# Load-Settlement Curve and Subgrade Reaction of Strip Footing on Bi-Layered Soil Using Constitutive FEM-AI Coupled Techniques

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

_{s}and υ

_{s}; thus

^{2}) while others such as the Biot, Terzaghi, Vesic, Meyerhof and Baike, Selvadurai and Bowles considered the rigidity and Poisson relations in their models for the Ks of footings rested on soils. Various other proposed numerical models have been presented on the evaluation of the Ks which agree with the Winkler model, and these models used the Plate Loading Test (PLT) technique to arrive at their constitutive relation [9]. Mughieda et al. [11] studied the behavior of a raft foundation structure with a response to the subgrade reaction (Ks) and found that Winkler is the most popular model used to study the interaction between the Ks and foundation, which is represented by a number of springs with a significant flaw based on the lack of coupling in springs and the non-linearity of settlement in soils. It further found that the value of the Ks has an effect on the pressure distribution on the soil below the footing. Extensive related research has been studied by Ziaie–Moayed and Janbaz [12] on the parameters that affect the subgrade reaction in clayey soils in which foundation size, shape, depth and rigidity effects were observed. The size effect was verified after conducting a 3D plaxis constitutive model for the load-settlement relation and it was found that the Terzaghi equation was not suitable for low, consistent clayey soils, in terms of the shape effect on Ks, while the Terzaghi equation was found to be suitable for stiff clayey soils. The conditions for depth embedment and rigidity effects were also proposed [12]. The coupled FEM-AI technique was used to predict the lateral behavior of free head piles in a multi-layered profile, and three Artificial Intelligence (AI) techniques (GP, EPR and ANN) were used to develop the predictive models [13]. Few numerical research studies have been evidently conducted on strip foundations underlain by multilayered soil arrangements based on a subgrade reaction and load settlement curve. Previously, the assumptions in the above constitutive methods have been that the soil is homogenous and finite layered and there has been no attempt to determine the load-settlement parameters “a” and “b” which underlie the constitutive load-settlement curve model and the subgrade reaction. In the present research work, a constitutive FEM approach was used to solve, simulate and generate a database for a strip foundation rested on a multiple bi-layered soil arrangement. The multiple databases were deployed using the smart learning abilities of AI-based techniques to predict the hybrid models of the load-settlement factors (a and b).

## 2. Artificial Intelligence (AI) Techniques

#### 2.1. Genetic Algorithm GA

#### 2.2. Genetic Programming (GP)

#### 2.3. Evolutionary Polynomial Regression EPR

^{2}+ Y

^{2}+ XY + X+Y + C), whereas a third-degree polynomial with three variables has 20 terms, a fourth-degree polynomial with four variables has 70 terms, and so on. It becomes harder to apply and less practical as the number of polynomial terms rises. Therefore, utilizing the GA approach, EPR technique seeks to maximize TPR by removing the less significant words and keeping only the most useful ones. Therefore, the chromosome is comprised of a list of polynomial terms. As a result, the population (solutions) consists of a collection of polynomials, the SSE is the fitting criterion and the chromosome is made-up of a list of polynomial terms, the length of which is determined by the number of terms. The most critical words are eliminated and the less critical ones accumulate in the survival chromosomes cycle after cycle [14].

#### 2.4. Artificial Neural Network (ANN)

## 3. Methodology

#### 3.1. Research Program

#### 3.1.1. Phase 1: Constitutive FEM Models

- Top layer (soil type S1 to soil type S6)
- Bottom layer (soil type S1 to soil type S6)
- Width of strip footing (B) (1.0 m to 5.0 m)
- Top layer thickness (h) (0.5 B to 1.0 B)
- Overburden stress (σ′v) (1.0 m to 3.0 m by the top density γ′t)

#### 3.1.2. Phase 2: Evaluate (a and b) Factors, Generate the Database and Conduct Statistical Analysis

- Cohesion, tangent of friction angle and effective density of top layer (Ct) kN/m
^{2}, tan (φt) and (γ′t) kN/m^{3}, respectively. - Top layer thickness (h) m,
- Cohesion, tangent of friction angle and effective density of bottom layer (Cb) kN/m
^{2}, tan (φb) and (γ′b) kN/m^{3}, respectively. - Strip footing width (B) m,
- Effective over burden stress at foundation depth (σ′v) kN/m
^{2}, - 1000 × hyperbolic factor (a),
- 1000 × hyperbolic factor (b).

