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A new genetic programming model for predicting settlement of shallow foundations Mohammad Rezania and Akbar A. Javadi
Abstract: In this paper, a new genetic programming (GP) approach for predicting settlement of shallow foundations is presented. The GP model is developed and verified using a large database of standard penetration test (SPT) based case histories that involve measured settlements of shallow foundations. The results of the developed GP model are compared with those of a number of commonly used traditional methods and artificial neural network (ANN) based models. It is shown that the GP model is able to learn, with a very high accuracy, the complex relationship between foundation settlement and its contributing factors, and render this knowledge in the form of a function. The attained function can be used to generalize the learning and apply it to predict settlement of foundations for new cases not used in the development of the model. The advantages of the proposed GP model over the conventional and ANN based models are highlighted. Key words: geotechnical models, foundation settlement, granular soils, evolutionary computation, genetic programming. Re´sume´ : Dans cet article, on pre´sente une nouvelle approche, base´e sur la programmation ge´ne´tique (« GP »), pour la pre´diction du tassement de fondations superficielles. Le mode`le « GP » est de´veloppe´ et ve´rifie´ au moyen d’une vaste base de donne´es obtenues d’histoires de cas impliquant des mesures de tassements de fondations superficielles. Les re´sultats du mode`le « GP » de´veloppe´ sont compare´s avec ceux d’un certain nombre de me´thodes traditionnelles utilise´es couramment et de mode`les base´s sur l’« ANN ». On montre que le mode`le « GP » peut apprendre avec grande pre´cision la relation complexe entre le tassement de la fondation et ses facteurs contributifs sous la forme d’une fonction. La fonction obtenue peut eˆtre utilise´e pour ge´ne´raliser la capacite´ de pre´dire le tassement des fondations pour de nouveaux cas non utilise´s dans le de´veloppement du mode`le. On a mis en lumie`re les avantages du mode`le « GP » propose´ par rapport aux me´thodes conventionnelles et aux mode`les base´s sur l’« ANN ». Mots-cle´s : mode`les ge´otechniques, tassement de fondation, sols pulve´rulents, calcul e´volutionniste, programmation ge´ne´tique. [Traduit par la Re´daction]
Introduction Two major criteria that control the design of shallow foundations on granular soils are the bearing capacity of the soil beneath the foundation and the settlement of the foundation. However, since excessive settlements often lead to serviceability problems, settlement usually governs the footing design process, particularly for shallow foundations wider than 1 m (Schmertmann 1970). Immediate settlement and consolidation settlement are the main components of shallow foundation settlement. In cohesive soils (silts and clays), as the load is applied the excess pore pressures dissipate slowly because of the low soil permeability. As a result, consolidation settlement may occur over a very long period of time. However, in coarse soils (sands and gravels), the focus of this paper, increases in pore pressures are dissipated rapidly owing to the high soil permeability and any settlement resulting from a change in loading occurs more or less immediately. Most of the immediate settlement Received 13 February 2006. Accepted 24 May 2007. Published on the NRC Research Press Web site at cgj.nrc.ca on 4 February 2008. M. Rezania1 and A.A. Javadi. Computational Geomechanics Group, Department of Engineering, School of Engineering, Computing and Mathematics, University of Exeter, Exeter, Devon, EX4 4QF, UK. 1Corresponding
author (e-mail:
[email protected]).
