and the corresponding output data without the need for predefined mathematical equations between these data. 2.1 Natural neural networks The structure and operation of natural neural networks (NNNs) have been described by many authors (Hertz et al., 1991; Zurada, 1992; Fausett, 1994; Neuralware Inc., 1997). NNNs, of which the brain is an example, consist of billions of densely interconnected nerve cells called neurons. Each neuron receives the combined output signals of many other neurons through the synaptic gaps by input paths called dendrites (Figure 1). The dendrites collect the output signals and send them to the
Figure 1. Typical structure of biological neuron. Source: (Neuralware Inc., 1997) 2
cell body, or the soma of the neuron, which sums the incoming signals. If the charge of the collected signals is strong enough, the neuron is activated and produces an output signal; otherwise the neuron remains inactive. The output signal is then transmitted to the neighbouring neurons through an output structure called the axon. The axon of a neuron divides and connects to dendrites of the neighbouring neurons through junctions called synapses. 2.2 Artificial neural networks Artificial neural networks (ANNs) are a form of artificial intelligence which, in their architecture, try to simulate the biological structure of the human brain. ANNs try to mimic the behaviour of the basic biological and chemical processes of NNNs. ANNs learn “by example” and therefore are well suited to complex processes where the relationship between the variables is unknown (Hubick, 1992). Many authors have described the structure and operation of ANNs (Hecht-Nielsen, 1990; Maren et al., 1990; Zurada, 1992; Fausett, 1994; Ripley, 1996; Neuralware Inc., 1997). ANNs consist of a number of artificial neurons (variously known as “processing elements”, “PEs”, “Nodes” or “Units”) representative of the neurons in NNNs. Each processing element has several input paths and one output path, as shown in Figure 2. An individual PE receives its inputs from many other processing elements via weighted input connections. These weighted inputs are summed and passed through a transfer function to produce a single activation level for the processing element, which is the node output.
Figure 2. Typical processing element (PE) in a neuron. (Source: Neuralware Inc., 1997)
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A typical structure of artificial neural networks consists of many processing elements that are arranged in layers: an input layer, an output layer, and one or more layers inbetween, called intermediate or hidden layers (Figure 3). Each processing element in a specific layer is interconnected to all the processing elements in the next layer via weighted connections. The scalar weights determine the strength of the connection between interconnected neurons. A zero weight refers to no connection between two neurons and a negative weight refers to a prohibitive relationship. The propagation of information starts at the input layer where the input data are presented. The inputs are weighted and received by each node in the next layer. The weighted inputs are then summed and passed through a non-linear transfer function to produce the node output, which is weighted and passed to the processing elements in the next layer. The network’s output is compared with the actual value and the error between the two values is calculated. This error is then used to adjust the weights until the network can find a set of weights that will produce the input-output mapping with the smallest possible error.
