1 Introduction

Generally, high rise structures like mooring systems for surface, transmission towers, or even high rise/tall buildings are subjected to large lateral loads, huge overturning moments due to waves, wind, or even both, in addition to vertical compressive load. Therefore, the usual bored piles (normally constructed from reinforced concrete) supporting these structures have to struggle with large uplift and lateral moments and loads. In most cases, these moments and loads (lateral, compressive, and uplift) act on the single piles simultaneously [1–3]. There are many theoretical methods available to assess pile behavior under different independent, lateral, vertical uplift, and vertical compressive loadings [4–6]. Under-reamed and/or enlarged base piles are normally constructed from concrete, with a mechanically formed under-reamed (also called belled pile) designed to increase bearing capacity of the pile. Typically, pile base geometry is in the form of an inverted cone. It can be used to provide uplift capacity by forcing a passive failure wedge of soil (normally because of weight of the soil placed above under-reamed pile) to mobilize for failure to occur and increase the compressive base capacity (end bearing capacity) [7].

Numerous field and laboratory works were performed to study the behavior of uplift loaded base pier. However, there are relatively few papers in the technical literature which deal with an under-reamed pier or belled pier resisting uplift load in sand [8–11]. For analyzing purposes, the limiting capacity of under-reamed pier embedded in sand can be carried out based on theoretical and semi empirical work [12, 13]. Full scale tests and laboratory tests on small models were performed in order to obtain appropriate solution [14, 15]. Furthermore, sometimes, a full-scale test was employed to verify the results based on laboratory tests [16]. Sego et al. [9] investigated on end bearing and total capacities of a pile after enlarging base of the pile. The pile was located on ice-rich permafrost. The results clearly proved that total capacity can be substantially improved through the use of belled piles. Honda et al. [17] applied a two-dimensional distinct element (DE) analysis to evaluate the uplift resistance of belled and multi-belled piles embedded in dense sand. As a result and based on the soil movements in DE analysis, a theoretical solution for predicting the uplift resistance of belled and multi-belled piles was generated from an upper bound limit analysis [17]. Chae et al. [18] investigated on the uplift capacity of belled pile in weathered sandstones in a coastal development on the Persian Gulf. They have conducted a series of full-scale pullout load test on belled tension piles in Abu Dhabi, the United Arab Emirates. After comparison of results from field, 3D finite element analysis, and the theoretical methods, it is found that common theoretical methods overestimate ultimate pull out resistance of the belled pile regardless of bell-shape considerations [18]. Yao and Chen [19] established an elastic-plastic analytical solutions for the uplift belled pile. Through the comparison between obtained results from generated solutions and results of theoretical calculation, it was found that elastic-plastic analytical solutions revealed a good approach to the field test results. Lin et al. [20] introduced a new method for estimating the ultimate uplift resistance of under-reamed pile embedded in sandy soil. The differences of new method with conventional methods is that the elastic/plastic behavior of soil around belled pile is being taken into account. Lin et al. [20] performed a series of experimental/field tests to gain development of the failure surface with curved slip-surface. Results illustrated that the starting location of failure surface is developed at a distance away from the pile surface. In several cases, researchers used extra reinforcement materials such as various types of geosynthetics to enhance the rate of uplift bearing capacity [21–23]. For instance, Tafreshi et al. [24] investigated on the enhancement of geocell reinforcement on uplift resistance of belled piles and also in the performance improvement of the uplift bearing capacity of belled pile under repeated tensile and compressive loads. It is observed that the geocell material allowed higher tensile loads or increased cycling. In addition, Tafreshi et al. [24] noted that required length of pile shaft can be reduced using the geocell reinforcement causes to reduce required pile’s material, backfill, and excavation.

There are also many studies on using artificial neural network (ANN) in order to predict bearing capacity of pile foundation [25–30], pile dynamic capacity [31, 32], pile setup [33, 34], and pile settlements [35, 36]. For example, Suman et al. [25] used artificial intelligence techniques to predict friction resistance of driven piles in clay using. The prediction models were presented using two recently developed artificial intelligence techniques, functional networks (FN) and multivariate adaptive regression splines (MARS). The models predict friction resistance of piles in clay soils based on experimental test results. Based on statistical performances, MARS and FN models are found to have a better predictive resistance than existing models. Rahman et al. [37] developed an ANN model to predict the uplift capacity of suction foundations using a database containing the results from a number of model and centrifuge tests. Ardalan et al. [38] optimized a group of data handling (GMDH) type neural networks using genetic algorithms (Gas). The model have been used to explore effects of effective cone point resistance and cone sleeve friction as input parameters on pile unit shaft resistance. Alavi et al. [39] utilized a classical tree-based genetic programming (TGP), linear genetic programming (LGP), and gene expression programming (GEP) to develop and formulate new prediction equations for the uplift capacity of suction caissons using an experimental database obtained from published literature. Cheng et al. [40] developed an intelligent fuzzy radial basis function (RBF) neural network inference model (IFRIM), which is a hybrid of RBF neural network, fuzzy logic (FL), and artificial bee colony (ABC) algorithm and has been used to predict the uplift capacity of suction caissons.

