This study investigates predicting the pullout capacity of small ground anchors using nonlinear computing techniques. The inputoutput prediction model for the nonlinear HammersteinWiener (NHW) and delay inputs for the adaptive neurofuzzy inference system (DANFIS) are developed and utilized to predict the pullout capacity. The results of the developed models are compared with previous studies that used artificial neural networks and least square support vector machine techniques for the same case study. The in situ data collection and statistical performances are used to evaluate the models performance. Results show that the developed models enhance the precision of predicting the pullout capacity when compared with previous studies. Also, the DANFIS model performance is proven to be better than other models used to detect the pullout capacity of ground anchors.
Light structures, which are built in open areas, are supported with the ground using small anchors. Such anchors are designed to resist tensile and uplift forces [
The numerical prediction models are used to detect the pullout capacity of small ground anchors based on inputoutput mapping for the in situ data. Shahin and Jaksa [
Nowadays, integrated system identifications are used to design nonlinear inputoutput prediction models [
The objectives of this study are the following: (
The MISO prediction models, NHM and DANFIS, are utilized in this study to extract the pullout capacity of small ground anchors. These models are described in the following subsections.
The NHW model is an integrated prediction model using nonlinear and linear transforming functions [
MISONHW model diagram structure.
In this study, four input variables are used to predict the pullout capacity of a MISO model. The trail and errors method is used to select the input and output nonlinearity functions. Therefore, the nonlinearity input function is applied to each input variable
The linear output block
In this paper, the prediction trials were performed with the Matlab command
The time delayed adaptive neurofuzzy inference system (DANFIS) is proposed in [
MISODANFIS model diagram structure.
The process of the ANFIS model can be found in [
Assuming first that
Rule
The second is normalizing the firing strength, as follows:
Based on (
To evaluate the developed models, the field data of 119 anchors are derived using an in situ test database from Shahin and Jaksa [
Database for field pullout tests (from Samui et al. [
The data are divided into training and testing subsets as presented in [
Statistical measurements for the training and testing datasets.
Statistical parameters 




IT 


Training dataset  
Max.  44.60  800.00  3.55  179.71  2.00  3.47 
Min.  25.00  400.00  0.95  12.22  1.00  0.29 

31.66  571.08  1.91  58.01  1.60  1.73 
SD  7.88  125.46  0.58  42.18  0.49  0.77 


Testing dataset  
Max.  44.60  800.00  3.03  178.26  2.00  3.80 
Min.  25.00  400.00  0.95  12.22  1.00  0.35 

28.85  594.44  1.98  56.62  1.56  1.80 
SD  7.00  101.26  0.55  36.70  0.50  0.78 
From Table
The data sensitivity is studied based on the previous models designed with the same database [
In this study, three criteria are used to evaluate the performance of the models design. The first criterion is the correlation coefficient
The scaled data are used in this section to evaluate the variables sensitivity. The correlations between the input and output variables are presented in Table
Correlation coefficient between input and output variables.
Variables 




IT 


0.15  0.44  −0.11  0.54  −0.26 
From Table
The simple regression model, as presented in (
Linear trend component and coefficient test for the regression models.
Model 






 

1 

5.5  2.5  7.6  −3.8  9.8  −2.8  0.77 
2 

2.4  2.6  6.2  —  7.5  —  0.70 
3 

4.9  2.6  8.3  −5.2  9.2  —  0.77 
4 

4.1  2.6  6.1  —  9.1  −4.4  0.74 
As a result of the models correlation and
Shahin and Jaksa [
Correlation coefficient (
From Figure
To assess the developed models, the models are programmed on Matlab. In the training phase, 83 datasets are selected and the coefficients of the models have been chosen by trial and error. In the NHW model, the same nonlinear functions for the inputs and output are used. The inputoutput nonlinear sigmoid functions and wavelet networks, saturation, onedimension polynomial, and piecewise functions are applied with 50 iterations. In addition, the order chosen of linear function (
Linear function order trails evaluation.






0.70 


0.95 


0.99 


0.90 
The presented values in Table
On the other hand, the DANFIS model is designed using the four input variables and onetimedelayed output; and the pullout capacity is the output value. Two MF functions for each variable are used in this case with 92 nodes and 62 model coefficients. Different MF types are evaluated with 50 iterations, and the best predicted pullout capacity (
DANFIS model design: (a) model application, (b) typical model architecture with five inputs, and (c) adjusted MF for the five inputs variables.
In this model, 32 fuzzy rules are used, and the numbers of linear and nonlinear coefficients are 32 and 30, respectively. The application of the model is presented in Figure
The performances of the designed NHW and DANFIS models are presented in Figure
Comparison between the developed models and the LSSVM [
Model  RMSE (KN)  MAE (KN) 


LSSVM [ 
0.22  0.19  0.94 
NHW 


0.99 
DANFIS 


0.99 
Training performance of the designed models (a) NHW and (b) DANFIS.
The observed and the predicted values of the pullout capacity by the NHW and the DANFIS models are shown in Figure
Comparison between designed models and LSSVM [
Model  RMSE (KN)  MAE (KN) 


LSSVM [ 
0.26  0.20  0.94 
NHW 


0.98 
DANFIS 


0.99 
Testing performance of the designed models (a) NHW and (b) DANFIS.
Finally, the models proposed, DANFIS and NHW, can be used to detect the pullout capacity with high accuracy with the DANFIS performing better than the NHW.
In this study, two models are developed using nonlinear integrated system, which are nonlinear HammersteinWiener (NHW) and delay inputs for the adaptive neurofuzzy inference system (DANFIS) to predict the pullout capacity of small ground anchors. The input variables sensitivity is studied to evaluate the variables effectiveness in prediction using polynomial regression model. The sensitivity analysis shows high effect of the equivalent anchor diameter, embedment depth, average cone resistance along the embedment depth, and average sleeve friction along the embedment depth variables in predicting the pullout capacity. The results of the developed models are evaluated using case study data and compared with previous studies. It is concluded that the two proposed models can be used to predict the pullout capacity with high accuracy. Moreover, the performance of the DANFIS outperforms the NHW model in training and testing dataset.
The authors declare that there are no conflicts of interest regarding the publication of this paper
This research was supported by PostDoctor Research Program in 2017 through the Incheon National University (INU), Incheon, South Korea.