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may be accommodated within the structure during or shortly after its construction by a considerable increase in the internal forces of the structure; unfortunately, this can result in cracks through structural elements, and may even end in structural failure. Determining shallow foundation settlement on granular soils is a complex geotechnical engineering problem owing to the heterogeneous nature of the soils and the large number of factors involved. Furthermore, the uncertainties in the stress–strain history of the soil, soil compressibility, and distribution of the applied stresses make the problem even more intricate (Shahin et al. 2002b). Because of this complexity, several researchers have attempted to model the settling phenomenon using different techniques. As a result, a considerable number of theoretical, empirical, and numerical methods have been developed to predict the settlement of shallow foundations on cohesionless soils. It has been reported that more than 40 different methods exist for predicting settlement in granular soils (Douglas 1986); this figure should now be even more, considering the more recent studies in this field. Current practice in predicting settlement of shallow foundations Because of the difficulties in obtaining undisturbed field samples of granular soil with which to conduct laboratory evaluations of soil compressibility characteristics, the use of in situ tests has been a common practice, and various test
doi:10.1139/T07-063
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methods have been developed based on field test results, including the standard penetration test (Alpan 1964; Schultze and Sherif 1973), cone penetration test (Schmertmann 1970; Sargand et al. 2003), dilatometer module test (U.S. Army Corps of Engineers 1990), seismic cone penetration test (Fioravante et al. 1998; Mayne 2000), and plate load test (Bond 1961). However, comparative studies of the available traditional methods have shown that settlement predictions can be very different for the same case depending on the method employed (Jeyapalan and Boehm 1986; Tan and Duncan 1991; Wahls 1997; Sivakugan and Johnson 2004). Consequently, alternative methods are needed to provide more accurate settlement predictions (Shahin et al. 2002b). In recent years, through pervasive developments in computational software and hardware, several alternative computer-aided pattern recognition and data classification techniques have emerged. The main idea behind pattern recognition systems such as neural network, fuzzy logic, or genetic programming (GP) is that they learn adaptively from experience and extract various discriminants, each appropriate for its purpose. Artificial neural networks (ANNs) are the most widely used pattern recognition procedures for determination of shallow foundation settlement based on field data (Sivakugan et al. 1998; Shahin et al. 2002a; Shahin et al. 2002b). These black-box models have the ability to operate on large quantities of data and learn complex model functions from examples, i.e., by training on sets of input and corresponding output data. The employed philosophy for model generation is similar to the one used for developing conventional statistical models. However, the greatest advantage of ANNs over traditional modeling techniques is their ability to capture nonlinear and complex interactions between variables of a system without having to assume the form of the relationship between input and output variables. In the context of foundation settlement determination, ANNs can be trained to learn the relationship between soil and footing characteristics and the amount of settlement, while requiring no prior knowledge of the form of the relationship. However, the main disadvantage of the artifical neural network (ANN) approach is the large complexity of the network structure, as it represents the knowledge in terms of a weight matrix together with biases that are not accessible to users. More recently, some researchers have attempted to overcome this problem by using alternative techniques such as neurofuzzy (Shahin et al. 2003a) or B-spline neurofuzzy models (Shahin et al. 2003b), in which network acquired knowledge can be translated into a set of fuzzy rules, or by proposing ANN based stochastic design charts (Shahin et al. 2005). However, these methods do not provide a well formulated scheme for settlement prediction that could be easily used by practicing engineers and non-academics. In this paper, a new GP approach for settlement prediction of shallow foundations on granular soils is presented. A database of 173 SPT based case histories, collected from seven independent studies and compiled by Shahin et al. (2002b) is used for the development and verification of the proposed GP model. The database incorporates a wide range of field measurements on shallow foundation settlements. The results of the proposed GP model are compared with those attained using a number of widely used traditional models, as well as a neural network based model and a neu-
