1 Introduction
In urban areas, construction activities are commonly restricted due to the high density of adjacent buildings. This leads to the requirement of retaining walls to reduce the excavation area impact on movement. A cantilever sheet wall is a regular retaining structure for the shallow excavation, and the stability of this structure relies on the balance of the active, and passive, soil pressures. The cause of failure primarily comes from the failure of maintaining this balance rather than exceeding the allowable stress in the wall. This failure may lead to the collapse of the adjacent building or buildings, and thus, accurately predicting the failure or stability of the cantilever is desired for not only the construction works itself, but also the surrounding environment. The horizontal displacement at the top of the cantilever sheet wall (Δ), can affect the working space within the excavation and be the alarm of the wall collapse, is a very complicated prediction because of its combinations of rotation due to imperfect supports and deflection of the wall. The ground movement, y(x), due to the excavation, is a potential failure thread to the adjacent building or their piles. Consequently, the soil settlement adjacent to a sheet wall is also a variable [1–5] of interest.
The finite element analysis (FEA), is a common approach to solve geotechnical problems and is commonly applied for the designing purpose as in Ref. [6]. However, FEA sometimes faces difficulty with solving the partial differential equations for complicated problems, and only yields to the prediction for a particular case without extracting the non-relationship of the input variables. It provides an approximation model [8]. Practical approaches to eliminate these drawbacks are commonly relegated to a machine learning (ML) technique: 1) applying ML techniques to solve the partial differential equations as in Refs. [9,10], and 2) using FEA to generate the database for the ML models [10–13].
The combination of FEA for developing the database, and predicting the output, with ML models as summarized in Ref. [15], has been conducted not only for structural analyses [15–18] but also for a variety of geotechnical problems. Examples are modeling soil behaviour [19–22], predicting pile capacity [23–25], site characterization [26–28], liquefaction [29–31], slope stability [32–34], and landslide assessment [35–37]. Within these studies, research on the retaining structures is a critical part [1,13,38–41]. In Ref. [38], the conventional reliability approach (i.e., First Order Reliability Method and Monte Carlo Simulation) is integrated with artificial neural network (ANN) for reliability and risk evaluation of deep excavations. In Ref. [49], the displacement of a braced retaining wall in clay soil is predicted with the combination of ANN and FEA. In Ref. [1], the prediction on the additional deformation of the soil due to the nearby underground excavation and the parameter study, is implemented. In Ref. [13], records from construction site are used as the database to develop the ANN model for approximation of diaphragm wall deflection. The same concerns are on the retaining wall of deep excavation, in Ref. [40], the adaptive Broyden-Fletcher-Goldfarb-Shanno algorithm, and the neural network, are combined to estimate the state-based wall deflection and maximum location of each value. In Refs. [41–43], the surface settlement due to a deep foundation construction process has been predicted by random forest (RF) and ANN.
In contradiction, the ML approaches have several flaws such as the uncertainty in finding the “best” hyper-parameters for the model. This leads to frequent use of a trial and error process and grid searching, or another optimization process is required in Refs. [7,44]. Furthermore, researcher who applied data-driven models for the geotechnical problems commonly focused on the single predicting task rather than proposing a system predicting various outputs regarding to the fact that the calculation be terminated with error. There is an absence of an initial classification model to observe soil-structure system failure possibility, and a shortage within current studies. Most studies have concentrated on the regression model where the variables of interest are continuous. Zhang et al. [45] summarized 92 studies with a soft computing approach, and the classification problem is rarely seen in this list, except for some on rock burst problems. Analogously, the review paper of Moayedi et al. [14], has shown this shortage.
In this study, a variable of interest, which is a binary variable, attached at the initial step of the predicting system, can be linguistically interpreted as “failure” or “not failure”. It is obvious that this category and other continuous variables, such as quantities of ground settlement, have a hierarchical order in the prediction process. The proposed system is a simplified network, which contains a sequence of a classification model, the random forest classification (RFC), and is chosen to predict the stability status of the cantilever sheet wall. Two other regression ANN models (RANN1 and RANN2), predicting the vertical settlement of the adjacent ground surface, and the horizontal displacement, are conducted separately. The uncertainty of the ML model is observed by a random sampling technique in the train/test data splitting process for the classification problem. A set of 1000 RFC models is obtained and selection, from this set, is implemented to choose the “best” model for predicting status of the soil-structure system. The configuration selection process for the ANN model is conventionally carried out based on the trial and error approach, and not provided in the context for the sake of simplicity. The parametric study is implemented for the regression models, and the practical application of the proposed data-driven based hierarchical system is illustrated.
2 Methodology
2.1 Simulating behaviour of cantilever sheet wall with finite element analysis
Geometric variables of the sheet wall structure provided in Fig.1(a), includes the depth of excavation H, length of the sheet wall L, and depth of the ground water table Hw. The addition of construction loads, such as vehicle load, or temporary weight load, are also modelled as a uniform load q, with the corresponding width of B along the x-axis. In some simulations, this load is not applied to widen the possible cases of the database. Material property (i.e., modulus of elasticity of steel, E) is combined with cross-section property of sheet wall (the moment of inertia, I) by using EI as an input variable. Cantilever sheet wall, and soil are modelled in Plaxis, as a 2D beam, and triangular two-dimensional deformation element, respectively. The meshing system includes the global mesh which is relatively coarse, and the mesh surrounds the sheet wall which is finer (Fig.1(b)).
