# Stiffness and Strength of Stabilized Organic Soils—Part II/II: Parametric Analysis and Modeling with Machine Learning

^{1}

^{2}

^{3}

^{4}

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## Abstract

**:**

## 1. Introduction

_{2}).

## 2. Background

## 3. Modeling

#### 3.1. Experimental Database

_{2}) incubators at three different temperatures (21 °C, 45 °C, and 60 °C) and three different relative humidity (70%, 80%, and 90%). Samples were cured to varying ages of 14, 28, 60, 90, 105, and 120 days before they were tested. Table 3 presents the control and response variables (the strength and stiffness measured from the tests) and their corresponding range of variation in the database. Figure 1 provides a graphical representation of the experimental design.

#### 3.2. Artificial Neural Networks

^{2}, the coefficient of determination between measured (actual) and predicted values, and the root mean square of error (RMSE) obtained from the discrepancy between measured and predicted values. To avoid overfitting, a fraction of the database was used to train the ANN models (training subset), and the rest was used to monitor and evaluate the generalization ability of the model (validation and test subsets).

#### 3.2.1. Radial Basis Functions

**X**is the input vector, and

**C**is the vector determining the center of the basis functions. σ is the parameter that specifies the spread of the basis functions and controls interpolation’s smoothness.

**Y**= output matrix (model response);

**W**= weight matrix (model parameters);

**X**= input matrix (predictor variables);

**C**= centers of Gaussian functions;

^{2}(R

^{2}

_{ave}) and RMSE (RMSE

_{ave}).

#### 3.2.2. Multilayer Perceptron

**Y**= output matrix (model response).

**W**= weight matrix of the hidden layer (model parameters).

_{1}**W**= weight matrix of the output layer (model parameters).

_{2}**X**= input matrix (predictor variables).

**b**= bias vector of the hidden layer (hyperparameters).

_{1}**b**= bias vector of the output layer (hyperparameters).

_{2}#### 3.2.3. Multiple Linear Regression (MLR)

**Y**= output matrix (model response).

_{0}, …, β

_{n}= regression coefficients (model parameters).

**X**= input matrix (predictor variables).

## 4. Analysis and Results

#### 4.1. RBF Network Analysis

^{2}ave over the test datasets is close to 0.8 for the prediction of E and above 0.9 for the prediction of UCS. Additionally, the RMSE is 13 MPa for E and 0.11 MPa for UCS. These results show that RBF models successfully captured the correlations between the control and the response variables within the training dataset and generalized the captured trends to the test dataset with acceptable accuracy. RBF models show better performance for the prediction of UCS compared to E.

^{2}values corresponding to the networks of the ensemble (RBF-Tot). As expected, the uncertainty of predictions over the training dataset is significantly lower than the test dataset. Additionally, the uncertainty of predictions for E is substantially higher than for UCS.

#### 4.2. MLP Network Analysis

_{0}= Number of input variables.

^{2}with respect to the number of hidden neurons. The optimum number of hidden neurons is 20 for predicting E and 25 for predicting UCS.

^{2}over the test dataset is 0.8 for UCS and 0.61 for E. The average RMSE is 18 MPa for E and 0.17 MPa for UCS, showing a reasonably good level of prediction accuracy (especially for UCS). Figure 6 presents the distribution of the R

^{2}associated with the networks of the MLP ensembles. As expected, and consistent with the RBF models’ results, the uncertainty of predictions over the training datasets was much lower compared to the other subsets. Additionally, the uncertainty of predictions for UCS was less than E.

#### 4.3. Stepwise Parameter Selection for ANNs

^{2}for the model becomes larger than the adjusted R

^{2}of the reduced model. In the backward elimination, the process starts with a network containing all input parameters, and the parameters are gradually removed one by one. A parameter is removed when the adjusted R

^{2}for the reduced model becomes larger than the present model.

^{2}provides the possibility of penalizing the model for the number of input parameters and is calculated based on the following equation:

^{2}

_{adj}= adjusted value.

^{2}= R

^{2}for the reduced model.

^{2}

_{adj}of the reduced model started to decrease. After this stage, the parameter that resulted in the smallest decrease in R

^{2}

_{adj}was added to the model at each step. This continued until all the parameters were added to the model.

#### 4.4. Linear Regression Analysis

^{2}ave of 0.42 for E and 0.62 for UCS over the test datasets.

#### 4.5. Model Comparison

^{2}> 0.95. The MLR model shows a significantly lower R

^{2}, 0.64 for prediction of E, and 0.73 for UCS.