To ensure serviceability and stability of the structures, bearing capacities against uplift and horizontal forces are taking into account in the design process of foundations. Unlike the lateral and compression resistance, the uplift capacities of piles with under-reamed has not yet been fully investigated; also, such piles are known to be very effective against uplift force, but research on their uplift behavior has been limited. In the present study, an attempt has been made to develop an optimized ANN-based predictive approach model to calculate the ultimate uplift capacity of under-reamed piles embedded in dry cohesionless soil. In order to develop the new design model, dataset consist of an extensive number of small scale laboratory tests on uplift capacity of under-reamed pile published literature which have been used. We have proposed a novel ANN model to model and predict ultimate uplift capacity of the under-reamed piles embedded in dry cohesionless soil. This model is introduced by us for the first time. This model is an accurate and very fast tool in comparison with the experimental or numerical methods. Also, using the proposed ANN model, an equation is proposed for the first time to relate inputs including base diameter, angle of enlarged base, shaft diameter and embedment ratio, and the output ultimate uplift resistance.

2 Data collection and methodology

Used data to make the ANN model in this study was obtained from the small scale laboratory work conducted in a geotechnical laboratory, performed by Nazir et al. [13]. The purpose of research carried out by Nazir et al. [12] was to determine the bearing capability of an enlarged base pier against uplift forces. Experiments were performed in a large plexiglass box. A total number of 432 laboratory experiments were undertaken in both dense and loose sand condition. Enlarged base piers with base angles of α = 30°, α = 45°, and α = 60°; base diameters between 75 and 150 mm; and pile diameter ranging from 30 to 50 mm were tested. An example of under-reamed base pile and key parameters considered for ANN model is depicted in Fig. 1 while Table 1 illustrated similar design parameters used in the small scale experimental plan as well as proposed ANN approach.

Fig. 1
figure 1

An example of under-reamed pile. a Key parameters. b Belling bucket

Full size image
Table 1 Physical quantities for the uplift resistance used in both ANN model and laboratory work
Full size table

3 Artificial neural network (ANN)

In the current paper, we optimized an ANN method to model and predict the ultimate uplift capacity of the under-reamed piles embedded in dense and loose dry sand. The new generated ANN model was developed by taking into account the key factors such as base diameter Db, angle of enlarged base (α), shaft diameter (Ds), and embedment ratio (L/Db) as the inputs and ultimate uplift resistance (Pul) as the output. For this means, a special type of feed forward network called the multi-layer perceptron (MLP) neural network is used, which is the most widely used ANN. Figure 2a shows the proposed MLP network used in this study. This structure consists of three layers as the input layer, the hidden layer, and the output layer. Each layer basically consists of interconnected neurons (basic processing elements of ANN), in which the synapses of biological neurons are modeled as the weights. The mathematic model of a neuron is shown in Fig. 2b, where U1 , U2 ,  .  .  .  .  , Ut are inputs, b is the bias term, f is the transfer function, and W1 , 1 , W1 , 2 ,  .  .  .  .  , W1 , t are the connection weights. The output of this neuron is also given by the following equation [41, 42]:

$$ O= f\left(\sum_{i=1}^t\left({U}_i\;{W}_{1, i}\right)+ b\right) $$
(1)
Fig. 2
figure 2

a Proposed ANN model. b Mathematic model of a neuron

Full size image

The neurons in input layer transfer inputs to the next layer without any changes. Also, in output layer, the transfer function f is equal to 1. For the hidden layers, the transfer function f is usually tansig function, which is given by the following equation for variable x:

$$ \mathrm{Tansig}(x)=\frac{2}{1+{e}^{-2 x}}-1 $$
(2)

The connection weights and bias can be adjusted during the training process using back-propagation training algorithm, which is an error-minimization technique.