For geotechnical engineering, it is critical to choose a proper model to simulate the soil behaviour. Hardening soil (HS), is the model chosen in this paper, has an advantage in accounting for the effect of stress to the stiffness of soil [46–48]. This characteristic is appropriate, especially for the excavating process with different phases. The HS is a hyperbolically elastic-plastic model which additionally uses the secant stiffness in standard drain triaxial test E50ref, tangent stiffness for primary oedometer loading Eoed, and unloading/reloading stiffness Eurref, to describe the stress-depend stiffness. It is commonly assumed that E50ref = Eoed, and Eurref = 3 × E50ref [46], and it is sufficient to account onlyEoed, to the ML models because of these relationships.
Along with Eoed, the friction angle , cohesion c, saturated weight γsat, and unsaturated weight γunsat, are the 5 primary properties of soil required for the analysis process. Detail of the ranges of soil properties is to be provided in Section 3. These soil properties are collected by the author’s team from various construction sites in Vietnam covering both cohesive and non-cohesive soils. Other additional properties of the soil such as natural water content, natural voids ratio, grain size distribution, degree of saturation, etc., are not always available in the collected data. These properties are not used to develop the prediction models, and can be found from their relationship with the 5 primary properties or reasonably assumed for the FEA. Soil with zero cohesion, will have this value set as 1 to ensure numerical stability in the computing process. Only homogeneous soil types, without stratification, and their long-term behaviour, are considered in this paper.
Each simulation starts with the first phase when the wall is inserted. The digging process is consequently taken place with 5 phases corresponding to soil layers, which equal to 1:5 of H (Fig.1(b)). Response of the structure, such as wall and soil displacement found in each phase, is used as the input for the succeeding phase.
To avoid the collapse of Plaxis from other non-failure causes, the sheet wall structure is considered to be in failure if the “soil body collapses” announcement is applied. This equates to that the current stiffness parameter, CSP, is less than 0.015 (as default). CSP is a parameter to measure the amount of plasticity occurring in the calculation, where the near zero value of CSP indicates failure of the model [49]. The “soil body collapses” error, thus implies the ultimate limit state of the soil is exceeded and will be used as the collapse criterion. Additionally, if the horizontal displacement at the top of sheet wall is significantly large, it is a warning of exceeding the deformation limit (i.e., serviceability limit state). In general, a sheet wall applied for an excavation will be considered as failure if it collapses or exceeds a threshold for Δ. Since there is no strict rule for this threshold, it is practically chosen at 200 mm based on expert opinions.
The status of the structure and horizontal displacement at the top of the sheet wall can be written as the function of input variables previously discussed:
Database for the vertical settlement of the ground is based on simulations but distinct from database for the status and Δ. In each Plaxis simulation, more than one sample can be found for y(x) because it is a x-dependent variable. Database for y(x) will be expanded to m × p samples where m is number of simulations and p is the number of chosen horizontal locations. Consequently, y(x) can be found by a function with inputs as in Eq. (3), and x as additional input variable:
2.2 Machine learning models
In this section, fundamentals of the chosen ML models, RF and ANN, are discussed. It is noted that attempts on both RF and ANN have been implemented for classification and regression problems. The RFC is chosen for the classification problem of sheet wall safety/failure prediction because of the high accuracy, and stable results. Meanwhile, the ANN models provide a more robust solution for the regression problems compared to that of RF. The model selection process is conventionally applied and not discussed in the study for simplicity.
The cross-validation technique is commonly conducted to observe the randomness of the models, which relies heavily on the accuracy of the data. The conventional method of data splitting into train set, and test set, is a common approach. Further techniques can be applied to capture the dependence of model to data (e.g., K-cross validation [50]). In this study, a set of 1000 RFC models are developed and each of them is developed based on the sampling process of taking 80% of the data from the database [51]. Critical results of the classification problem, which are: accuracy, precision, and recall, for each model on the test set (e.g., the remaining 20% of the database), are calculated. Consequently, the distributions of these metrics are observed to evaluate the dependence of model quality on the database. This process is illustrated in Fig.2.
2.2.1 The random forest classification
In this paper, the RFC model, predicts the failure status of the sheet wall. The RF method relies on the classification and regression tree algorithm, CART [52], to develop a voting process. Starting with a root node contains the database, CART splits the parent node into the left and the right nodes (or the children nodes) by the best of a pair of tk and k, where k is the kth feature and tk is a threshold corresponds to kth feature. The chosen pair of tk and k must have the minimum following cost function :
where is the impurity of the left/right subset; is the sample of the left/right subset. The Gini is used in this paper as the impurity measure. With the 2 classes, (0) corresponding to the failure or (1) not failure , the Gini index of a node can be written as:
where is the ratio of instances, belongs to failure class (0) over the total instances in the left/right node. Denominator, is the number of instances in the left/right node and, and is the number of instances, belongs to failure and not failure classes, in the left/right node, respectively.