#### 3.1.3. Phase 3: Predicting (a and b) Values Using AI Techniques

## 4. Results and Discussion of the Predictive Models

#### 4.1. Results Presentation

#### 4.1.1. Model (1)—Using (ANN) Technique

^{2}) values were 0.974 & 0.932. The relations between calculated and predicted values are illustrated in Figure 7c and Figure 8c.

#### 4.1.2. Model (2)—Using GP Technique

^{2}) values for his model were 50%, 0.575 and 49%, 0.572, respectively.

#### 4.1.3. Model (3)—Using EPR Technique

^{5}, 330 terms for X

^{4}, 120 terms for X

^{3}, 36 terms for X

^{2}, 8 terms for X

^{1}and 1 term for X

^{0}(total 1287 terms) as follows:

#### 4.2. Results Discussion

## 5. Conclusions

- The developed formulas using the GP technique showed a limited accuracy of 50%. All input factors were utilized, except the cohesion of both top and bottom soils (Ct), (Cb).
- EPR technique generated two seven term polynomials out of 1287 possible terms. The accuracy is better than the GP models (65%). In addition, all input factors except the overburden pressure (σ′v) and the cohesion of both the top and bottom soils (Ct), (Cb) were generated.
- Finally, ANN technique presents the best accuracy of 80% and used all the input factors. The relative importance of each factor is indicated by the size of the blocks in Figure 5, and, accordingly, all factors have almost the same effect on the load-settlement curve except (B), tan (φt) and tan (φb), which have a slightly higher effect.

- Both GP and EPR could not capture the influence of soil cohesion on the load-settlement curve, which gives the advantage to the ANN model.
- The developed GP model is not recommended because of its limited accuracy.
- Although the ANN model showed the best accuracy and utilised all input factors, its model is too complicated to be manually handled.
- The developed EPR model could be used for manual calculations, while the ANN model is suitable for computerized calculations
- The developed models should be used within the factor values considered in the study. The prediction accuracy must be verified beyond this range.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 7.**Relation between predicted and calculated (1000a) values using the developed models, (

**a**) using GP, (

**b**) using EPR and (

**c**) using ANN.

**Figure 8.**Relation between predicted and calculated (1000b) values using the developed models, (

**a**) using GP, (

**b**) using EPR (

**c**) using ANN.

**Table 1.**Different proposed formulas to calculate the subgrade reaction, Ks based on elastic parameters.

No | Investigator | Year | Suggested Formula |
---|---|---|---|

1 | Winkler | 1867 | $\frac{\mathrm{q}}{\mathsf{\delta}}$ |

2 | Biot | 1937 | $\frac{0.95{\mathrm{E}}_{\mathrm{s}}}{\mathrm{B}\left(1-{\mathsf{\upsilon}}_{\mathrm{s}}^{2}\right)}{\left[\frac{{\mathrm{B}}^{4}{\mathrm{E}}_{\mathrm{s}}}{\left(1-{\mathsf{\upsilon}}_{\mathrm{s}}^{2}\right)\mathrm{EI}}\right]}^{0.108}$ |

3 | Terzaghi | 1955 | ${\mathrm{K}}_{\mathrm{sf}}={\mathrm{K}}_{\mathrm{sp}}\left(\frac{\mathrm{B}-{\mathrm{B}}_{1}}{2\mathrm{B}}\right)$ |

4 | Vesic | 1961 | $\frac{0.65{\mathrm{E}}_{\mathrm{s}}}{\mathrm{B}\left(1-{\mathsf{\upsilon}}_{\mathrm{s}}^{2}\right)}\sqrt[12]{\frac{{\mathrm{E}}_{\mathrm{s}}{\mathrm{B}}^{4}}{\mathrm{EI}}}$ |

5 | Meyerhof and Baike | 1965 | $\frac{{\mathrm{E}}_{\mathrm{s}}}{\mathrm{B}\left(1-{\mathsf{\upsilon}}_{\mathrm{s}}^{2}\right)}$ |

6 | Selvadurai | 1984 | $\frac{0.65}{\mathrm{B}}\ast \frac{{\mathrm{E}}_{\mathrm{s}}}{\left(1-{\mathsf{\upsilon}}_{\mathrm{s}}^{2}\right)}$ |