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ral network based formula, and the advantages of the proposed approach over the existing methods are described in detail. It is shown that the proposed model is much more accurate than the traditional models. The GP model is also more accurate than the neural network based models, and much easier to use in practice. Genetic programming Background Genetic programming, which was introduced in the early 1990s by Koza (1992), is an evolutionary computing method that generates a transparent and structured representation of the data provided. Evolutionary algorithms (EAs) are search techniques based on computer implementations of some of the evolutionary mechanisms found in nature (such as selection, crossover, and mutation) to solve a function identification problem. The function identification problem is to search for a function in a symbolic form that fits a set of experimental data. Genetic algorithm (GA) and GP are the major types of evolutionary algorithms, and GP is a generalization and extension of GA. Genetic algorithms are generally used in parameter optimization to evolve the best values for a given set of model parameters, whereas GP gives the basic structure of the approximation model together with the values of its parameters. While a GA uses a string of numbers to represent the solution, the GP combines a high level symbolic representation with the search efficiency of the GA to form the best possible model for the system. Representation schemes in GP are composed of nodes, which are elements from either a terminal set, (this includes constants like 2, or variables like x1, x2, etc., or both), and a functional set, (this includes mathematical operators that generate the regression model (e.g., ± and xy, etc)). A typical genetic programming tree representing the simple algebraic expression (2/x1 + x2)2 is shown in Fig. 1. The functional set can be subdivided into binary nodes, which take any two arguments (such as occur in addition), and unary nodes, which take one argument (as in a square root). The solution domain is created by the recursive composition of elements from the functional set for any internal node, and from the terminal set for any external nodes (leaf nodes). Whenever a node in a tree is created from the functional set, a number of links equal to the number of arguments the function takes is created to radiate out from that node. The result of this process is a set of random trees of different sizes and shapes, each exhibiting a different fitness with the objective function. If the set of applied functions is sufficiently rich, tree structures are capable of representing hierarchical programs of any complexity. These functions may include arithmetic operators (like +, , or –), mathematical functions (like sin(), cos(), or ln()), Boolean operators (like AND or OR), logical expressions (like IF or THEN), iterative functions (like DO, CONTINUE, or UNTIL), or other user defined functions (Sette and Boullart 2001). Overview of genetic programming process The nature of GP allows the user to gain additional information on how the system performs, i.e., it gives an insight into the relationship between input and output data. Once a #
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further use; however, their selection is less probable. Programs which are successfully selected for reproduction then enter the mating pool. Crossover To develop new genetic information, and hopefully improve the fitness of the population, the tree structures in the mating pool undergo crossover. Crossover is one of the basic genetic operators that help evolve the model structures. In genetic programming, the crossover operator creates new offspring from the genetic material taken from the parents, and this is implemented in the following way: . .
.
population of computer programs has been randomly created, the process of evolving the population proceeds using the simple principles as for GAs, with the minor difference that it is the strings of functions and terminals that are reproduced, crossed over, and mutated rather than the strings of binary codes. Evolutionary algorithms maintain a population of structures that evolve according to the rules of natural selection, using some operators inspired from natural genetics, such as reproduction or crossover. Each individual in the population receives a measure of its fitness with the current environment. The fitness criteria are calculated by the objective function, which measures how good the individual is at competing with the rest of the population. At each generation, a new population is created by selecting individuals according to their fitness and breeding them together using the genetic operators of crossover and mutation. The existing population will then be replaced with the new population. This procedure continues until the termination criterion, which can be either the maximum number of generations or a particular allowable error, is satisfied. After the termination criterion is met, the best program in the final population is designated the result of the GP process. A typical flow diagram for a genetic programming procedure is illustrated in Fig. 2. The basic genetic programming operators (reproduction, crossover, and mutation) are described in the following sections. Initial population The first step in genetic programming is to create an initial population of n tree structures (computer programs) by randomly selecting functions and terminals from the user defined sets. The fitness of each structure is then evaluated according to how well it performs the desired objective, and a corresponding fitness value is assigned. Reproduction The second stage in the process involves selecting a proportion of the initial population to be copied to the next generation, which is done probabilistically according to the fitness of each tree structure. The method of selection is known as Roulette Wheel selection, which is essentially a random selection that allows for some bias based on fitness score. So, while choices are made randomly, those trees with high fitness scores have a higher probability of being selected. Those with lower scores can also be selected for
Two trees are selected from the population. One node is selected randomly from within each of these trees. The sub-trees under the selected nodes are swapped, thus two offspring belonging to the new population are obtained. A typical crossover operation is shown in Fig. 3.