The splitting process is recursively implemented from the root node to generate its descendants until the stop condition(s) is reached. The nodes in the last generation are the leaf nodes. In this paper, the main stopping condition are the minimum samples required to split a node, is 2. The minimum samples required to be at a leaf node is 1.
The forest in the RF method, is composed by many CARTs where each of the trees is established based on a part of the FEA database, and n subsets of database are randomly selected from database via boot-strapping (i.e., bagging), or pasting sampling processes, with or without replacement, respectively. The result of the RF is obtained by a voting process, which aggregates the results of all the decision trees. Throughout, the feature importance levels of each input variable, are found based on its influence on establishing the decision trees within a “forest” of trees. To be specific, feature importance level is found from, the average depth in which the feature appeared within the decision trees [53].
2.2.2 The Artificial Neural Network for Regression
The idea of ANN was introduced by McCulloch and Pitts [55] who tried to simulate the mechanics of neurals in the brain. The ANN has a long history of development with various key studies such as image recognition [56], Restricted Boltzmann Machines [57], Deep Belief Net [58], and auto-encoders [59].
A conventional feed-forward ANN is trained by feeding data to the input layer, and each node in this layer takes its corresponding feature of each sample. Every neuron combined its input signals from nodes in the previous layer by weight-summing them. With an activate function (e.g., ReLu, sigmoid or Hyperbolic tangent), this neuron generates a signal to the nodes in the next layer until the output layer is reached. After feeding by so-call batch size b samples, the loss function in Eq. (12) is calculated, and the Back-Propagation algorithm is applied to update the weights of neuron-to-neuron connections.
There are two ANN models, trained separately in this study. The first regression ANN model, RANN1, approximates the lateral displacement of the sheet wall Δ, and the second regression ANN model RANN2, predicts the soil vertical settlement y(x).
2.3 The Proposed Hierarchic System with a Mixture of Classification and Regression models
Once the models successfully developed, a mixture of models is applied as a hierarchic system for designing the process with a logic gate, after the prediction with RFC is conducted (Fig.3). To avoid the conflict of the final outputs, the conventional binary values of 1 and 0 are used to check the condition. If the sample is labeled as 0 after the RFC, the computing process will stop with the –1 values be assigned for the outputs of RANN1 and RANN2 as the collapse of the sheet wall without actual predictions by these ANN models. Otherwise, if the sample is labeled as 1, the logic gate will trigger the next two regression models, RANN1 and RANN2, to predict lateral displacement at the top of the sheet wall Δ, and the quantity of soil vertical settlement, y(x), respectively. It is noted that the inputs of RANN2 are different from RFC and RANN1 because of the additional x input variables presented for the location along x-axis that are of interest. The output system of the framework in this paper only contains Δ and y(x) and can be developed by adding other outputs of interest such as maximum moment in the sheet wall.
3 Numerical results
3.1 Validation of hardening soil model and experimental data
A simulation is first conducted with the input variable of the experiment from [60] (listed in Tab.1 where H = 5.83 m). The results from Plaxis are provided in a contour plot (Fig.4). It can be seen in Fig.4(a) that the maximum disruption of the soil is 100 mm at the bottom of the excavation, and the maximum settlement of the ground is 280 mm near the top of the sheet wall. The adjacent ground surface near the sheet wall (i.e., on the left) settles the most due to the excavation. The more the distance from the top of the sheet wall is, the less the vertical settlement is of the ground surface. In Fig.4(b), it is reasonable to observe a maximum of the horizontal displacement at the top of the sheet wall is at 380 mm, and most of points with the significant vertical displacement are concentrated at the back of the wall. In general, results from this observation agree with the intuition on the behaviour of the soil-sheet wall structure.
Tab.1 Inputs of the validating cases (to compare with experimental case) and control case (for parametric study) |
| No. | variable | unit | validating cases (from Ref. [60]) | control case (for parametric study in Subsection 3.4) |
|---|---|---|---|---|
| 1 | L | m | 10 | 12 |
| 2 | EI | kN·m2/m | 6540.7 | 44982 |
| 3 | Hw | m | 0 | 2 |
| 4 | H | m | 3.05; 4.05; 5.05; 5.53; 5.83 | 4 |
| 5 | q | kN/m2 | 0 | 5 |
| 6 | B | m | 0 | 5 |
| 7 | γunsat | kN/m3 | 16 | 18.0 |
| 8 | γsat | kN/m3 | 19.85 | 19.0 |
| 9 | Eoed | kN/m2 | 10000 | 9958 |
| 10 | c | kN/m2 | 0 | 1 |
| 11 | phi | ° | 39.4 | 27 |
The results from 5 experiments in Ref. [60] versus the present models are provided in Fig.5. Despite the existence of differences between the experiment and simulation results, the simulations in these studies provide reasonable values and they closely follow the experimental results. In general, the overall shapes of the corresponding lines are well matched, along with the minor differences between results of simulation and experiment. For instance, differences between displacements at the top of the sheet wall of experiment and simulation are roughly estimated at +20, +10, 15, –25, and –25 mm for the cases of H = 3.05, 4.05, 5.05, 5.53, and 5.83 m, respectively. This comparison validates the hardening soil model applied in this study which is used for data generating process in the next section.