Figure 3. Typical structure of ANN (Source: Neuralware Inc., 1997)
2.3 Network type Back-propagation neural networks are adopted in this work, as they have a high capability of data mapping (Hecht-Nielsen, 1990). Back-propagation neural networks have been applied to a wide range of areas including classification, estimation, prediction, and functions synthesis (Moselhi et al., 1992) and they are currently the most widely used neural network. The topology and algorithm details of back-propagation neural networks are beyond the scope of this report and can be found in many publications (Hertz et al., 1991; Zurada, 1992; Fausset, 1994; Picton, 1994; Ripley, 1996). 4
3. NEURAL NETWORK MODEL FOR SETTLEMENT PREDICTION 3.1 Computer program The PC-based software package NeuralWorks Predict Release 2.1 (Neuralware Inc., 1997) is used in this work to simulate the artificial neural network operation. The package works within the framework of Microsoft Excel (Microsoft Corp., 1997). 3.2 Model database A database was constructed, which formed the basis for the artificial neural network model. The data used in this report were collated from the literature and incorporate field measurements for settlement of shallow foundations as well as the corresponding information regarding the footings and soil. The database covers a wide range of variation in footing dimensions (mainly isolated footings and raft foundations) and soil density (generally sands and gravels). The database comprises a total of 272 individual cases and can be found in the literature as summarised in Table 1. Table 1. Data base for ANN model Reference No. of cases Vargas, 1961 2 Levy and Morton, 1974 46 Burland and Burbidge, 1985 114 Picornell and del Monte, 1988 1 Papadopoulos, 1992 83 Wahls, 1997 21 Maugeri et al., 1998 5 Total 272
3.3 Model development The steps in developing ANN models, as outlined by Maier and Dandy (2000b), are used as a guide in this work. There are a number of factors (e.g. network geometry, weight optimisation method, parameters controlling the optimisation method, Stopping criteria), which can have an impact on the results obtained. Detailed investigation of these parameters is considered beyond the scope of this report. The default values suggested in the software package used are adopted unless stated otherwise. Use of the default parameters is considered reasonable as this best reflects an actual modelling situation. Details of the default parameters are given below and discussed in Neuralware Inc. (1997). In subsequent phases of this research, these parameters will be modified to determine their effect on the predictions. The following sections describe the procedure that is used for developing the neural network model. 5
3.3.1 Data division Due to reasons discussed later, the database is randomly divided into three sets: training, testing and validation. In total, 220 cases are used for training, 25 cases for testing and 27 cases for validation. For good model prediction, it is essential that the data used for training, testing and validation represent the same population (Maier and Dandy, 2000a). Consequently, the three sets of data, for each input and output variable, are selected in such a way that they are statistically consistent and thus represent the same population. The statistical parameters used include the mean, standard deviation, skewness, maximum, minimum and range, as shown in Table 2. Despite trying numerous random combinations of training, testing and validation sets, there are still some inconsistencies in the statistical parameters for the training, testing and validation sets that are most closely matched (Table 2). This can be attributed to the fact that the data contain singular, rare event, which cannot be replicated in all 3 data sets. However, on the whole, the statistics are in good agreement and all three data sets may be considered to represent the same population. Table 2. Statistical parameters of the model training, testing and validation sets Model variables and data sets Mean Std. dev. Footing width, B (m) All data 7.4 12.1 Training set 7.7 12.7 Testing set 6.6 10.4 Validation set 6.1 7.1 Footing length, L (m) All data 14.9 22.1 Training set 15.6 23.3 Testing set 11.2 15.9 Validation set 13.0 17.2 Footing applied pressure, q (kPa) All data 225.4 140.1 Training set 223.0 143.7 Testing set 247.5 125.5 Validation set 224 125.1 Average blow count, SPT-N All data 26.2 14.6 Training set 26.1 14.4 Testing set 27.2 15.2 Validation set 26.0 16.3
Statistical parameters Skewness Max. Min.
Range
6.7 6.9 1.9 2.1
150 150 36 29.7
0.3 0.3 0.3 0.3
149.7 149.7 35.7 29.4
3.7 3.7 1.9 3.2
195 195 63.8 86.2
0.3 0.3 0.3 0.3
194.7 194.7 63.5 85.9
0.9 0.9 0.6 0.9
697 697 500 500
6.3 6.3 74 35
690.7 690.7 426 465
0.9 0.7 1.8 1.7
80 80 80 80
4 4 7 6
76 76 73 74
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Table 2. Statistical parameters of the model training, testing and validation sets (continued) Model variables and data sets Mean Std. dev. Measured settlement, Sm (mm)
Statistical parameters Skewness Max. Min.
