Although it is a regular duty of geotechnical engineers to evaluate how much shallow foundation settles in the granular soil, there is no well-approved formula for this task. The intent of this research is to develop a formula that is adequately simple to be used in routine geotechnical engineering work but complete enough to address the behavior of granular soil associated with the settlement issue.
Cone penetration test and foundation load test data were used to generate a formula that can predict the settlement. Genetic Programming (GP) based Symbolic Regression (GP-SR) and artificial neural networks were used to develop an optimized formula. Settlements were also calculated using the finite method and compared to the results of the developed formula.
Two formulas were developed using SR, and several models were developed using ANN. ANN model 1 has the highest R^{2} value (0.93) and the lowest MSE (0.16) among all developed ANN and GP-SR models. FEM settlements were almost double the measured ones in some instances.
Open Peer Review Details | |||
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Manuscript submitted on 05-10-2018 |
Original Manuscript | Intelligent Computing Based Formulas to Predict the Settlement of Shallow Foundations on Cohesionless Soils |
The foundation system carrying the loads of any structure can be either shallow or deep.
Shallow foundations are considered a feasible alternative because of their cost effectiveness, short construction time, and environmental friendliness. Typically, their design is dominated not by bearing capacity but by settlement. Because performance data is limited and predictions of settlements inflated, shallow foundations are frequently underutilized. Therefore, estimating the settlement is an important criterion to be considered in the design stage of shallow foundations.
Different methods, ranging from purely empirical to complex nonlinear finite elements, have been tried but has failed to produce accurate predictions of settlements [1H.G. Poulos, "Common procedures for foundation settlement analysis-Are they adequate?", Proceeding of the 8th Australia New Zealand Conference on Geomechanics, pp. 3-25.Hobart]. Several researchers have studied the behavior of shallow foundations on cohesionless soils [2E.E. Jorden, "Settlement in sand-methods of calculating and factor affecting", Ground Eng., vol. 10, no. 1, pp. 30-37.-12B. Tarawneh, J. Nusairat, and Y. Hakam, "Load testing and settlement of shallow foundation on desert sands", Proceedings of the Institution of Civil Engineers-Geotechnical Engineering, vol. 171, no. 1, pp. 52-63.
[http://dx.doi.org/10.1680/jgeen.16.00109] ]. Most of them had examined how the results of calculated and field measured settlements of shallow foundations are compared to one another.
The principle factor in predicting settlement is the subsurface exploration data, specifically, its quality and quantity. Current in-situ tests notably the Cone Penetration Test, Standard Penetration (SPT), and Dilatometer Modulus (DMT) tests attempt to estimate the parameters of soils in the subsurface. Actually, in addition to its strong theoretical background, the CPT has several advantages: speed, economy, near continuousness, and repeatability. As Robertson [13P.K. Robertson, "Interpretation of cone penetration tests-a unified approach", Can. Geotech. J., vol. 46, no. 11, pp. 1337-1355.
[http://dx.doi.org/10.1139/T09-065] ] pointed out that these advantages have steadily advanced the CPT’s world-wide use and application. Furthermore, CPT data is useful in predicting settlement behaviors by applying the methods proposed by Schmertmann [14J.H. Schmertmann, "Static cone to compute static settlement over sand", J. Soil Mech. Found. Div., vol. 96, no. 3, pp. 1011-1043.], Meyerhof [15G.G. Meyerhof, "State-of-the-art of penetration testing in countries outside Europe", Proceeding of the 1st European Symposium on penetration testing, .Stockholm], and DeBeer [16E.E. DeBeer, "Bearing capacity and settlement of shallow foundations on sand", Proceedings of Symposium on Bearing Capacity and Settlement of Foundations, Duke University: Durham, N.C, pp. 315-333.].