The case with H = 4.05 m in Fig.6 is further used to investigate the effect of element size to the output of FEA. The available element types provided by Plaxis are observed, including: the very coarse, coarse, the medium, fine and very fine meshing systems. It can also be seen from Fig.6 that the results of FEA are converged with the fine sized elements with minor difference between the corresponding lines. The meshing system with finest elements almost matches with the experiment data from Ref. [60]. The horizontal displacement at the top of the cantilever sheet wall Δ, of the very fine and the fine mesh type, are 84.767 and 86.625 mm, respectively. The differences between the finest (Δ = 84.767 mm) and the largest element size (Δ = 98.533 mm) are significant, at 16.240% delta. This indicates that the quality of the FE model depends heavily on the meshing system of the element size. Obviously, a trade-off is unavoidable when the computing time for the very coarse mesh is 357 s, and that of very fine mesh is almost double, with 642 s. To maintain quality of the FEA database, the study consistently uses the very fine meshing system for data generation.
3.2 The database and data allocation for training and testing processes
Practically, there is no fixed threshold for the size of the database for an engineering problem, and a sufficient size for database depends on the specific scenarios. In many cases, a few hundred samples in the FEA database are adequate to train a ML model. Examples are, Verma et al. [10] used a database with 100 samples to predict the factor of safety for slopes; 272 data points are used in Ref. [61] to predict settlement of foots; or Duong et al. [16] used a data set of 150 samples to develop ANN model predicting capacity of rectangular concrete-filled steel tube short columns. In this study, the developed database contains the results of 200 FEA samples. It is noted that the database can be expanded if satisfied evaluation metrics of models are not obtained.
A part of the database provided in Tab.2 and Tab.3 and the ranges of the input are in Tab.4. These ranges are extremely important for ML prediction because they imply the boundaries of the models. Only if the input variables satisfy such boundaries, the prediction is valid. This is analogous to the extrapolation, if a set of input variables contained value(s) lies outside the corresponding ranges, an inappropriate prediction will occur. Consequently, the developed models are applicable to predict the response of soil-sheet wall with the depth of the excavation is up to 5 m. Other input ranges can be found in Tab.4. Fig.7 provides the histogram of the Δ and y(x) of the labeled database.
Tab.2 Database developed by FEA for stable and Δ |
| No. | L (m) | EI (kN·m2/m) | Hw (m) | H (m) | q (kN/m2) | B (m) | γunsat (kN/m3) | γsat (kN/m3) | Eoed (kN/m2) | c (kN/m2) | phi (° ) | stable | Δ (mm) | not failure? |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 8 | 44982 | 40 | 5 | 0 | 0 | 18 | 19 | 23000 | 41 | 15 | 1 | 33.61 | 1 |
| 2 | 9 | 44982 | 40 | 5 | 0 | 0 | 18 | 19 | 23000 | 41 | 15 | 1 | 30.12 | 1 |
| 3 | 10 | 44982 | 40 | 5 | 0 | 0 | 18 | 19 | 23000 | 41 | 15 | 1 | 28.69 | 1 |
| 4 | 12 | 44982 | 40 | 5 | 0 | 0 | 18 | 19 | 23000 | 41 | 15 | 1 | 28.14 | 1 |
| 5 | 16 | 44982 | 40 | 5 | 0 | 0 | 18 | 19 | 23000 | 41 | 15 | 1 | 27.97 | 1 |
| 6 | 8 | 44982 | 40 | 5 | 0 | 0 | 19 | 20 | 9958 | 1 | 27 | 0 | – | 0* |
| 7 | 9 | 44982 | 40 | 5 | 0 | 0 | 19 | 20 | 9958 | 1 | 27 | 0 | – | 0* |
| 8 | 10 | 44982 | 40 | 5 | 0 | 0 | 19 | 20 | 9958 | 1 | 27 | 1 | 211.68 | 0* |
| 9 | 12 | 44982 | 40 | 5 | 0 | 0 | 19 | 20 | 9958 | 1 | 27 | 1 | 164.33 | 1 |
| 10 | 16 | 44982 | 40 | 5 | 0 | 0 | 19 | 20 | 9958 | 1 | 27 | 1 | 155.18 | 1 |
| 198 | 10 | 44982 | 4 | 5 | 10 | 10 | 19 | 19.5 | 12000 | 32 | 22 | 1 | 40.67 | 1 |
| 199 | 12 | 44982 | 4 | 5 | 10 | 10 | 19 | 19.5 | 12000 | 32 | 22 | 1 | 39.1 | 1 |
| 200 | 16 | 44982 | 4 | 5 | 10 | 10 | 19 | 19.5 | 12000 | 32 | 22 | 1 | 39.14 | 1 |
*Note: These zero values then switched into –1 in the framework to avoid confusing with the value of regression predictions. |
Tab.3 Database developed by FEA for y(x) |
| sample # | 1 | 2 | … | 199 | 200 | ||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| x (m) | y(x) (m) | x (m) | y(x) (m) | … | x (m) | y(x) (m) | x (m) | y(x) (m) | |||||
| 1 | 0.0000 | 0.0082 | 0.0000 | 0.0061 | … | 0.0000 | 0.0140 | 0.0000 | 0.0134 | ||||