In order to find the optimized ANN model, many ANN structures (i.e., using MLP) were tested using 1, 2, and 3 hidden layers. Different number of neurons were considered in each hidden layer. This process continued to obtain an accurate ANN model with the minimum error in both training and testing data set. Besides, the optimized ANN model should have a simple and less complicated structure. After testing large number of ANN structures, the best ANN model structure is found with one hidden layer and 10 neurons in its hidden layer. Noteworthy, developed model possesses 5 neurons in input layer, while 10 neurons in hidden layer, and 1 neurons in output layer. To train and test the proposed ANN model, required data set was obtained from the small scale laboratory tests. These data were divided into two sets, i.e., about 70% for training MLP models and the rest (about 30%) considered for testing the trained network. MATLAB 7.0.4 software was used for this process. Therefore, the most appropriate proposed ANN model is shown in Fig. 2a.

4 Results and discussion

As stated earlier, in training process algorithm, many different MLP structure were tested and trained by changing number of epochs, number of hidden layers, number of neurons in each hidden layer, and the training algorithm. Results from the predicted and measured data for both training and testing dataset are compared. At the end of this process, the most suitable ANN model was selected with respect to the minimum error calculated from three common error evaluation methods: (i) root mean square error (RMSE), (ii) mean absolute error (MAE), and (iii) correlation factor (CF). In addition, the connection weights and bias were thoroughly adjusted. Therefore, by using Eqs. (1) and (2), we can obtain an equation between base diameter Db, angle of enlarged base (α), shaft diameter (Ds), and embedment ratio (L/Db) as inputs and ultimate uplift resistance (Pul) as output. The following terms are shown in Eq. (3); DS, alpha (α), DT, ER, Db, and Pul stand for the shaft diameter (mm), angle of enlarged base (degree), density condition (dense and/or loose), embedment ratio (L/Db), base diameter (Db), and ultimate uplift resistance (kN), respectively. Noteworthy, the term DT = 10 is used to represent for loose sand where DT = 20 is used to introduce dense sand. Also, “tansig” is the tansig function and “abs” is a MATLAB function that returns the absolute value. Following Eq. (3), it can be directly run in MATLAB software.

$$ \begin{array}{l}{D}_{\mathrm{s}}=50;\hfill \\ {}\mathrm{Alpha}=60;\hfill \\ {}\mathrm{DT}=20;\hfill \\ {}\mathrm{ER}=5;\hfill \\ {}{D}_{\mathrm{b}}=150;\hfill \end{array} $$
(3)

Pul = abs(−1.3419*tansig(0.012753*Ds − 1.4692*Alpha − 0.12898*DT + 1.2804*ER + 0.65553*Db + 10.8701) − 0.010131*tansig(0.27276*Ds − 0.0022046*Alpha + 0.4184*DT + 1.1763*ER − 0.16275*Db − 5.9077) + 0.00083048*tansig(1.7705*Ds − 1.194*Alpha + 2.0979*DT − 0.17825*ER − 0.57311*Db + 2.3419) − 1.9967*tansig(−0.00096448*Ds + 0.00073819*Alpha − 0.45228*DT − 0.47458*ER + 0.037422*Db − 0.13683) + 2.7963*tansig(−0.0030848*Ds − 0.0017212*Alpha − 0.071306*DT + 0.34076*ER + 0.011644*Db − 2.1228) − 1.3461*tansig(0.002462*Ds + 1.5202*Alpha − 0.26247*DT + 3.2369*ER − 0.98471*Db + 15.6301) − 1.8572*tansig(0.00080372*Ds + 0.0018985*Alpha − 0.20589*DT − 0.67674*ER + 0.035409*Db + 1.6114) − 1.8427*tansig(−0.0033276*Ds − 0.0016549*Alpha + 0.49151*DT + 0.54666*ER + 0.025507*Db − 10.5262) − 1.8289*tansig(−0.00047678*Ds − 0.0018453*Alpha − 0.20311*DT + 0.62268*ER − 0.038398*Db + 6.8981) − 0.46146*tansig(0.0041671*Ds + 0.0040608*Alpha + 0.18664*DT − 2.7405*ER − 0.11086*Db + 3.3236) + 2.0822)

4.1 Discussion on ANN data

Figure 3 shows the training and testing results of proposed ANN model in comparison with the experimental data. In Fig. 3, the blue line is a reference line to show the equal relation between the measured and predicted (ANN) results and the square markers reveal exact relation between them. The absolute error (AE) is given by the following equation:

$$ \mathrm{AE}=\left|{X}_i\left(\mathrm{Exp}\right)-{X}_i\left(\mathrm{Pred}\right)\right| $$
(4)
Fig. 3
figure 3