7 | Bowles | 1988 | $\frac{{\mathrm{E}}_{\mathrm{s}}}{{\mathrm{B}}_{1}\left(1-{\mathsf{\upsilon}}_{\mathrm{s}}^{2}\right){\mathrm{mI}}_{\mathrm{s}}{\mathrm{I}}_{\mathrm{F}}}$ |

_{1}= side dimension of square base used in the plate load test. B = width of footing. ksp = the value of subgrade reaction for 0.3 × 0.3 (1 ft wide) bearing plate. K

_{sf}= value of modulus of sub-grade reaction for the full-size foundation. Es = modulus of elasticity. υ

_{s}= Poisson’s ratio. EI = flexural rigidity of footing, m = takes 1, 2 and 4 for edges, sides and center of footing, respectively. I

_{S}and I

_{F}= influence factors depend on the shape of footing.

Soil Type | Soil Description | C (kN/m ^{2}) | φ (°) | γ (kN/m ^{3}) | E (MN/m ^{2}) | υ |
---|---|---|---|---|---|---|

S 1 | loose Sand | 0.0 | 29 | 16 | 9.0 | 0.350 |

S 2 | Dense Sand | 0.0 | 38 | 20 | 50.0 | 0.300 |

S 3 | Soft Clay | 25 | 0.0 | 14 | 1.5 | 0.450 |

S 4 | Stiff Clay | 100 | 0.0 | 20 | 10.0 | 0.350 |

S 5 | Soft Silt | 25 | 5 | 18 | 6.0 | 0.400 |

S 6 | Stiff Silt | 100 | 20 | 20 | 30.0 | 0.330 |

Ct | tan (φt) | γ′t | h | Cb | tan (φb) | γ′b | B | σ′v | 1000a | 1000b | |
---|---|---|---|---|---|---|---|---|---|---|---|

kN/m^{2} | - | kN/m^{3} | m | kN/m^{2} | - | kN/m^{3} | m | kN/m^{2} | kN/m^{2} | kN/m^{2} | |

Training set | |||||||||||

Min. | 0.1 | 0.0 | 14.0 | 0.5 | 0.1 | 0.0 | 14.0 | 1.0 | 18.0 | 0.343 | 0.086 |

Max. | 100 | 1 | 20 | 5 | 100 | 1 | 20 | 5 | 54 | 6.340 | 1.880 |

Avg. | 35.6 | 0.4 | 18.1 | 2.1 | 37.4 | 0.3 | 17.8 | 3.0 | 35.0 | 1.850 | 0.327 |

SD | 46.0 | 0.3 | 2.2 | 1.3 | 41.2 | 0.3 | 2.4 | 1.6 | 14.7 | 1.610 | 0.272 |

VAR | 1.29 | 0.87 | 0.12 | 0.61 | 1.10 | 1.02 | 0.13 | 0.54 | 0.42 | 0.870 | 0.832 |

Validation set | |||||||||||

Min. | 0.1 | 0.0 | 14.0 | 0.5 | 0.1 | 0.0 | 14.0 | 1.0 | 18.0 | 0.372 | 0.094 |

Max. | 100 | 1 | 20 | 5 | 100 | 1 | 20 | 5 | 54 | 6.450 | 1.920 |

Avg. | 40.6 | 0.3 | 18.4 | 1.8 | 39.5 | 0.3 | 17.6 | 2.7 | 38.4 | 1.880 | 0.347 |

SD | 46.9 | 0.3 | 2.0 | 1.2 | 39.9 | 0.3 | 2.5 | 1.5 | 14.5 | 1.540 | 0.263 |

VAR | 1.15 | 0.97 | 0.11 | 0.67 | 1.01 | 1.18 | 0.14 | 0.56 | 0.38 | 0.819 | 0.758 |

Ct | tan (φt) | γ′t | h | Cb | tan (φb) | γ′b | B | σ′v | 1000a | 1000b | |
---|---|---|---|---|---|---|---|---|---|---|---|