| 2 | 0.2951 | 0.0113 | 0.2951 | 0.0092 | … | 0.2951 | 0.0232 | 0.2951 | 0.0234 | ||||
| 3 | 0.2951 | 0.0113 | 0.2951 | 0.0092 | … | 0.2951 | 0.0232 | 0.2951 | 0.0234 | ||||
| 4 | 0.6264 | 0.0123 | 0.6264 | 0.0104 | … | 0.6264 | 0.0251 | 0.6264 | 0.0254 | ||||
| 5 | 0.6264 | 0.0123 | 0.6264 | 0.0104 | … | 0.6264 | 0.0251 | 0.6264 | 0.0254 | ||||
| 53 | 55.2219 | 0.0009 | 55.2219 | 0.0009 | … | 55.2219 | 0.0019 | 55.2219 | 0.0020 | ||||
| 54 | 57.6109 | 0.0009 | 57.6109 | 0.0009 | … | 57.6109 | 0.0019 | 57.6109 | 0.0019 | ||||
| 55 | 60.0000 | 0.0009 | 60.0000 | 0.0009 | … | 60.0000 | 0.0019 | 60.0000 | 0.0019 | ||||
Tab.4 Ranges of variables in the database |
| No. | variable | unit | count | min | max |
|---|---|---|---|---|---|
| 1 | L | (m) | 200 | 8 | 16 |
| 2 | EI | (kN·m2/m) | 200 | 44982 | 110250 |
| 3 | Hw | (m) | 200 | 1 | 40 |
| 4 | H | (m) | 200 | 2.5 | 5 |
| 5 | q | (kN/m2) | 200 | 0 | 15 |
| 6 | B | (m) | 200 | 0 | 15 |
| 7 | γunsat | (kN/m3) | 200 | 17 | 19 |
| 8 | γsat | (kN/m3) | 200 | 17.6 | 20 |
| 9 | Eoed | (kN/m2) | 200 | 5479 | 78000 |
| 10 | c | (kN/m2) | 200 | 1 | 41 |
| 11 | ° | 200 | 15 | 34 |
As in the above discussion, a sample is classified as failure if Plaxis warns an error due to the collapse of computing process or the horizontal displacement at the top of the sheet wall exceeds 200 mm. There are 50 samples that are interrupted during the computation process, and 15 samples exceed the threshold for Δ, leading to the failure and not failure categories, containing 65 and 135 samples, respectively. The ratio number of failure/total number of stable samples is (200–135)/200 = 0.325 indicating an unbalanced database or more samples are in safe condition. The labeled database is first split in to the train set, and test set, with 80% of the database as the train set. This train set (160 samples) is then used for the training process to develop the model, and the test set (40 samples) is used for validating the model.
To predict Δ, only 135 samples which are labeled as stable category are used to develop the RANN1. This data set is randomly split again into train set (108 samples) and test set (27 samples). Analogous to this of RANN1, data for RANN2 is generated from 135 non-failure samples. For each of these 135 simulations, y(x) at a set of 31 horizontal locations, x is recorded. To validate the effectiveness of developed model, 5 samples are randomly chosen for the final illustration, which are the 1st, 33rd, 96th, 101st, and 140th samples. These 5 samples provide the validation of RANN2 on an unfamiliar data set. The rest of the database (130 simulations) provides 130 × 31 = 4030 samples and then splits into the train and test sets with the ratio of 8:2, respectively. The train set thus contains 104 × 31 = 3224 samples and the test set contains 26 × 31 = 806 samples. Details of the data allocation are provided in Tab.5.
Tab.5 Detail of data splitting for 3 developed models |
| No. | model | data ID | predicted variable | unit | valid count | min | max | samples for training | samples for testing | samples for graphical evaluation |
|---|---|---|---|---|---|---|---|---|---|---|
| 1 | RFC | data1 | Statusa) | – | 200 | –1 | 1 | 160 | 40 | – |
| 2 | RANN1 | data2 | Δb) | mm | 135 | 8.2 | 200 | 108 | 27 | – |
| 3 | RANN2 | data3 | y(x)c) | mm | 135 × 31 = 4185d) | –13.1 | 854.7 | 104 × 31 = 3224 | 26 × 31 = 806 | 5e) × 31 = 155 |
Notes: a) Unstable if Δ > 200 mm or error in computation process. b) Only Δ ≤ 200 mm is used for developing the RANN1. c) Only Δ ≤ 200 mm is used for developing the RANN2. d) The 31 selected horizontal locations are: 0.0, 0.5, 1.0, 1.5, 2.0, 2.5, 3.0, 3.5, 4.0, 4.5, 5.0, 5.5, 6.0, 6.5, 7.0, 7.5, 8.0, 8.5, 9.0, 9.5, 10.0, 12.0, 14.0, 16.0, 18.0, 20.0, 25.0, 30.0, 40.0, 50.0, 60.0 (m). e) Samples: 1st, 33rd, 96th, 101st, and 140th. |
Tab.6 provides the insight relationships of the input and output variables within the database developed from FEA. It is worthy to note that each row of this table is calculated from different databases corresponding to the data1, data2, and data3 designated in Data ID column Tab.5. The Kendall correlation coefficients in Tab.6 are slightly low, with many values close to zero and the maximum absolute value is 0.4753 as compared to the maximum at 1.0000 for this factor. However, various papers focusing on ML, have developed quality models with inputs, which have correlation coefficients that are less than 0.5000 such as Ref. [62].