The goal of this study was to develop formulas to predict the settlement of shallow foundations on cohesionless soil using Genetic Programming (GP) based Symbolic Regression (GP-SR) and Artificial Neural Networks (ANNs). CPTs and foundation load tests data have been used, leading to a numerical formula that enables the accurate prediction of shallow-foundation settlement on granular soils. To develop and augment the formula, ANNs and GP-SR have been used with a database compiled from the results of 44 shallow foundation field load tests. There is a little application of GP-SR technique to predict the settlement of shallow foundation, using CPT and load tests, in the literature.
A database was compiled from the results of 44 square shallow foundation load tests (270 data point). All load tests were carried out post ground improvement work using Dynamic Compaction (DC) or Rapid Impact Compaction (RIC). Menard and Broise [17L. Menard, and Y. Broise, "Theoretical and practical aspects of dynamic compaction", Geotechnique, vol. 25, no. 3, pp. 3-18.
[http://dx.doi.org/10.1680/geot.1975.25.1.3] ] were the first to expound the virtues of DC, and since then it has become a popular technique because it is simple, cost-effective, and reaches to significant depths. Moreover, although primarily testing granular fills and sandy materials, it works well with many other soil types and conditions, as well [18P. Mayne, J. Jones Jr., and J. Dumas, "Ground response to dynamic compaction", J. Geotech. Eng., vol. 110, no. 6, pp. 757-774.
[http://dx.doi.org/10.1061/(ASCE)0733-9410(1984)110:6(757)] ]. RIC applies low energy to compact sandy soil at shallow depths, thus filling the void between the DC and shallow compaction methods such as roller compaction [19B. Tarawneh, and M. Matraji, "Ground improvement using rapid impact compaction: case study in Dubai", Gradevinar, vol. 66, no. 11, pp. 1007-1014.].
All tests were performed according to the ASTM D1196/D1196M-12. Foundations were loaded to 150% of the design load. The applied testing load was produced by a hydraulic jack that had enough capacity for applying the maximum load needed and was equipped with a precisely calibrated gauge to point out the magnitude of the applied load. Reaction to the hydraulic jack was provided by a platform carrying concrete blocks. The platform was supported by an array of secondary steel beams and the main girder. Deflections (settlements) were measured about the fixed reference beams using dial gauges. Such deflections were monitored for each loading increment using four dial gauges. The average readings of the four dial gauges were taken as the settlement for the load increment. Before load testing, CPTs were performed at each location to estimate the soil parameters. The data collected at each test location included the width of the footing (B) in m, applied pressure (P) in kPa, settlement (S) in mm, and the average cone tip resistance (q_{c}) in MPa for a 2B depth below the footing’s bottom.
Based on the work by Lunne et al. [20T. Lunne, and P.K. Robertson, Cone penetration testing in Geotechnical practice., Blackie Academic, EF Spon/Routledge Publishing, .New York, USA] and Robertson [13P.K. Robertson, "Interpretation of cone penetration tests-a unified approach", Can. Geotech. J., vol. 46, no. 11, pp. 1337-1355.
[http://dx.doi.org/10.1139/T09-065] ], the soil was classified at each test location according to CPT test results regarding Soil Behavior Type (SBT). The classifications were either clean sand-to-silty or gravelly sand-to-sand. It is to be noted that no cohesive soil was found in any of the test locations. The depth of the water table ranged from 0.5m to 1.75m below the bottom of the footing. This type of soil has a high permeability, and the pore water pressure is very low. Therefore, the groundwater table was not considered as an influencing factor in this research. The groundwater is important in soft, fine-grained soils where in-situ moisture takes a longer time to dissipate. Table 1 provides a summary of the collected data from each load test.