#### 4.6. Sensitivity Analysis

- UCS shows a slight increase with w, reaching maximum at a range between 800% to 1000%, opposing the generally expected trend. Such a trend has occurred primarily in the field when the initial water content is very low, also when adding large quantities of dry binders that prevent proper mixing and hydration [81].
- Increasing G/S increases both E and UCS up to a certain value. In this analysis, the optimum range for G/S is between 2 and 2.5.
- The optimum range for W/B is between 0.6 to 0.7, and larger values of W/B cause a considerable decrease in E and UCS.
- D/H does not show a significant impact on UCS; however, E shows to increase with D/H slightly.
- Age of the specimen is positively correlated with E and UCS. E and UCS rapidly increase during the first 90 days (a time when much of the hydration process developed). After that, the rate of increase decays.
- Both E and UCS decay with an increase in curing temperature (T), opposing the behavior of common stabilized mineral soils, where the temperature increases the strength and stiffness. The negative effect of temperature on the stiffness and strength of organic soils is related to several factors, such as gradual loss of the initial evaporable water in the mix, dehydration during chemical reactions, and porosity changes [20].
- RH = 0.8 is shown to be the optimum value for both E and UCS, although the overall trend suggests that relative humidity is not a significant parameter.
- CO
_{2}shows a minimal effect on E and UCS, with a slight developing trend observed for UCS.

## 5. Conclusions

^{2}> 0.8 for E and R

^{2}> 0.9 for UCS). The most relevant input parameters for ANN models were identified through the stepwise parameter selection method, and the redundant parameters were eliminated.

- Part I and II of this study together provide comprehensive details and descriptions of both experimental and computational investigations. Unlike most of the other studies, the experiments were designed to generate a well-populated database suitable for application of ML. This is essential for developing robust, reliable predictive models on any experimental database. Full access to the database and descriptive statistics are provided in Part I.
- The mechanical behavior of stabilized organic soils has not been comprehensively addressed by other studies. In this study, using a hybrid experimental and ML approach, the stiffness and strength as the two critical engineering design parameters were investigated, and the impacts of the relevant factors on the stiffness and strength of stabilized organic soils were evaluated.
- This study investigated various types of soils (low and high plasticity clays) and binders (both cement and non-cement based).
- Using a novel ML approach, the most influential parameters (control variables) were identified, the trends of strength and stiffness variation with these parameters (with 50% CI) were developed, and the optimum ranges were identified, allowing for an optimal mixture design.
- The two most prominent ANN algorithms were successfully applied to predict the stiffness and strength, and the full details of the architecture development and training methods were provided. Comparing their performance with other ML methods recently applied in other studies showed that these ANN algorithms are still highly competent.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 8.**Response graphs showing the trends of E and UCS variations with the control variables, including 50% CI: (

**a**) OC, (

**b**) w, (

**c**) QB, (

**d**) G/S, (

**e**) W/B, (

**f**) D/H, (

**g**) t, (

**h**) T, (

**i**) RH, (

**j**) CO

_{2}.

**Figure 9.**Response graphs for the control variables (input parameters) with respect to the scale of variation.

Organic Soil | Density (kg/m^{3}) | OC (%) | w (%) |
---|---|---|---|

Irish Moss Peat (Pt) | 294 | 94 | 210 |

446 | 500 | ||

1014 | 1000 | ||

Medium Organic Clay (OH-1) | 1219 | 30 | 180 |

Low Organic Clay (OH-2) | 1471 | 4 | 85 |

Binders | Binder Mixtures | Ratio |
---|---|---|

Portland Cement (PC) | PC | |

Blast Furnace Slag (BFS) | PC + BFS | 1:2 |

Pulverized Fuel Ash (PFA) | PC + PFA | 1:1 |

Lime (L) | PC + PFA + L | 3:6:1 |

Magnesium Oxide Cement (MgO-C) | PC + PFA + MgO-C | 2:6:2 |

Gypsum (G) | L + G + BFS | 1:1:1 |

No. | Variables | Range of Variation | Mean | Standard Deviation |
---|---|---|---|---|

Control Variables (Input Parameters) | ||||

1 | Organic Content of Soil (OC) | 4 (OH-2), 30 (OH-1), 94 (Pt) (%) | 69.2 (%) | 33.2 (%) |

2 | Water Content of Soil (w) | 85, 180, 210, 500, 1000 (%) | 400.2 (%) | 258.6 (%) |

3 | Ratio of Binder for Portland Cement (RB-PC) | 0, 0.2, 0.3, 0.33, 0.5, 1 | 0.749 | 0.339 |

4 | Ratio of Binder for Blast Furnace Slag (RB-BFS) | 0, 0.33, 0.67 | 0.180 | 0.292 |

5 | Ratio of Binder for Pulverized Fuel Ash (RB-PFA) | 0, 0.5, 0.6 | 0.045 | 0.154 |