Range
All data Training set Testing set Validation set
3.1 3.1 2.8 3.1
99.8 99.5 91.3 89.8
13.6 13.5 13.7 14.2
17.5 16.6 22.2 20.3
100 100 91.6 90
0.2 0.5 0.3 0.2
3.3.2 Model inputs An important step in developing the neural network model is to select the input variables that have significant influence on settlement. Consistent with Burland and Burbidge (1985), the following variables are considered to have the greatest effect on settlement of shallow foundations on cohesionless soil and are selected as input variables for the back-propagation neural network: • • • •
Width of footing, B; Length of footing, L; Footing applied pressure, q; and Average SPT blow count/300 mm, SPT-N, over the depth of influence of the foundation, as a measure of the soil compressibility.
The settlement of shallow foundations is the only output variable in the neural network, as shown in Figure 4.
Settlement (Output)
Neural Network hidden layer
Footing width
Footing length
Footing pressure
SPT-N
(Inputs) Figure 4. General neural network model 7
3.3.3 Model architecture The software uses a constructive method for determining the optimal number of hidden nodes rather than assuming a fixed number of hidden nodes in advance. This constructive method is called Cascade learning and was developed Fahlman and Lebiere (1990). This method is characterised by the following steps (Neuralware Inc., 1997): • Initially, the network is trained without hidden nodes and with direct connection between the input layer and the output layer; • Hidden PEs are added randomly one or a few at a time; • New hidden PEs have connections from both the input layer and previously established hidden nodes; and • Construction is stopped when performance on an independent set shows no further improvement. Using the above method, a network with one hidden layer and 41 hidden neurons was found to perform best. 3.3.4 Model optimisation The default parameters used for model optimisation are (Neuralware Inc., 1997): • • • • •
Learning rule: Adaptive gradient learning rule; Learning rate: 100 for the hidden layer and 0.01 for the output layer; Transfer function for hidden layer: Tanh transfer function; Transfer function for output layer: Sigmoid transfer function; and Initial weights: The software uses weights from a uniform population with initial weight distribution inversely proportional to the square root of the number of weights.
3.3.5 Stopping criteria The cross-validation technique (Stone, 1974) is used as the stopping criteria in this work, as it is considered that sufficient data are available to create training, testing and validation sets. It is the most valuable tool to ensure overfitting does not occur (Smith, 1993). 3.3.6 Model validation Once the training phase of the model has been successfully accomplished, the performance of the trained model has to be validated for an independent data set. One of the most important criteria that has to be considered when assessing model performance is that good performance during training can always be attained. However, it is also important for the model to perform well for a set of data 8
previously unseen by the model. Consequently, it is essential to check that the model performs consistently on all three data sets (i.e. training, testing and validation). The coefficient of correlation, r, the root mean square error, RMSE, and the mean absolute error, MAE, are the major criteria that are used to evaluate the performance of the model. Smith (1986) suggested the following guide for values of |r| between 0.0 and 1.0: • |r| ≥ 0.8 • 0.2 < |r| < 0.8 • |r| ≤ 0.2
Strong correlation exists between two sets of variables; Correlation exists between the two sets of variables; and Weak correlation exists between the two sets of variables.
The root mean square error, RMSE, is the most popular measure of error and it has the advantage that large errors receive much greater attention than small errors (Hecht-Nielsen, 1990). Root mean square error, RMSE, and mean absolute error, MAE, are desirable when the data evaluated are smooth or continuous (Twomey and Smith, 1997). 4. RESULTS The best result obtained by the neural network is summarised in Table 3. The plots of the measured and predicted settlements for training, testing and validation sets are shown in Figures 5a, 5b and 5c respectively. The results indicate that the model performs well, as it has high coefficients of correlation and low root mean square errors and mean absolute errors between the measured and predicted settlements. Table 3 shows that the results obtained for the model during validation is generally consistent with those obtained during training and testing. This is to be expected (Table 2), as the input and output variables of this model are statistically consistent. Some of the errors as measured by RMSE and MAE for the training set are greater than those obtained for the testing and validation sets (Table 3). This can be attributed to the fact that all extreme values are contained in the training set.