Two intelligent computing techniques, Artificial Neural Networks and Symbolic Regression (SR), led to the development of a formula to predict the settlement of the shallow foundation. The input variables are cone tip resistance, applied pressure, and footing width. In order to evaluate the developed formula, a performance criterion was established. Three statistical measurements were utilized, the Mean Square Error (MSE), Mean Absolute Error (MAE), and the coefficient of determination (R^{2}), with a goal of developing a formula with higher R^{2} value and the least MAE and MSE. The equations of these performance criteria are given below:
(1) |
(2) |
(3) |
Where n is the total count of measurements, e_{i} is the differences between actual (measured) values and predicted settlement values, S_{i} is actual settlement values, and is the average of the measured settlement values.
Symbolic Regression can find mathematical formulas by minimizing errors. It is a function-finding technique for modeling numeric multivariate datasets. It is different from traditional regression in that it builds mathematical functions by searching the parameters and different forms of equations [21M.D. Schmidt, and H. Lipson, Co-evolving fitness predictors for accelerating and reducing evaluations, Genetic Programming Theory and Practice IV, Genetic and Evolutionary Computation., vol. 5, Springer, ., 22D. Karaboga, C. Ozturk, N. Karaboga, and B. Gorkemli, "Artificial bee colony programming for symbolic regression", Inf. Sci., vol. 209, pp. 1-15.
[http://dx.doi.org/10.1016/j.ins.2012.05.002] ]. It approaches a specific modeling problem by exploring nonlinear equation forms alongside their parameters, usually finding a mathematical function which can clarify the relationship between dependent and independent variables [22D. Karaboga, C. Ozturk, N. Karaboga, and B. Gorkemli, "Artificial bee colony programming for symbolic regression", Inf. Sci., vol. 209, pp. 1-15.
[http://dx.doi.org/10.1016/j.ins.2012.05.002] , 23J. Kleijnen, and R. Sargent, "A methodology for fitting and validating meta models in simulation", Eur. J. Oper. Res., vol. 120, no. 1, pp. 14-29.
[http://dx.doi.org/10.1016/S0377-2217(98)00392-0] ]. SR is an application of genetic programming, and, because it requires no special knowledge to create free-form mathematical models from collected data, it is an appealing option to the standard regression method. A Genetic Programming GP-based Symbolic Regression (GP-SR) software, Eureqa [24M. Schmidt, and H. Lipson, "Distilling free-form natural laws from experimental data", Science, vol. 324, no. 5923, pp. 81-85.
[http://dx.doi.org/10.1126/science.1165893] [PMID: 19342586] ], was utilized in this research to develop a formula that can predict the settlements.
A set of input variables (P, q_{c}, B), with its experimental settlement results (S), was used for the GP-SR. Mathematical operators were defined for use in the developed formula. Various combinations of operators and variables were then generated via a genetic algorithm in order to develop a symbolic equation that reflects an appropriate approximation. Developed equations were subsequently rated according to complexity and fit, based on the R^{2}, MSE, and MAE. The results of the symbolic regression analysis are shown in Table 2). Two equations were developed to predict the settlement (S) using P, q_{c} and B. R^{2.} Values for those models were 0.84 and 0.78. MSE values were 1.62 and 2.13, while the MAE values were 0.44 and 0.62.
Many previous investigations into geotechnical engineering have used artificial intelligence and ANN [25M. Mosallanezhad, and H. Moayedi, "Developing hybrid artificial neural network model for predicting uplift resistance of screw piles", Arab. J. Geosci., vol. 10, no. 22, pp. 479-485.
[http://dx.doi.org/10.1007/s12517-017-3285-5] -31B.K. Al Tarawneh, Inspection, durability, and risk assessment of highway culverts., Ph.D. Thesis. Ohio University: Athens, Ohio, .].
Shi et al. [32J. Shi, J. Ortigao, and J. Bai, "Modular neural networks for predicting settlement during tunneling", J. Geotech. Geoenviron. Eng., vol. 124, no. 5, pp. 389-395.