Tab.6 Input-output correlation matrix |
| variable | L | EI | Hw | H | q | B | γunsat | γsat | Eoed | c | x | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| stable | 0.3674 | –0.0642 | 0.1327 | –0.0832 | 0.1362 | 0.1184 | –0.0698 | –0.1439 | 0.4293 | 0.4100 | –0.0055 | – |
| Δ | 0.3235 | –0.0447 | 0.0012 | 0.1555 | –0.1327 | –0.1248 | 0.0838 | 0.1510 | –0.4126 | –0.4753 | 0.0798 | – |
| y(x) | 0.0614 | –0.0077 | –0.0771 | -0.0374 | 0.2938 | 0.2575 | –0.2399 | –0.2714 | –0.3055 | –0.3830 | 0.3683 | –0.3926 |
It can be seen from Tab.6 that critical inputs for predicting the stable and Δ share similarity to some level. To begin, L, c, and Eoed, are the most significant inputs with the values within 0.3 to 0.45. Width of the applied load, B, and saturated weighted, γsat, are at the medium level to their counterparts with the absolute correlation factors ranges within 0.1 to 0.15. Ground water table, Hw, and depth of excavation, H, have the medium correlation to stable and Δ, respectively. Meanwhile, stiffness of the sheet wall, EI, and friction angle, , have almost no correlation to outputs.
On the other hand, the vertical displacement of a point on the surface is strongly correlated to x. The quantity and length of applied load on the surface, q and B, tangent stiffness for primary oedometer loading, Eoed, friction angle, , saturated weight γsat and unsaturated weight γunsat, have the highest correlation to y(x). In contrast to the cases of stable and Δ, the length of sheet wall, L, and depth of excavation, H, have weak correlations to the vertical displacement.
3.3 The developed data-driven models
3.3.1 The random forest classification model
As in the above discussion, the data splitting process is repeated randomly to obtain 1000 RFC models with ratio 0.8/0.2 and the critical hyper-parameters of the models are: no maximum of depth of a decision tree is set, minimum samples per leaf is 1, minimum sample for a split (in the decision tree) is 2, and the number of decision trees is 100. Results of distribution of accuracy, precision, and recall, for these RFC, are provided in Fig.8. It can be seen that the distribution of these metrics are strongly left skewed to the natural boundary of 1.0000. Due to this boundary, the median is a better option for expectation observation of these random variables as compared to the mean. It can be seen in Fig.8 that the recall is significantly lower than its counterparts. This implies the Type I error or false negative is more significant than the Type II error of fault positive for models developed with this database. These lead to the un-conservative side of such models when more actual failure is predicted safety, than actual safety predicted failure. This can be explained by the unbalanced database with more safety samples than failure samples, discussed earlier. For this reason, the recall is the priority evaluation metric in the classification problem, from both practice and database points of view. In the next steps, a model with accuracy, precision, and recall, all equal at 1.0000, is chosen out of the 1000 developed RFC models. The confusion matrix of this chosen model is provided in Fig.9.
3.3.2 Regression models with artificial neural networks
Since there are no rules to obtain the best network [63], various ANN configurations have been tried with, 104 epochs, “adam” optimization algorithm, and Mean Absolute error is the loss function [53]. The batch sizes are 50 and 500 for RANN1 and RANN2, respectively. Details of the chosen configurations for RANN1 and RANN2 are provided in Tab.7 and Tab.8. As can be seen in Tab.7, the RANN1 contains 625 nodes, with 3 hidden layers, and 50561 trainable weights. Meanwhile, the RANN2 needs 317 nodes, distributed in 6 hidden layers, with 15729 trainable weights (Tab.8).