Ct | 1.00 | ||||||||||

tan (φt) | −0.66 | 1.00 | |||||||||

γ′t | 0.59 | 0.05 | 1.00 | ||||||||

h | −0.21 | 0.40 | 0.07 | 1.00 | |||||||

Cb | 0.06 | 0.08 | 0.13 | 0.04 | 1.00 | ||||||

tan (φb) | 0.09 | 0.02 | 0.02 | 0.05 | −0.55 | 1.00 | |||||

γ′b | 0.09 | 0.08 | 0.15 | 0.08 | 0.43 | 0.32 | 1.00 | ||||

B | 0.02 | −0.01 | −0.01 | 0.88 | 0.02 | 0.01 | 0.04 | 1.00 | |||

σ′v | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 1.00 | ||

1000a | −0.12 | −0.50 | −0.55 | −0.30 | −0.20 | −0.29 | −0.38 | −0.07 | −0.18 | 1.00 | |

1000b | −0.15 | −0.08 | −0.30 | 0.41 | −0.19 | −0.25 | −0.47 | 0.56 | 0.01 | 0.13 | 1.00 |

Hidden | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

H1 | H2 | H3 | H4 | H5 | H6 | H7 | H8 | H9 | H10 | |||

Input Layer | (Bias) | −0.96 | 1.31 | −0.01 | −1.42 | −0.34 | 0.29 | −0.39 | 0.17 | −0.59 | −1.54 | |

Ct | −0.27 | 0.63 | 0.75 | −0.07 | 0.12 | 0.22 | −0.15 | −0.41 | 0.00 | −0.1 | ||

tan (φt) | 0.31 | 0.36 | 0.41 | 1.16 | 0.17 | −0.03 | −0.45 | 0 | −0.3 | −0.36 | ||

γ′t | −0.74 | 0.51 | −0.16 | −0.63 | 0.00 | −0.28 | −0.09 | 0.41 | −0.34 | −0.49 | ||

h | 0.15 | −0.65 | 0.23 | 0.02 | −0.04 | −0.22 | −0.11 | 0.4 | −0.35 | −0.15 | ||

Cb | 0.32 | −0.4 | 0.44 | 0.33 | −0.78 | 0.00 | 0.24 | −0.06 | 0.08 | −0.37 | ||

tan (φb) | 1.24 | −0.94 | 0.56 | 0.47 | −0.8 | −0.32 | −0.9 | −0.71 | −0.49 | −0.27 | ||

γ′b | −0.26 | −0.83 | 0.26 | 0.61 | −1.04 | 0.32 | −1.11 | −0.42 | 0.6 | −0.17 | ||

B | 0.1 | 0.25 | −0.15 | 0.15 | 0.37 | 0.5 | 0.02 | −0.55 | 0.5 | −0.01 | ||

σ′v | 0.45 | 0.26 | −0.21 | −0.26 | −0.01 | −0.24 | −0.6 | 0.37 | 0.02 | −0.27 | ||

Hidden | ||||||||||||

H1 | H2 | H3 | H4 | H5 | H6 | H7 | H8 | H9 | H10 | (Bias) | ||

Output | 1000a | −0.81 | −1.33 | −0.51 | −1.28 | −0.16 | 0.15 | 0.12 | 0.09 | 0.03 | 0.92 | −0.49 |

1000b | 0.33 | 0.73 | −0.07 | 0.78 | 0.62 | 0.04 | −0.15 | −0.33 | 0.46 | 0.06 | −0.18 |

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**MDPI and ACS Style**

Ebid, A.M.; Onyelowe, K.C.; Salah, M.
Load-Settlement Curve and Subgrade Reaction of Strip Footing on Bi-Layered Soil Using Constitutive FEM-AI Coupled Techniques. *Designs* **2022**, *6*, 104.
https://doi.org/10.3390/designs6060104

**AMA Style**

Ebid AM, Onyelowe KC, Salah M.
Load-Settlement Curve and Subgrade Reaction of Strip Footing on Bi-Layered Soil Using Constitutive FEM-AI Coupled Techniques. *Designs*. 2022; 6(6):104.
https://doi.org/10.3390/designs6060104

**Chicago/Turabian Style**

Ebid, Ahmed M., Kennedy C. Onyelowe, and Mohamed Salah.
2022. "Load-Settlement Curve and Subgrade Reaction of Strip Footing on Bi-Layered Soil Using Constitutive FEM-AI Coupled Techniques" *Designs* 6, no. 6: 104.
https://doi.org/10.3390/designs6060104