Mutation In GP the mutation procedure involves selecting a casual node within a randomly selected tree and replacing it with any other randomly selected node from the same function or terminal set. The selected nodes must have the same number of arguments, and cannot replace themselves. Also, a functional node replaces a functional node, and a terminal node (variable or constant) replaces a terminal one. This procedure, which is shown in Fig. 4, is called the allele mutation. Genetic programming model for predicting the settlement of shallow foundations In this paper, a GP based procedure for prediction of settlement of shallow foundations on granular soils is introduced. The procedure leads to the development of a polynomial function that describes the relationship between foundation settlement and its main related foundation and soil parameters. A database of SPT based case histories collected from seven independent studies and compiled by Shahin et al. (2002b) is used for development and verification of the proposed GP model. The database includes 173 entries of field measurements for settlement of shallow foundations, and covers a wide range of footing dimensions (mostly isolated footings and raft foundations), soil densities, and structures. It includes soil types varying from fine sand to gravel, and foundations with widths, lengths, and depths of 0.9–55 m, 0.9–101 m, and 0–10.7 m, respectively. The average SPT blow count for the soils varies between 4 and 60 counts. The general information about the data sources, the number of cases taken from each reference, and the corresponding soil type, soil density, and structures is summarized in Table 1. The database contains the main parameters used to predict the settlement of shallow foundations on granular soils, as will be discussed later. Implementation of the evolutionary procedure A physical system having an output y, dependent on a set of inputs X and parameters a, can be mathematically formulated as #
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Fig. 2. Typical flow diagram for a genetic programming procedure. M, number of generations; Pm, probability of mutation; Pc, probability of crossover; Ps, probability of reproduction.
½1
where F(X,a) is a function in an m-dimensional space where m is the number of inputs. Genetic programming generates a population of expressions for F(X,a) coded in tree structures of variable size, and performs a global search of the best fit expression for F(X,a). The general form of expression used in GP can be presented as ½2
N X
y ¼ FðX; aÞ
y¼
n X
F½X; f ðXÞ; aj þ a0
j¼1
where y is the estimated vector of output of the process; aj is a constant; F is a function constructed by the process; X is the matrix of input variables; f is a function defined by the user; and n is the number of terms of the target expression. Initially, in the GP procedure, a number of potential models are randomly generated. Each model is trained and tested using the training and testing cases, respectively. The fitness of each model is evaluated by minimizing the sum of the squared differences as the objective function in the form of
½3
f ðSci Þ ¼
ðSci Þ2
i¼1
N
where N is the number of cases, and Sc is the difference between the predicted foundation settlement computed using the model and the actual measurement. If errors calculated using eq. [3] for all the models in the population do not meet the termination criteria, the evolutionary process continues to create a new generation of models. The evolutionary process is carried out by using the GP principles as discussed above. The procedure is continued until the best model is obtained. Factors affecting settlement of shallow foundations It is generally accepted that five parameters have the most significant effect on the settlement of shallow foundations on granular soils (Burland and Burbidge 1985; Shahin et al. 2002b): breadth of the foundation (B), net applied pressure on the foundation (q), soil compressibility within the depth #
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N is greater than 15, a correction should be made to the N value to account for the effect of the water table, as proposed by Terzaghi and Peck (1948) ½4
N 0 ¼ 15 þ 0:5ðN 15Þ
while for gravel or sandy gravel, the N value should be corrected as ½5
N 0 ¼ 1:25N
where N’ is the corrected value of N. Burland and Burbidge (1985) also recommended that no correction should be applied to N for the effect of overburden pressure as it is understood that these corrections have already been implemented in the database (Shahin et al. 2002b).
Fig. 4. Typical mutation operation in genetic programming showing genetic program (a) before mutation, and (b) after mutation.