6 | Ratio of Binder for Lime (RB-L) | 0, 0.1, 0.33 | 0.012 | 0.056 |

7 | Ratio of Binder for Magnesium Oxide (RB-MgO-C) | 0, 0.2 | 0.005 | 0.032 |

8 | Ratio of Binder for Gypsum (RB-G) | 0, 0.33 | 0.009 | 0.054 |

9 | Quantity of Binder (QB) | 100–500 (kg/m^{3}) | 265.9 (kg/m^{3}) | 76.6 (kg/m^{3}) |

10 | Grout to Soil Ratio (G/S) | 0.14–3.38 | 0.856 | 0.568 |

11 | Water to Binder ratio (W/B) | 0.5, 0.8, 1 | 0.937 | 0.157 |

12 | Specimen Diameter/Height (D/H) | 0.5, 1 | 0.920 | 0.183 |

13 | Time (t) | 14–180 (days) | 61 (days) | 36 (days) |

14 | Temperature (T) | 21, 45, 60 (°C) | 32.9 (°C) | 15.7 (°C) |

15 | Relative Humidity (RH) | 70, 80, 90 (%) | 87.6 (%) | 4.9 (%) |

16 | Carbonation (CO_{2}) | 0, 20 (%) | 2.9 (%) | 7.1 (%) |

Response Variables (Target Parameters) | ||||

Unconfined Tangent Modulus (E) | 0.83–214.62 (MPa) | 34.03 (MPa) | 31.44 (MPa) | |

Unconfined Compression Strength (UCS) | 0.04–2.09 (MPa) | 0.50 (MPa) | 0.40 (MPa) |

Model | All | Training | Test | |||
---|---|---|---|---|---|---|

R^{2}_{ave} | RMSE_{ave} | R^{2}_{ave} | RMSE_{ave} | R^{2}_{ave} | RMSE_{ave} | |

RBF-Tot-E | 0.89 | 2.78 | 0.92 | 0.03 | 0.77 | 13.77 |

RBF-Tot-UCS | 0.95 | 0.09 | 0.96 | 0.08 | 0.92 | 0.11 |

Model | All | Training | Validation | Test | ||||
---|---|---|---|---|---|---|---|---|

R^{2}_{ave} | RMSE_{ave} | R^{2}_{ave} | RMSE_{ave} | R^{2}_{ave} | RMSE_{ave} | R^{2}_{ave} | RMSE_{ave} | |

MLP-20-E | 0.81 | 13.58 | 0.95 | 6.15 | 0.50 | 20.74 | 0.61 | 18.47 |

MLP-25-UCS | 0.90 | 0.12 | 0.99 | 0.04 | 0.73 | 0.20 | 0.80 | 0.17 |

RBF-E | RBF-UCS | |||||
---|---|---|---|---|---|---|

Order of Addition/Elimination | FSS | BSE | Ranking/Sequence of Addition | FSS | BSE | Ranking/Sequence of Addition |

1 | Binder Type | w | Binder Type | G/S | W/B | G/S |

2 | G/S | OC | G/S | Binder Type | Binder Type | |

3 | t | QB | t | t | t | |

4 | D/H | W/B | D/H | T | T | |

5 | T | T | RH | RH | ||

6 | RH | RH | CO_{2} | CO_{2} | ||

7 | CO_{2} | CO_{2} | D/H | D/H | ||

8 | w | w | ||||

9 | OC | OC | ||||

10 | QB | QB | ||||

11 | W/B | W/B |

MLP-E | MLP-UCS | |||||
---|---|---|---|---|---|---|

Order of Addition/Elimination | FSS | BSE | Ranking/Sequence of Addition | FSS | BSE | Ranking/Sequence of Addition |

1 | Binder Type | W/B | Binder Type | G/S | QB | G/S |

2 | G/S | G/S | Binder Type | Binder Type | ||

3 | t | t | t | t | ||

4 | D/H | D/H | T | T | ||

5 | T | T | RH | RH | ||

6 | RH | RH | CO_{2} | CO_{2} | ||

7 | CO_{2} | CO_{2} | W/B | W/B | ||

8 | QB | QB | D/H | D/H | ||

9 | w | QB | QB | |||

10 | OC | OC | OC | |||

11 | W/B | w |

OC | w | PC | BFS | PFA | L | MgO-C | G | QB | G/S | W/B | D/H | t | T | RH | CO_{2} | E | UCS | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

OC | 1.00 | 0.69 | 0.32 | −0.17 | −0.23 | −0.16 | −0.13 | −0.13 | 0.16 | 0.61 | −0.30 | −0.33 | −0.07 | −0.16 | 0.11 | −0.09 | 0.08 | 0.19 |

w | 1.00 | 0.26 | −0.17 | −0.16 | −0.12 | −0.09 | −0.09 | 0.14 | 0.00 | −0.81 | −0.29 | −0.05 | −0.17 | 0.11 | −0.10 | −0.19 | −0.09 | |