[http://dx.doi.org/10.1061/(ASCE)1090-0241(1998)124:5(389)] ] offered a study of neural networks for predicting settlements of tunnels. Using SPT data, Shahin et al. [33M.A. Shahin, M.B. Jaksa, and H.R. Maier, "Predicting the settlement of shallow foundations on cohesionless soils using back-propagation neural networks", Research Report No. R 167, the University of Adelaide, Adelaide, Australia., .] presented an artificial neural network model designed to predict the settlement of shallow foundations on cohesionless soils. Nejad et al. [34F.P. Nejad, M.B. Jaksa, M. Kakhi, and B.A. McCabe, "Prediction of pile settlement using artificial neural networks based on standard penetration test data", Comput. Geotech., vol. 36, no. 7, pp. 1125-1133.
[http://dx.doi.org/10.1016/j.compgeo.2009.04.003] ] developed an ANN model to predict pile settlement based on standard penetration testing. The ANN developed by Tatari et al. [35O. Tatari, S.M. Sargand, T. Masada, and B. Tarawneh, "Neural network approach to condition assessment of highway culverts: Case study in ohio", J. Infrastruct. Syst., vol. 19, no. 4, pp. 409-414.
[http://dx.doi.org/10.1061/(ASCE)IS.1943-555X.0000139] ] assesses the condition of culverts based on inventory data presented by Masada et al. [36T. Masada, S.M. Sargand, and B. Tarawneh, "B., G.F. Mitchell, and D. Gruver, "New inspection and risk assessment methods for metal highway culverts in Ohio", Transp. Res. Rec., vol. 1976, no. 1, pp. 141-148.
[http://dx.doi.org/10.3141/1976-17] , 37T. Masada, S.M. Sargand, B. Tarawneh, G.F. Mitchell, and D. Gruver, "Inspection and risk assessment of concrete culverts under Ohio’s highways", J. Perform. Constr. Facil., vol. 21, no. 3, pp. 225-233.
[http://dx.doi.org/10.1061/(ASCE)0887-3828(2007)21:3(225)] ]. Tarawneh [38B. Tarawneh, "Pipe pile setup: database and prediction model using artificial neural network", Soil Found., vol. 53, no. 4, pp. 607-615.
[http://dx.doi.org/10.1016/j.sandf.2013.06.011] ] and Tarawneh and Imam [39B. Tarawneh, and R. Imam, "Regression versus artificial neural networks: predicting pile setup from empirical data", KSCE J. Civ. Eng., vol. 18, no. 4, pp. 1018-1027.
[http://dx.doi.org/10.1007/s12205-014-0072-7] ] developed ANN models using dynamic load testing that can predict pile setup for three pile types (pipe, concrete, and H-pile). Tarawneh and Nazzal [40B. Tarawneh, and M. Nazzal, "Optimization of resilient modulus prediction from FWD results using artificial neural network", Periodica Polytechnica, vol. 58, no. 2, pp. 143-154.
[http://dx.doi.org/10.3311/PPci.2201] ] employed ANN to optimize the prediction of subgrade resilient modulus design input from falling weight deflectometer test results.
The functioned approximation techniques of ANNs can be used in a nonlinear, complex contact nature between input(s) and output(s). ANN is considered a soft computing method that imitates how the human brain processes information-transfer [41J.H. Garrett, "Where and why artificial neural networks are applicable in civil engineering", J. Comput. Civ. Eng., vol. 8, no. 2, pp. 129-130.
[http://dx.doi.org/10.1061/(ASCE)0887-3801(1994)8:2(129)] ]. Among the different types of ANNs, the multilayer feed forward ANN is the most frequently used. It consists of input, hidden, and output layers which are connected by different connection weights. For the optimum outcome, ANNs should be trained through learning algorithms, such as the most widely used back-propagation [42G. Dreyfus, Neural Networks: Methodology and Application., Springer Berlin Heidelberg: Germany, .]. The basis for that algorithm is a gradient descendent optimization procedure that minimizes the MSE (average mean squared error) between the predicted and desired values/outputs.