of influence of the footing (which can be represented by SPT average blow count number N), foundation length (L), and foundation embedment (Df). There are two other parameters, the thickness of soil layer beneath the footing and the depth of groundwater, which are less effective (Burland and Burbidge 1985). The effect of the thickness of soil layer beneath the foundation is not considered in this study because of the lack of related information in the current dataset. It is also assumed that the effect of groundwater has already been included in the observed penetration resistance (i.e., in the SPT blow count number N) (Meyerhof 1965; Shahin et al. 2002b). The SPT is one of the most commonly used in situ tests for estimating the compressibility of cohesionless soils. In this study, the average SPT blow count per 300 mm over the depth of influence of the foundation is used as a measure of soil compressibility and is represented by N. Burland and Burbidge (1985) suggested that, for very fine and silty sand below the ground water level, if the recorded
Data division and processing One of the advantages of the GP model is that there is no need to introduce new (artificial) parameters, or use functional (e.g., logarithmic, etc.) values affecting parameters; all parameters can be fed into the model as they are and without any normalization or calibration. Therefore, the aforementioned parameters (i.e., foundation width (B), foundation length (L), foundation net applied pressure (q), average SPT blow count (N), and foundation embedment (Df)) are presented to the GP process as model input variables. Settlement is the single output variable. Generally, in pattern recognition procedures (e.g., neural network, fuzzy logic, or GP) the model construction is based on adaptive learning over a number of cases, and the performance of the constructed model is then evaluated using an independent validation dataset. In GP modeling, the way in which the data are divided into training and validation sets has a significant effect on the results (Javadi et al. 2006). In this study, the dataset is divided into several random combinations of training and validation sets until a robust representation of the whole population is achieved for both training and validation sets. To select the most robust representation, a statistical analysis is performed on the input and output parameters of the randomly selected training and validation sets. The aim of the analysis is to ensure that the statistical properties of the data in each of the subsets are as close to each other as possible, and therefore represent the same statistical population. After the analysis, the most statistically consistent combination is used for construction and validation of the GP model. The parameters used in statistical analysis include the maximum, minimum, mean, and standard deviation. The results of the statistical analysis of the finally selected combination are shown in Table 2. Considering that the data contain singular, rare events that cannot be replicated in all cases of the dataset, there may still be some minor inconsistencies in the statistical parameters for the training and validation sets. The proposed model From the total 173 case histories of recorded settlements of shallow foundations in the database, 141 cases (81.5%) were used for training the GP model, while the remaining 32 cases (18.5%) were used for validation of the trained GP model. In this study, the mathematical operators of addition, multiplication, division, square, and negation, and a population size of 1000 individuals were used in the evolutionary proc#
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Table 1. Summary of the records withing the database. Reference Briaud and Gibbens (1999) Maugeri et al. (1998) Wahls (1997) Picornell and del Monte (1988) Burland and Burbidge (1985) Burbidge (1982) Bazaraa (1967)
No. of cases 4 2 30 1 109 22 5
Soil type Silty fine sand Gravely sand Sand Silty gravel/silty sand Fine sand to coarse sand with gravel Fine/medium sand to sand/gravel Clayey sand to sand
Soil density Medium dense — — Loose to medium dense Very loose to very dense Very loose to very dense Medium dense
Foundation structure Test foundation Building foundation Bridge foundation Test foundation Various types of foundation Mainly building foundation Building foundation
Table 2. Statistics of parameters for the training and testing subsets used in development of the genetic programming model. Input and output parameters
Statistical parameters Max. Min. Mean Standard deviation
Subset Training Testing Training Testing Training Testing Training Testing
Foundation width, B (m) 55 25 0.9 0.9 7.72 6.7 8.78 6.53
Foundation length, L (m) 101 84 0.9 0.9 15.72 17.34 18.66 18.53
ess. To simplify the evolved expressions, the power of variables was restricted to integer values from –2 to 2. The analysis was repeated with various combinations of different functions, multiobjective optimization strategies, and number of generations and terms to obtain the most suitable form for the model. The solutions were analyzed to determine the simplest generated model that conformed as closely as possible to the engineering understanding of the settlement mechanism of shallow foundations. After analysis of different alternative models and practical considerations, the following expression was found to be the most robust and practical model for settlement prediction: ½6
Sc ¼
qð1:80B þ 4:62Þ 346:15Df þ L N2
L ¼
11:22L 11:11 L
Foundation net applied pressure, q (kPa) 697 584 18.32 52 195.6 164.2 127.9 110
Average STP blow counts, N 60 50 4 6 25.71 19.69 13.99 10.35
Foundation embedment, Df (m) 10.7 6.7 0 0 2.428 2.155 1.984 1.267
Foundation settlement, Sc (mm) 121 92 0.6 1.5 18.7 20.96 25.29 25.52
Fig. 5. Results of the genetic programming model in (a) training and (b) testing.