Training and testing results of the proposed ANN model

Full size image

where “X(Exp)” and “X(Pred)” stand for the experimental and predicted values, respectively. Also, RMSE, MAE, and CF are given by:

$$ \mathrm{RMSE}={\left[\frac{\sum_{i=1}^N{\left({X}_i\left(\mathrm{Exp}\right)-{X}_i\left(\mathrm{Pred}\right)\right)}^2}{N}\right]}^{0.5} $$
(5)
$$ \mathrm{MAE}=\frac{1}{N}\sum_{i=1}^Z\left|{X}_i\left(\mathrm{Exp}\right)-{X}_i\left( \Pr \kern0.5em \mathrm{ed}\right)\right| $$
(6)
$$ \mathrm{CF}=1-\left[\frac{\sum_{i=1}^N{\left({X}_i\left(\mathrm{Exp}\right)-{X}_i\left(\mathrm{Pred}\right)\right)}^2}{\sum_{i=1}^N{\left({X}_i\left(\mathrm{Exp}\right)\right)}^2}\right] $$
(7)

where N is the number of data. For the proposed ANN model, the obtained minimum and maximum AE for training data are 0.00003 and 0.2037, respectively. Also, for the testing data, the obtained minimum and maximum AE are 0.00059 and 0.262, respectively. From these results, it is clear that the proposed ANN model results are close to the experimental with a minimum error.

4.2 Discussion on measured data

In this section, results of uplift load capacity for a single under-reamed pile embedded in dense and loose sand are presented. The measured uplift resistance (Pul) from the laboratory tests (also called LAB) along with the predicted values from the ANN approach are tabulated in Tables 2 and 3, respectively. The dataset is provided for various shaft diameter (Ds), enlarged base diameter (Db), and L/Db in dense and loose sand are. From these two tables, it can be found that the ultimate upload capacity of the pile (Pul) increases as the L/Db and Db increases. These two terms are dominant factor on the uplift capacity of the under-reamed pile. Moreover, results reveal that the embedment ratio of the under-reamed pile is almost the most significant parameters. For instance, for the ANN model with Db = 150 mm and Ds = 30 mm, the uplift load capacity of L/Db = 1, 2, 3, 4, and 5, in dense and loose sand were 0.15, 0.67, 1.71, 3.26, and 4.32, and 0.10, 0.37, 0.83, 1.58, and 2.69 kN, respectively. The under-reamed piles are very effective way to improve the pullout forces. The presented data in Tables 2 and 3 is a good evidence to prove that the new ANN approach method shows very good agreement with measured data.

Table 2 Measured and predicted (ANN) ultimate uplift resistance, Pul (kN), for various Ds, Db, and L/Db in dense sand
Full size table
Table 3 Measured and predicted (ANN) ultimate uplift resistance, Pul (kN), for various Ds, Db, and L/Db in loose sand
Full size table

4.3 Comparison of ANN model performance

This section illustrates evaluation of ANN performance of the developed model for calculation of the uplift capacity (Pul). Figures 4 and 5 present basis of comparison where results from ANN were directly compared and related to the measured data. The ultimate uplift force (Pul) for measured and predicted model, based on the change in various key parameters such as shaft diameter (Ds), under-reamed base diameter (Db), and angle (α = 30°, α = 45°, and α = 60°) along with different L/Db ratio, are shown and compared for both of the dense and loose sand. Hence, it is first a comparison between LAB and ANN outputs for Db equal to 75, 100, 125, and 150 mm in dense and loose sand, respectively. A simple regression on the predicted and measured results for both loose and dense sand presents R-square of almost 1.0. This reveals the accuracy of the provided ANN model. The presented model is reliable and simple where excellent agreement with the measured output in which proved that this method can be used as reliable and flexible tools to predict the ultimate uplift capacity (Pul) of under-reamed base piles.

Fig. 4
figure 4

Comparison between measured and predicted results for various Db in dense sand. aDb = 75 mm. bDb = 100 mm. cDb = 125 mm. dDb = 150 mm

Full size image
Fig. 5
figure 5

Comparison between measured and predicted results for various Db in loose sand. aDb = 75 mm. bDb = 100 mm. cDb = 125 mm. d Db = 150 mm

Full size image

5 Conclusions

In this paper, the effects of shaft diameter, base diameter, angle of enlarged base, and embedment ratio on performance of the under-reamed single concrete piles were investigated using computational intelligence. For this purpose, a particular ANN (MLP) structure was developed. The main objective was to estimate the ultimate uplift capacity through the above input parameters with an optimized ANN model. Based on the obtained results, proposed ANN model was able to accurately predict the output with least error. The maximum AE for the training and testing data were obtained 0.2037 and 3.5000, respectively. The findings from this study prove that generated ANN model can be used as flexible and reliable solution due to their fast speed and high accuracy; therefore, it can be applied to predict the experiments precisely.