In this research, the ANN model input variables are applied pressure (P), CPT tip resistance (q_{c}), and width of square footing (B). The ANN model output is Settlement (S). The data was divided into three sets: training, cross-validation, and testing. Seventy percent of the data points were selected for training, 15% for cross-validation, and 15% for testing the network. The training data points were used to train the network and compute the weights of the inputs. The test data points established the performance level of the selected ANN model. Cross validation measured test-set error during the period in which the network was going through the training set [30B. Tarawneh, "Predicting standard penetration test N-value from cone penetration test data using artificial neural networks", Geoscience Frontiers, vol. 8, no. 1, pp. 199-204.
[http://dx.doi.org/10.1016/j.gsf.2016.02.003] ].
It is essential that the data used for training, cross-validation, and testing characterize the same population and its statistical properties. Constructing the best model possible necessitates a training set that includes all of the patterns that the data contains. Likewise, the test set determines when to stop training; therefore, it should be representative of the training set and should contain all patterns existing in the available data [43M.A. Shahin, H.R. Maier, and M.B. Jaksa, "Predicting settlements of shallow foundations using artificial neural networks", J. Geotech. Geoenviron. Eng., vol. 128, no. 9, pp. 785-793.
[http://dx.doi.org/10.1061/(ASCE)1090-0241(2002)128:9(785)] ]. To achieve this, numerous random combinations of the training, cross validation, and testing sets were engaged until a statistically reliable data set was obtained.
Numerous network structures, with different numbers of hidden layers and nodes in the hidden layer, were trained and tested to find the model with the best performing network architecture. Because it has been shown that a network with one hidden layer can approximate any continuous function [44K. Hornik, and M. Stinchcombe, "Multilayer feed forward networks are universal approximators", Neural Network, vol. 2, no. 5, pp. 359-366.], in this research one hidden layer was used. Typically, the structure of ANNs includes processing elements (PEs), or nodes, that are arranged in three layers input, output, and hidden (Fig. 1). Every PE in each layer is joined, whether totally or partly, to other PEs by weighted connections. The input from each PE in the previous layer (I_{i}) is multiplied by an adjustable connection weight (w_{ij}). The weighted input signals are summed at each PE, at which point a bias (Ɵ_{i}) is added. This combined input (I_{i}) is then passed through a sigmoidal transfer function or other nonlinear transfer function to produce the PE’s output (y_{i}), which provides the input to the PEs in the next layer. Equations 4 and 5 show this process, and Figs. (1 and 2) describe it.
(4) |
(5) |
To develop optimal network geometry, ANNs with a single hidden layer and a different number of nodes in the hidden layer were trained with sigmoid (Sig.) and hyperbolic tangent (tanh) transfer functions for the hidden and output layers. Combinations of the number of elements in the hidden layer and types of the transfer function that yielded the most accurate predictions of settlement (S) are shown in Table 3.
Table 3 shows that model 1 has the greatest R^{2} value and the least MSE and MAE values for the testing data set. The plot of the measured and predicted settlements for the model’s testing data set is shown in Fig. (3). This model is the best performer among the developed ANNs and SR. The mathematical expression of the ANN algorithm is presented below:
A separate shallow foundation three load test was used to verify the selected ANN model. Settlements were estimated using the Finite Element Method (FEM) and compared with the outcomes from the ANN model and the measured settlements of those load tests.
The load test was performed in agreement with the ASTM D1196/D1196M-12. The tests were carried out using 2.5m-wide footing on a sandy soil. Table 4 shows the average tip resistance (q_{c}) within 5m below the bottom of the footing, pressure, and measured settlements.
Settlements were estimated using FEM. The conventional, Mohr-Coulomb soil model was used in the analysis process. This elastic perfectly plastic model relies on a combination between Hooke’s law and the Coulomb’s failure criterion. That is, it has five input parameters, including Young’s modulus and Poisson’s ratio for soil elasticity, angle of friction and cohesion for soil plasticity, and the angle of dilatancy. Soil input parameters were estimated based on CPT data as presented by Robertson [13P.K. Robertson, "Interpretation of cone penetration tests-a unified approach", Can. Geotech. J., vol. 46, no. 11, pp. 1337-1355.