where ½7
Equation [6] describes settlement of shallow foundation as a function of the main contributing factors listed above. Figure 5a shows the results of the training of the GP model. From these, it is evident that the GP model was able to learn the complex relationship between foundation settlement and its main contributing factors with a very high accuracy. After training, the performance of the trained GP model was tested using the validation data, which had not been used as a part of the model building process. The purpose of validation was to examine the capabilities of the trained model to generalize the training and apply its learning to unseen cases within the limits of the training data. In the validation process, eq. [6] was used to predict the foundation settlement for 32 case histories not used in the training proc#
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Can. Geotech. J. Vol. 44, 2007 Table 3. Summary of artificial neural network based model and traditional methods for settlement prediction. Reference Shahin et al. (2002a)
Schmertmann et al. (1978) Schultze and Sherif (1973)
Method Sc ðmmÞ ¼ 0:6 þ
120:4 1þeð0:3120:725tanðhÞx1 þ2:984tanðhÞx2 Þ
i
and
x1 ¼ 0:1 þ 103 ½3:8B þ 0:7q þ 4:1N 1:8ðL=BÞ þ 19ðDf =BÞ x2 ¼ 103 ½0:7 41B 1:6q þ 75N 52ðL=BÞ þ 740ðDf =BÞ X Zj Sc ðmÞ ¼ C1 Ct q IZj Ej j pffiffiffi 0:5521 B q Sc ðcmÞ ¼ f 0:87 Df ðNÞ
Meyerhof (1965)
h
1þ0:4 B
Sc ðcmÞ ¼ 0:19 Nq where B < 1:2 m B 2 0:284q where B > 1:2 m Sc ðcmÞ ¼ N Bþ0:33
Note: Sc, foundation settlement; B, foundation width (m); q, foundation net loading (kPa); N, average standard penetration test (SPT) blow count; Df foundation embedment (m); C1, correction factor for embedment; Ct, creep correction factor; IZj, strain influence factor; Ej, Young’s modulus (kPa) at the middle of the jth layer of thickness Zj (m); f, coefficient (obtained from chart).
ess, and the results are shown in Fig. 5b. Here it is seen that the evolved model is also able to predict, with excellent accuracy, the foundation settlement for other (unseen) case histories that were not introduced to the GP as a part of the model building process. The performance of the proposed GP model was also evaluated in terms of coefficient of determination (R2 ), root mean square error (RMSE), and mean absolute error (MAE). These coefficients are defined as X X ðXm Þ2 ðXm Xp Þ2 N N X ½8 R2 ¼ ðXm Þ2 N
½9
RMSE ¼
vX ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u ðXm Xp Þ2 t N
N X ðXm Xp Þ
½10
MAE ¼
N
N
where N is the number of case histories and Xm and Xp are the measured and predicted values (of foundation settlement in this case), respectively. Using training data, the values of R2, RMSE, and MAE for the GP model were 0.94, 7.71, and 5.77 mm, respectively. The GP model also performed very well for the validation set, achieving an R2 value of 0.96, RSME of 6.86 mm, and MSE of 4.92 mm. It is shown that the results obtained for the validation set are consistent with those obtained during the training. This also indicates the excellent generalization capability of the developed GP model when used within the range of the data used for training. Results and comparison with traditional methods To assess the performance of the proposed GP model in predicting settlement of shallow foundations, the results of the model are compared with those obtained using a number of commonly used empirical methods, including the methods proposed by Schmertmann et al. (1978), Schultze and
Sherif (1973), and Meyerhof (1965). The results are also compared with those of a neural network based model (Shahin et al. 2002b), as well as a neural network based formula (Shahin et al. 2002a). The aforementioned empirical methods are selected because they are commonly used to predict settlement of foundations, and because the database used in this study contains the parameters that allow for prediction of settlement using these models. Also, ANNs are the most widely used pattern recognition methods for predicting the settlement of shallow foundations. A backpropagation neural network was trained and validated using the same training and validation datasets as those used in the development of the GP model. The program was written within MATLAB1 using the Neural Network ToolboxTM. The neural network model consisted of one hidden layer with five neurons. The structure and the parameters of the neural network were kept the same as those proposed by Shahin et al. (2002b). A brief description of the conventional methods is provided in Table 3. The results of settlement prediction (for the case studies used in the validation of the GP model), obtained using the aforementioned empirical methods and the ANN are presented in Fig. 6. The results show that the method of Schmertmann et al. (1978) appears to overpredict small settlements of about 0–20 mm, and underpredict larger settlements. The method of Schultze and Sherif (1973) generally provides reasonable predictions for small settlements of up to 20 mm but underestimates larger settlements. The method of Meyerhof (1965) overestimates the settlements. It is shown that, in some cases, applying the traditional methods could lead to errors of more than 300%. The ANN model performs better than the traditional methods, although it has some problems with accurate prediction of small settlements. Comparison of Figs. 5b and 6 shows that the proposed GP model outperforms the traditional methods. It also provides an improvement over both the ANN model and the ANN based formula. Table 4 shows a comparison of the performances of the models in terms of R2, RMSE, and MAE. The coefficients R2, RMSE, and MAE for the results of the GP model (for the validation cases) are 0.96, 6.86, and 4.92 mm, respectively. Comparing the values of these coefficients with those corresponding with the traditional and #
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Fig. 6. Predicted versus measured settlement using (a) artificial neural network model, (b) artificial neural network formula, (c) Schmertmann method, (d) Schultze and Sherif method, and (e) Meyerhof method.