Table 3. Neural network results Data set Correlation coefficient, r RMSE (mm) 0.94 6.2 All data 0.92 6.7 Training set 0.98 3.4 Testing set 0.98 3.7 Validation set
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MAE (mm) 3.6 3.9 2.7 2.6
Predicted settlement (mm)
125
Training set r=0.92
100
75
50
25
0 0
25 50 75 100 Measured settlement (mm)
125
Figure 5a. Comparison of measured and predicted settlement for ANN training set
Predicted settlement (mm)
125
Testing set r=0.98
100
75
50
25
0 0
25 50 75 100 Measured settlement (mm)
125
Figure 5b. Comparison of measured and predicted settlement for ANN testing set
10
Predicted settlement (mm)
100
Validation set r=0.98
80
60
40
20
0 0
20
40 60 80 Measured settlement (mm)
100
Figure 5c. Comparison of measured and predicted settlement for ANN validation set
5. COMPARISON WITH CONVENTIONAL METHODS The results obtained by utilising ANNs are compared with three of the most commonly used deterministic methods: Meyerhof (1965), Schultze and Sherif (1973) and Schmertmann (1970). For the 27 cases of the validation set, only 13 have complete details of soil, which are required for the calculation of settlement using these conventional methods. As a result, these 13 are considered for comparison. Table 4 shows the measured and predicted settlement for the three deterministic methods and from the ANN. A statistical summary of the comparison for the 13 cases considered is presented in Table 5. The results in Table 5 show that the ANN method performs better than the conventional methods for all three performance measures considered. The coefficient of correlation, r, the RMSE and MAE obtained using the ANN model are 0.99, 3.9 mm and 2.6 mm respectively. In contrast, these measures ranged from 0.33 to 0.86, from 23.8 mm to 45.2 mm and from 11.1 mm to 29.5 mm, respectively, when the conventional methods were used. Table 5 also shows that the method of Shultze and Sherif (1973) performed best out of the conventional methods evaluated.
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Table 4. Measured and predicted settlements Predicted settlement (mm) ANN Meyerhof Schultze & Sherif
Measured settlement (mm) 15.5 10.6 10 7.8 9.5 10.1 2.3 5.2 90 87.5 0.2 1.3 1.2 1.1 6 11.8 10.2 12.7 15.5 10.3 21.2 9 9.3 6.5 2.7 3.5 15.4 10.1 10 5.5 8.7 9.4 74 70.9 3.1 8.6 11 9.7 2.1 3.7 12.7 11.5 10.5 10 6 7 14 13.2 3.4 3.3 12.9 11.7 7 8.4 * Incomplete soil description
11.8 17 28 50 24.3 24 12.5 22.5 31.8 13.8 5 21.3 30 9.3 20 19.8 55 5.8 8.5 11.8 51 10.8 61.8 70.3 17.3 34.3 11.3
* * * 2.4 18.5 0.7 1.1 * * * 4.5 9.6 5.1 * 8.3 7.4 29.7 * * * * 8.1 * * * 11.4 5.8
Schmertmann * * * 20.2 143.9 8 29.1 * * * 12.6 27.2 0.63 * 149.9 28.4 107.2 * * * * 26.9 * * * 28.5 29.9
Table 5. Statistical summary of comparison for 13 cases of validation set ANN Meyerhof Schultze & Sherif Schmertmann Category Correlation, r 0.99 0.33 0.86 0.70 RMSE (mm) 3.9 27.0 23.8 45.2 MAE (mm) 2.6 20.8 11.1 29.5
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6. SUMMARY AND CONCLUSION A back-propagation neural network was used to demonstrate the feasibility of ANNs to predict the settlement of shallow foundations on cohesionless soils. More than two hundred cases of actual field measurements for settlement of shallow foundations on cohesionless soils were used for model development and verification. The predicted settlements obtained by utilising ANNs and three other conventional deterministic methods were compared with the measured settlements. The results indicate that back-propagation neural networks have the capability of predicting the settlement of shallow foundations on cohesionless soils with a high degree of accuracy. The results also demonstrate that the ANN method outperforms the conventional methods for an independent validation set. The ANN method has another advantage over the conventional methods in that once the model is trained; the model can be used as an accurate and quick tool for estimating the settlement of shallow foundations. In contrast with the conventional methods, the ANN method does not need any manual work such as using tables or charts to calculate the settlement.