PC | 1.00 | −0.79 | −0.37 | −0.42 | −0.27 | −0.36 | 0.15 | 0.18 | −0.30 | −0.32 | −0.07 | −0.13 | −0.01 | −0.02 | 0.18 | 0.30 | ||

BFS | 1.00 | −0.18 | 0.05 | −0.10 | 0.08 | −0.13 | −0.11 | 0.25 | 0.27 | 0.00 | 0.33 | −0.11 | 0.12 | −0.07 | −0.14 | |||

PFA | 1.00 | 0.13 | 0.60 | −0.05 | −0.06 | −0.12 | 0.12 | 0.13 | 0.10 | −0.22 | 0.14 | −0.12 | −0.16 | −0.27 | ||||

L | 1.00 | −0.03 | 0.95 | −0.04 | −0.08 | 0.08 | 0.09 | 0.07 | −0.16 | 0.10 | −0.09 | −0.13 | −0.15 | |||||

MgO−C | 1.00 | −0.03 | −0.03 | −0.07 | 0.07 | 0.07 | 0.05 | −0.13 | 0.08 | −0.07 | −0.08 | −0.16 | ||||||

G | 1.00 | −0.03 | −0.07 | 0.07 | 0.07 | 0.05 | −0.13 | 0.08 | −0.07 | −0.10 | −0.11 | |||||||

QB | 1.00 | 0.46 | −0.28 | −0.19 | −0.01 | −0.16 | 0.10 | −0.09 | 0.40 | 0.59 | ||||||||

G/S | 1.00 | 0.22 | −0.20 | −0.05 | −0.12 | 0.08 | −0.07 | 0.45 | 0.56 | |||||||||

W/B | 1.00 | 0.37 | 0.02 | 0.30 | −0.20 | 0.17 | 0.16 | 0.02 | ||||||||||

D/H | 1.00 | 0.02 | 0.33 | −0.21 | 0.18 | 0.11 | −0.21 | |||||||||||

t | 1.00 | −0.03 | 0.02 | −0.01 | 0.09 | 0.11 | ||||||||||||

T | 1.00 | −0.31 | 0.35 | −0.21 | −0.29 | |||||||||||||

RH | 1.00 | 0.20 | −0.02 | 0.05 | ||||||||||||||

CO_{2} | 1.00 | −0.08 | −0.09 |

Model | R^{2}_{ave} | RMSE_{ave} | ||||
---|---|---|---|---|---|---|

All | Training | Test | All | Training | Test | |

LR-E | 0.49 | 0.51 | 0.42 | 22.33 | 22.04 | 23.48 |

LR-UCS | 0.66 | 0.67 | 0.62 | 0.23 | 0.23 | 0.25 |

**Table 10.**Ranking of the control variables (input parameters) based on the sensitivity analysis using the profile method.

Ranking | Parameter | Max. E Variation (MPa) | Parameter | Max. UCS Variation (MPa) |
---|---|---|---|---|

1 | QB | 80.59 | G/S | 0.99 |

2 | G/S | 53.07 | OC | 0.81 |

3 | OC | 34.97 | QB | 0.71 |

4 | T | 32.91 | W/B | 0.55 |

5 | W/B | 29.93 | T | 0.41 |

6 | t | 25.61 | w | 0.33 |

7 | D/H | 20.88 | t | 0.22 |

8 | CO_{2} | 4.25 | CO_{2} | 0.12 |

9 | RH | 3.94 | RH | 0.10 |

10 | w | 3.74 | D/H | 0.08 |

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**MDPI and ACS Style**

Yousefpour, N.; Medina-Cetina, Z.; Hernandez-Martinez, F.G.; Al-Tabbaa, A.
Stiffness and Strength of Stabilized Organic Soils—Part II/II: Parametric Analysis and Modeling with Machine Learning. *Geosciences* **2021**, *11*, 218.
https://doi.org/10.3390/geosciences11050218

**AMA Style**

Yousefpour N, Medina-Cetina Z, Hernandez-Martinez FG, Al-Tabbaa A.
Stiffness and Strength of Stabilized Organic Soils—Part II/II: Parametric Analysis and Modeling with Machine Learning. *Geosciences*. 2021; 11(5):218.
https://doi.org/10.3390/geosciences11050218

**Chicago/Turabian Style**

Yousefpour, Negin, Zenon Medina-Cetina, Francisco G. Hernandez-Martinez, and Abir Al-Tabbaa.
2021. "Stiffness and Strength of Stabilized Organic Soils—Part II/II: Parametric Analysis and Modeling with Machine Learning" *Geosciences* 11, no. 5: 218.
https://doi.org/10.3390/geosciences11050218