Tab.7 Configuration of the RANN1 |
| layer | number of nodes | trainable weights |
|---|---|---|
| input layer | 12 + 1 (bias) | – |
| hidden layer 1 | 64 + 1 (bias) | (12+1) × 64 = 832 |
| hidden layer 2 | 512 + 1 (bias) | (64+1) × 512 = 33280 |
| hidden layer 3 | 32 + 1 (bias) | (512+1) × 32 = 16416 |
| output layer | 1 | (32+1) × 1 = 33 |
| total | 625 | 50561 |
Tab.8 Configuration of the RANN2 |
| layer | number of nodes | trainable weights |
|---|---|---|
| input layer | 12 + 1(x) + 1 (bias) | – |
| hidden layer 1 | 64 + 1 (bias) | (13+1) × 64 = 896 |
| hidden layer 2 | 64 + 1 (bias) | (64+1) × 64 = 4160 |
| hidden layer 3 | 64 + 1 (bias) | (64+1) × 64 = 4160 |
| hidden layer 4 | 64 + 1 (bias) | (64+1) × 64 = 4160 |
| hidden layer 5 | 32 + 1 (bias) | (64+1) × 32 = 2080 |
| hidden layer 6 | 8 + 1 (bias) | (32+1) × 8 = 264 |
| output layer | 1 | (8+1) × 1 = 9 |
| total | 317 | 15729 |
The evaluation metrics for both RANN1 and RANN2 are provided in Tab.9. It can be seen that errors on both test set and train set of RANN1 and RANN2 are all minor compared to the mean values of y(x), and Δ (57.7836 and 18.264, respectively). With a much larger database, it is reasonable to observe that R2 on test set of RANN2 (0.9988) is much higher than this of RANN1 (0.95212). Fig.10 illustrates the converging process of the ANN models, versus the number of epochs in logarithmic scale. It can be seen from Fig.10 that the error of RANN1 model (e.g., MAE) on the normalized test set approaches its stable level (MAE is around 0.2000) at 103 epochs. Meanwhile, this error value is roughly stable after 2 × 103 epochs in the case of RANN2. Further training for this database may lead to the severe overfitting, when the MAE on the train sets of both models continuously decrease without any improvement of error on the test sets.
Tab.9 Evaluation metrics for Regression models |
| metric | equation | RANN1 | RANN2 | |||
|---|---|---|---|---|---|---|
| on train set | on test set | on train set | on test set | |||
| mean absolute error | 0.201697 | 4.531986 | 0.000288 | 0.00103 | ||
| mean squared error | 0.11290 | 49.2773 | 7.2152e−07 | 9.6209e−06 | ||
| coefficient of determination | 0.9999 | 0.95212 | 0.9999 | 0.9988 | ||
It can be seen from Fig.11, which is a scatter plot of predicted and simulated values of the two models, that the test data points of the two models highly concentrate around 1:1 lines, except a few data points. This is the graphical illustration of the high evaluation metrics of the models in Tab.9. Another illustration of the RANN2 is provided in Fig.12, where the predicted and simulated ground settlement values for sample 1st, 33rd, 96th, 101st, 140th. According to this figure, the RANN2 model successfully predicts ground settlement in both trend, and quantities, despite of some minor errors. The different shapes of the settlement lines are also successfully predicted.
3.4 Feature importance and parametric studies
To evaluate the impact of inputs to the prediction of RFC model, the feature importance analysis is conducted with normalized results provided in Fig.13. It can be seen from Fig.13 that the length of the sheet wall, L, has the critical role with the normalized feature importance score of 0.3186, followed at a distance by Eoed (0.1335), and Hw (0.1021). It is reasonable to observe the second ranked position of a variable is related to the soil stiffness, Eoed, due to the importance of soil property for maintaining the stability of the retaining wall. Other soil properties (i.e., c, , γusat, γsat) have low to medium impact to the output with importance factor, ranging from 0.0834 (c) to 0.0540 (γsat). Meanwhile, the changes of applied load with corresponding inputs, B and q, and the stiffness of the sheet wall, EI, have minor effect to the prediction, with normalized feature importance levels that are less than 0.0500.
For further observation of inputs to outputs in the developed ML models, the parametric study on RANN1 and RANN2 is implemented. Inputs for the control case for these analyses are given in Tab.1. Fig.14 and Fig.15 illustrate the parametric studies for RANN1 and RANN2, respectively.
For RANN1 in Fig.14, the length of sheet wall, L, reinforced its strong effect to the output (i.e., Δ in this scenario) with a clear separation of 8, 10 and 12 m in all sub-figures. For longer walls with L = 14 and 16 m, the differences are minor. In some cases, the corresponding lines of L = 12, 14, and 16 m tend to be overlapped, which implies that the use of these walls has the analogous effects to the system. Figure 16 is intentionally provided with the same scale on y axis with Δ ranging from 0 (mm) to its maximum value of 200 mm. Except for the length of sheet wall, L, the depth of excavation, H, is reasonably having the strongest and positive correlation, to the displacement of the wall with the exponential lines appeared in the H versus Δ sub-figure. Soil properties have the reversed trends where the increases of Eoed, c, γunsat, and , lead to the decrease of Δ. This is predictable because the improvement of soil properties naturally reduce the displacement of the sheet wall. In contrast to the RFC, the changes of ground water table (hw), width and quantity of applied load (B and q) result in the slight fluctuations of Δ. This implies the less importance of such input variables compared to the above-mentioned inputs of RANN1. Sheet wall stiffness, EI, and saturated weight, γsat, are in the group of features that have the lowest impact to the output.