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7. REFERENCES Abu Kiefa, M. A. (1998). “General regression neural networks for driven piles in cohesionless soils.” J. Geotech. & Geoenv. Engrg., ASCE, 124(12), 1177-1185. Burland, J. B. and Burbidge, M. C. (1985). “Settlement of foundations on sand and gravel.” Proc., Institution of Civil Engineers, Part I, 78(6), 1325-1381. Ellis, G. W., Yao, C., Zhao, R. and Penumadu, D. (1995). “Stress-strain modelling of sands using artificial neural networks.” J. Geotech. Engrg., ASCE, 121(5), 429435. Fahlman, S. E. and Lebiere, C. (1990). “The cascade-correlation learning architecture.” In: Touretzky, D. S. (Ed.), Advances in Neural Information Processing Systems 2, Morgan Kaufmann, San Mateo, CA. Fausett, L. V. (1994). “Fundamentals neural networks: Architecture, algorithms, and applications.” Prentice-Hall, Inc., Englewood Cliffs, New Jersy. Gifford, D. G., Wheeler, J. R., Kraemer, S. R. and McKown, A. F. (1987). “Spread footings for highway bridges.” Final Report, FHWA/RD-86/185, Federal Highway Administration, Washington, DC, 222p. Goh, A. T. C. (1994). “Seismic liquefaction potential assessed by neural network.” J. Geotech. & Geoenv. Engrg., ASCE, 120(9), 1467-1480. Hecht-Nielsen, R. (1990). “Neurocomputing.” Addison-Wesely Publishing Company. Hertz, J. A., Krogh, A. and Palmer, R. G. (1991). “Introduction to the theory of neural computation.” Addison-Wesely Publishing Company, Red Wood City, California. Hubick, K. T. (1992). “Artificial neural networks in Australia.” Department of Industry, Technology and Commerce, Commonwealth of Australia, Canberra. Jeyapalan, J. K. and Boehm, R. (1986). “Procedures for predicting settlements in sands.” Proc., Settlement of Shallow Foundations on Cohesionless Soils: Design and Performance, Geotech. Special Publication No. 5, ASCE, Seattle, Washington, 1-22. Lee, I. and Lee, J. (1996). “Prediction of pile bearing capacity using artificial neural networks.” Computers and Geotechnics, 18(3), 189-200. Levy, J. F. and Morton, K. (1974). “Loading tests and settlement observations on granular soils.” Conf. Settlement of Structures, Cambridge, 43-52. Maier, H. R. and Dandy, G. C. (2000a). “Neural networks for the prediction and forecasting of water resources variables: a review of modelling issues and applications.” Environmental Modelling & Software, 15 (2000), 101-124. Maier, H. R. and Dandy, G. C. (2000b). “Application of artificial neural networks to forecasting of surface water quality variables: issues, applications and challenges.” In: Artificial Neural Networks in Hydrology, edited by R. S. Govindaraju and A. R. Rao, Kluwer, Dordrecht, The Netherlands, in press. Maren, A., Harston, C. and Pap, R. (1990). “Handbook of neural computing applications.” Academic Press, Inc., San Diego, California. 14
Maugeri, M., Castelli, F., Massimino, M. R. and Verona, G. (1998). “Observed and computed settlements of two shallow foundations on sand.” J. Geotech. & Geoenv. Engrg., 124(7), 595-605. McClelland, J. L. and Rumelhart, D. E. (1988). “Explorations in parallel distributed processing.” Cambridge, MA: MIT Press. McCulloch, W. S. and Pitts, W. (1943). “ A Logical calculus of ideas imminent in nervous activity.” Bulletin and Mathematical Biophysics, 5, 115-133. Meyerhof, G. G. (1965). “Shallow foundations.” J. Soil Mech. & Found. Div., ASCE, 91(SM2), 21-31. Microsoft Corporation (1997). “Microsoft Excel 97.”