ANN methods again indicates that the GP model performs considerably better than the traditional methods, and it also outperforms the ANN models.
Discussion Estimating settlement of shallow foundations on cohesionless soil is a complex engineering problem. The traditional methods for settlement analysis of shallow foundations suf-
fer from a lack of understanding of the physical nature of the problem, and the simplifying assumptions that are usually made in the development of these models. The application of some of these methods in some cases may lead to very large errors. In recent years, a number of alternative pattern recognition methods (such as neural networks) have begun to be used in engineering problems, including the settlement analysis of shallow foundations. These methods have the advant#
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Can. Geotech. J. Vol. 44, 2007 Table 4. Comparison of performances of the genetic programming and existing methods. Performance criteria
Method GP model ANN model Shahin et al. (2002a) Schmertmann et al. (1978) Schultze and Sherif (1973) Meyerhof (1965)
Coefficient of determination, R2 (%) 95.69 90.56 92.23 34.57 61.24 34.92
Root mean square error (mm) 6.86 10.15 9.2 26.72 20.56 26.64
Mean absolute error (mm) 4.92 6.97 6.65 18.2 10.62 19.49
Note: GP, genetic programming; ANN, artificial neural network.
Table 5. Ranges of values for parameters used in the development of the genetic programming model. Parameter Foundation width, B (m) Foundation length, L (m) Foundation net applied pressure, q (kPa) Average standard penetration test blow count, N Foundation embedment, Df (m) Measured settlement, Sc (mm)
age that they do not require any simplifying assumptions in developing the model. However, the neural network models suffer from a number of shortcomings including (i) their inability to present an explicit relationship between the input and output parameters, (ii) their requirement that the structure of the neural network (e.g., number of inputs, kernel type, transfer functions, number of hidden layers, etc.) be identified a priori, and (iii) their reliance on trial and error to obtain the optimum structure and network parameters. In this paper, a new approach has been presented for the analysis of settlement of shallow foundations on cohesionless soils using genetic programming. One of the advantages of the GP approach is that the number and combination of the terms, as well as the values of the coefficients for different parameters (i.e., the form of the relationship), can all be evolved during construction of the model (training). The nature of GP permits global exploration of expressions and allows the user to gain additional information on how the system performs; i.e., it gives an insight into the relationship between input and output parameters. Another interesting feature of the GP approach is the possibility of getting more than one model for a complex problem, and the high level of interactivity between the user and the methodology. The user insight and physical understanding of the problem can be used to select the appropriate structure for function f (X,aj) (i.e., eq. [1]). Selecting an appropriate objective function by assuming preselected elements in eq. [1] based on engineering judgment and working with dimensional information enable refinement of a final model. In the GP approach, no preprocessing of the data is required and there is no need to normalize or scaling of the data. Another major advantage of the GP approach, which is also found in the ANN approach, is that as more data become available, the quality of the pre-
Minimum value 0.9 0.9 18.32 4 0 0.6
Maximum value 55 101 697 60 10.7 121
diction can be improved by retraining the GP model using the new data. Considering the limited amount of case histories and the variability, uncertainty, and potential noise present in the existing data, additional new case histories will be valuable in improving the quality of prediction for any model. The proposed GP based model was trained and validated using a database of case histories involving settlement of shallow foundations on cohesionless soils. The prediction capabilities of the GP model were compared with a number of traditional as well as ANN based models. The results show that the GP model presented in this paper provides a significant improvement over the existing models. Furthermore, the proposed model generates a simple structured representation of the system. For design purposes, it is easy to use and provides results that are more accurate than the existing methods in settlement analysis of shallow foundations. It should be noted that solutions with better fitness than the proposed model (i.e., eq. [6]) were produced, but were rejected because of their excessive length. Simplicity is a requirement in such engineering applications;because, as the complexity of the model increases, its ability to generalize can be affected by the risk of overfitting the data. For new cases where one or more parameters fall outside the range of the parameters used in training of the GP model (see Table 5), the predicted results should be taken with caution and allowance should be made for the uncertainty. Settlements larger than 100 mm are not well constrained by the case history data, because there are only four cases with settlement greater than 100 mm in the database. It can be suggested that any predicted settlement greater than 100 mm is beyond the predictive range of the proposed model, and should be taken as an indication that large settlements are possible. #
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Fig. 7. Results of parametric study. Effects of (a) foundation width, (b) foundation length, (c) foundation net applied pressure, (d) average standard penetration test (SPT) number, and (e) foundation embedment on foundation settlement.