In Fig.15, results of the parametric study for the RANN2 are provided. The horizontal location of the prediction, x, is the natural key feature of the model with a clear separation of y(x) lines corresponding to the locations adjacent to the sheet wall (e.g., x = 2.5 and 5 m). With further surface points, the differences are minor. This indicates the fading effect of sheet wall existence along with the increase of distance from this structure. The depth of excavation, H, and length of sheet wall, L, and the internal friction angle, , are the most critical inputs with dramatic changes of y(x) with the increases of these inputs. While H has the positive correlation with y(x), , and L, have the negative effect to the outputs, with the increase of these variables resulting in the decrease of y(x). Hw, c, q, γunsat, and EI are in the intermediate group which establishes a clear relationship to output. Even though the relationship of Eoed and y(x) is apparent, curved-shaped lines are observed with fluctuations. The trends in the B versus y(x) are not clear, with the trends reversed there exists a crossover of x = 2.5 m and x = 5 m lines. The γsat with the corresponding lines that are almost parallel to the horizontal axis, seems to be the least important variable.
3.5 Results of the proposed hierarchical system
An example is provided of the output system for those proposed in Fig.3, and the implementation of the 3 developed models, RFC, RANN1 and RANN2, are in Tab.10 with the comparison from FEA. This output system is not only classifying the failure status of the soil-cantilever sheet wall structure, but also predicting the variables such as Δ and y(x) continuously. For simulation number 1st, 2nd, 3rd, 4th, 5th, 9th, 10th, outputs are positive real values, and imply a not-fail status of these samples. Meanwhile, simulations number 6th, 7th, and 8th are considered as failure. Among those failure cases, the 6th and 7th simulations are failed status due to the collapse of soil, and the 8th simulation fails by the exceedance of the condition of Δ ≤ 200 mm. Even though the y(x) of the 8th simulation is available, it is not provided in the input as a warning of large displacement.
Tab.10 Example of output system with proposed framework (Fig.3) |
| No. | stable | Δ (FEA) (mm) | x* (m) | y(x) (FEA) (mm) | not failure? | output 1 (Δ) (mm) | output 2 (y(x)) (mm) |
|---|---|---|---|---|---|---|---|
| 1 | 1 | 33.610 | 2.5 | 12.700 | 1 | 30.939 | 13.200 |
| 2 | 1 | 30.120 | 3.5 | 10.100 | 1 | 29.084 | 98.800 |
| 3 | 1 | 28.690 | 4 | 9.100 | 1 | 27.399 | 9.900 |
| 4 | 1 | 28.140 | 0.5 | 8.700 | 1 | 26.209 | 8.300 |
| 5 | 1 | 27.970 | 10 | 4.400 | 1 | 27.932 | 4.700 |
| 6 | 0 | – | – | – | 0 | –1 | –1 |
| 7 | 0 | – | – | – | 0 | –1 | –1 |
| 8 | 1 | 211.680 | 2.5 | 22.840 | 0 | –1 | –1 |
| 9 | 1 | 164.330 | 6.5 | 29.500 | 1 | 149.577 | 28.900 |
| 10 | 1 | 155.180 | 1.5 | 107.100 | 1 | 130.018 | 112.900 |
*Note: x location is chosen based on interest. A list of discontinuous x within a range can provide the profile of the settlement on the ground surface. |
4 Conclusions
In this study, a hierarchical system composed from data-driven models is proposed to predict the behaviour of the soil–cantilever sheet wall structure. In this framework, a classification is conducted to predict the status of soil–sheet wall structure with a trained RF model, RFC. Once the non-failure condition is satisfied, predictions on the variables of interest behaviour of sheet wall and soil, are conducted for the horizontal displacement at the top of the sheet wall Δ with RANN1 model, and the vertical displacement of the surface of the adjacent ground y(x) with RANN2 model.
The use of HS model in Plaxis has been compared with other experiment data before being used as the data generating tool to obtain 200 FEA simulations, and to develop the data-driven models. The classification model has been developed with RF method. The random sampling for the train/test splitting the database has been conducted with 1000 sub-training and 1000 sub-testing sets to develop 1000 RFC models. The distributions of the evaluation metrics of RFC models are observed. To be specific, the medians for accuracy/precision/recall are 1.000/1.000/0.9286; those of means are 0.9564/0.9775/0.9338; and those of standard deviation are 0.05674/0.05680/0.0729, respectively. The negative skewed data toward the absolute value of 1.0000 provides insightful observation of the effect of randomness of the database to the performance of developed models. A RFC model was chosen from the best models of this set of 1000 RFC models, having an accuracy of 1.0 on both train set and test set. With the developed database, RANN1 and RANN2, are the ANN models developed for predicting continuous variables, the Δ, and y(x), respectively. Evaluation metrics of these models on test set are appropriated with low errors and high R2 (0.9521 for RANN1 and 0.9988 for RANN2).
The analyses on importance level of the inputs for RFC model and parametric studies for ANN models are conducted. The critical inputs for each model are not analogous. Length of the sheet wall, L, Eoed and Hw, are the most critical inputs for RFC model. Meanwhile, the set of [L and H] and the set of [x, L, H, and ] are the most important inputs for RANN1 and RANN2, respectively.
To the point, a hierarchical system is proposed and the illustration on how such a system is applied in the engineering problem is provided. It is worth noting that various additions or adjustments may be applied for this system. For example, the check of the Ultimate limit state (model collapse), and the Serviceability limit state (Δ ≤ 200 mm), can be separated by 2 single models. Other possibilities can be the addition of other interested variables, e.g., maximum moment in the sheet wall. These variations all require an expansion on the database and thus can be improved in future work.