. Moselhi, O., Hegazy, T. and Fazio, P. (1992). “Potential applications of neural networks in construction.” Cand. J. Civil Engng., 19, 521-529. Neuralware Inc. (1997). “NeuralWorks predict release 2.1.”. Papadopoulos, B. P. (1992). “Settlements of shallow foundations on cohesionless soils.” J. Geotech. Engrg., ASCE, 118(3), 377-393. Picornell, M. and del Monte, E. (1988). “Prediction of settlements of cohesive granular soils.” Proc., Measured Performance of Shallow Found., Geotech. Special Publication No. 15, ASCE, Nashville, Tennessee, 55-72. Picton, P. D. (1994). “Introduction to neural networks.” The MacMillan Press Ltd. Ripley, B. D. (1996). “Pattern recognition and neural networks.” Cambridge University Press. Rumelhart, D. E., Hinton, G. E. and Williams, R. J. (1986a). “Learning internal representations by error propagation.” In: D. E. Rumelhart & J. L. McMlelland, Eds., Parallel Distributed Processing, 1, Chapter 8, Reprinted in Anderson & Rosenfeld [1988], 675-695. Rumelhart, D. E., Hinton, G. E. and Williams, R. J. (1986b). “Learning representations by back-propagation error.” Nature, 323:533-536, Reprinted in Anderson & Rosenfeld [1988], 696-699. Schmertmann, J. H. (1970). “Static cone to compute static settlement over sand.” J. Soil Mech. & Found. Div., ASCE, 96(SM3), 7302-1043. Schultze, E. and Sherif, G. (1973). “Prediction of settlements from evaluated settlement observations for sand.” Proc., 8th Int. Conf. On Soil Mech. & Found. Engrg., 1(3), 225-230. Smith, G. N. (1986). “Probability and statistics in civil engineering: An Introduction.” Collins, London. Smith, M. (1993). “Neural networks for statistical modelling.” Van Nostrand Reinhold, New York, N. Y. Stone, M. (1974). “Cross-validatory choice and assessment of statistical predictions.” J. the Royal Statistical Society, B 36, 111-147. Twomey, J. M. and Smith, A. E. (1997). “Validation and verification.” In: Kartam, N., Flood, I., and Garrett, J. H.: Artificial neural networks for civil engineers: Fundamentals and applications, Published by ASCE, New York, 44-64. Vargas, M. (1961). “Foundations of tall buildings on sand in Sao Paulo.” Proc., 5th Int. Conf. On Soil Mech. & Found. Engrg., Paris, 1, 841-843. 15
Wahls, H. E. (1997). “Settlement analysis for shallow foundations on sand.” Proc., Third Int. Geotech. Engrg. Conf., Cairo, 7-28. Zhou, Y. and Wu, X. (1994). “Use of neural networks in the analysis and interpretation of site investigation data.” Computer and Geotechnics, 16, 105-122. Zurada, J. M. (1992). “Introduction to artificial neural systems.” West Publishing Company, St. Paul.
APPENDIX – NOTATION The following symbols are used in this report: AI ANNs B CPT DMT L MAE NNNs PEs q r RMSE Sm Sp SPT SPT-N
= abbreviation for artificial intelligence; = abbreviation for artificial neural networks; = footing width; = abbreviation for cone penetration test; = abbreviation for dilatometer modulus test; = footing length; = abbreviation for mean absolute error; = abbreviation for natural neural network; = abbreviation for processing elements; = footing applied pressure; = correlation coefficient; = abbreviation for root mean square error; = measured settlement; = predicted settlement; = abbreviation for standard penetration test; and = average SPT blow count/300 mm to a depth of influence of foundation.
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