One the other hand, the settlement of most civil engineering structures is usually limited to a maximum of 25 mm, which is well within the predictive range of the proposed model. Despite the superior performance of the GP model over other alternative models, there are still cases in which predictions of settlement are unconservative. This is common to all models of settlement prediction, and a comparison of the results shows that the number of cases in which (and the degree to which) the results are unconservative is far less in the GP model than the other models. In any case, the large factor of safety that is commonly adopted in geotechnical engineering applications should provide a sufficient safety margin for these cases. Sensitivity analysis A parametric study was carried out to evaluate the predictive capabilities of the proposed formula as well as the effects of varying different input parameters on the output. This was done through a basic approach to sensitivity analysis which is to fix all but one input variables to the their mean values, and to vary the remaining one within the range of its maximum and minimum values. This procedure was re-
peated consecutively for all input parameters and the results are shown in Fig. 7. The results indicate that: (1) The settlement increases with increasing width and applied pressure, and with decreasing foundation depth and average SPT blow count; (2) Parameters N, B, and q have the most effect on settlement; (3) The effect of length of foundation, L, on settlement appears to be negligible. There is a slight increase in settlement with increasing length up to about 5B, but any further increase in length has no effect on settlement. This is consistent with the traditional methods (that usually ignore the effect of length of foundation), and with the practical assumption that for L/B ratios greater than five, the foundation acts as a continuous footing with theoretically infinite length. (4) The proposed GP model is consistent with traditional models of Meyerhof (1965); Schultze and Sherif (1973), and Schmertmann et al. (1978) in terms of the form of the relationships, in that in all these relationships, settle#
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ment is directly proportional to q and indirectly proportional to N (or E in the Schmertmann’s model).
Summary and conclusion Analyzing the settlement of shallow foundations on cohesionless soils is a complex geotechnical engineering problem owing to the heterogeneous nature of soils and the participation of a large number of factors involved. Using the traditional methods to predict settlement of shallow foundations could lead to very large errors. Furthermore, for the same case, settlement predictions using different methods can be very different depending on the method employed. Consequently, alternative methods are needed to provide more accurate settlement predictions. In this paper, a new approach was presented, based on genetic programming (GP), for the analysis of settlement of shallow foundations on cohesionless soils. A GP model was trained and validated using a database of case histories involving settlement of shallow foundations. The results of this model were compared with those obtained using three commonly used traditional methods and neural network based models. Comparison of the results shows that predictions made by the proposed GP model provide significant improvements over the traditional methods and also outperform the ANN models. It is shown that the model is capable of learning, with a very high accuracy, the underlying relationship between foundation settlement and the contributing factors, which is presented in the form of a polynomial function,and generalizing the training to predict settlements for new cases. The results show that the genetic programming model provides a very efficient method for the analysis of settlement of foundations.
Acknowledgements The authors would like to thank Professor Mark B. Jaksa, of Adelaide University, for providing the database on the case histories.
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