[http://dx.doi.org/10.1139/T09-065] ]. Soil parameters are shown in Table 5. Midas Soil Works FEM software was utilized to perform the analyses.
A comparison was made between the results of the finite element analysis and the measured settlements and predicted values by ANN model as shown in Figs. (4 to 6). Those figures clearly demonstrate the superior performance of the developed ANN model over the FEM. Settlements estimated by ANN are comparable to those measured from load tests. In all tests, FEM over-predicted the settlements, giving conservative predictions. FEM settlements are almost double the measured ones in some cases, as shown in those figures. It should be noted that ANN is over-predicting the settlements by a value of less than one millimetre and half millimetre for load test-1 and 2, respectively.
Fig. (1) Structure of the developed ANNs. |
Fig. (2) Structure of a neural cell. |
Fig. (3) Comparison between measured and predicted settlement for ANN model 1 testing set. |
(6) |
Where A_{1}, A_{2}, and A_{3} can be calculated using the equations below:
(7) |
(8) |
(9) |
This paper has presented the results of a study that was conducted to evaluate the use of SR and ANNs to develop a formula that can accurately estimate settlement of shallow foundations on cohesionless soils. From 44 square shallow foundation load tests, post ground improvement, a database was compiled. At all test locations, soil was classified as either clean sand-to-silty sand or gravelly sand-to-sand. There was no cohesive soil present in any of the test locations. The depth of the water table was 0.5m to 1.75m below the bottom of the footing.
ANN and SR were employed to develop a formula that can reliably predict the settlement. The input variables were cone tip resistance, applied pressure, and footing width. To evaluate the developed formula, a performance criterion was established using MSE, MAE, and R^{2}. The goal was to develop a formula with highest R^{2} value and least MAE and MSE. Two formulas were developed using SR (Table 2), and several models were developed using ANN. As shown in Table 3, ANN model 1 has the highest R^{2} value (0.93) and the lowest MSE (0.16) and MAE (0.2) among all developed ANN and SR models.
A separate shallow foundation three load tests verified the selected ANN model. Further, settlements were estimated using FEM and compared with the results of the selected ANN model and the measured settlements of those load tests. In all tests, FEM over predicted the settlements, giving conservative predictions. FEM settlements were almost double the measured ones in some instances.
Fig. (4) Comparison between FEM, ANN, and measured settlements for load test-1. |
Fig. (5) Comparison between FEM, ANN, and measured settlements for load test-2. |
Fig. (6) Comparison between FEM, ANN, and measured settlements for load test-3. |
It can be concluded that the ANN model is a satisfactory predictor of the settlements of shallow foundation on cohesionless soils. A benefit of ANNs is that, once the formula is trained, it can be utilized as a quick method for settlement estimation. Conversely, the main disadvantage of ANNs is insufficient theory to foster their development and their limited ability to explain the method by which they analyze the available data to achieve a solution. It should be noted that the ANN modeling hinges on experimental data and is applicable for use in an interpolative sense. As an empirical formula, the scope of its applicability is controlled by the data used to build and calibrate the model. To update the model and increase its accuracy, it is important to include additional data to enable the model to be re-trained.
Despite the above-mentioned shortcomings, the results of this research point out that ANNs have several significant powerful and practical benefits that make them a valuable tool in predicting the settlement of shallow foundations on cohesionless soils. It should be noted that the developed formulas can only be used for similar type of soils.
Not applicable.
The authors declare no conflict of interest, financial or otherwise.
The authors would like to acknowledge the collaboration and assistance of both Menard Vibro Middle East and RIC Middle East in Dubai, UAE.
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