# Nonlinear Settlement Calculation of Composite Foundation Based on Tangent Modulus Method: Two Case Studies

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Calculation Principle of Tangent Modulus Method

#### 2.1. Settlement Calculation Method of Composite Foundation—Layerwise Summation Method

#### 2.2. Tangent Modulus of Shallow Soil

_{u}. ${R}_{f}$ is the damage ratio coefficient, and ${E}_{t}$ is the tangent modulus of the soil under a different stress level.

#### 2.3. Tangent Modulus of Deep Soil

## 3. Test Case 1—Settlement of Multi-Layer Soil Composite Foundation

#### 3.1. Project Profile

^{2}.

#### 3.2. Vertical Static Load Test of Dynamic Compaction Composite Foundation

^{2}. The load–displacement p–s curve obtained from the shallow plate loading test is shown in Figure 5. It can be seen from Figure 5 that the cumulative settlement of the plate loading is 18.16 mm when loaded to 840 kPa, and the rebound after unloading is 4.66 mm, with a rebound rate of about 25.7%.

#### 3.3. Tangent Modulus Calculation

#### 3.4. Settlement Analysis of Composite Foundation under Flexible Uniform Load

#### 3.4.1. Settlement Calculation Using Modulus of Compression

_{1}of the midpoint of the raft boundary line is 35.18 mm.

#### 3.4.2. Settlement Calculation by Tangent Modulus

- The gravel layer:

- The completely weathered calcareous conglomerate:

_{u}of the completely weathered conglomerate is calculated by the Terzaghi formula ${P}_{u}=\frac{1}{2}\gamma b{N}_{\gamma}+q{N}_{q}+c{N}_{c}$, where c and $\phi $ are 32 kPa and 20°, respectively. The tangent modulus of the completely weathered calcareous conglomerate at different depths can be calculated according to Formula (5). Figure 7 shows the p–s curves of the midpoint and corner of the boundary line calculated by the compression modulus and tangent modulus, respectively, under the uniform load, and the measured curve of corner points 1 and 9 and the midpoint 3 and 7 are presented.

#### 3.5. Finite Element Analysis of Foundation Settlement

#### 3.5.1. Establishment of Finite Element Model

#### 3.5.2. Comparison of the Test and Numerical Results

## 4. Test Case 2—Multi-Layered Soil Foundation with Rigid Pile

#### 4.1. Project Profile

^{2}.

#### 4.2. Vertical Static Load Test of Soil

#### 4.3. Finite Element Analysis of Foundation Settlement

#### 4.3.1. Establishment of Finite Element Model

#### 4.3.2. Comparison of the Test and Numerical Results

## 5. Conclusions

- (1)
- The tangent modulus method is mainly suitable for the hard plastic soil with a strong structure, such as silty clay, completely weathered rock, granite residual soil, and so on, whose SPT blow count N is more than 8.
- (2)
- The tangent modulus takes into account the influence of soil stress and can better reflect the nonlinear settlement characteristics of the foundation. The measured settlement results of the two cases show that the calculation accuracy of the tangent modulus is significantly better than the traditional compression modulus.
- (3)
- The Chinese foundation code adopts the empirical coefficient of the settlement calculation to adjust the settlement of low-compressibility soil, and the minimum empirical coefficient of the settlement is 0.2. However, the value of the empirical coefficient is greatly influenced by the experience of the engineers. The tangent modulus method directly obtains the tangent modulus of the soil by an in-situ pressure plate test, and no empirical coefficient is needed in the settlement calculation process.
- (4)
- Owing to the high cost of the deep plate load test, for deep hard plastic soil, it is a common geological exploration method to determine the deformation modulus of deep soil by the SPT blow count, and there are corresponding empirical formulae for different types of soil in different regions. In this paper, it is assumed that the ratio of the initial tangential modulus to the deformation modulus is the same as the ratio of the unloading and reloading modulus ${E}_{\mathrm{ur}}^{\mathrm{ref}}$ to the secant modulus ${E}_{50}^{\mathrm{ref}}$ obtained from the triaxial test of the sampled soil, and then the initial tangential modulus can be derived from this ratio. The related method can greatly reduce the cost of obtaining the tangent modulus of deep soil. The analysis results of two cases prove that the method has high accuracy. More deep plate loading tests are required to further verify whether the method is universal.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Zhang, L.; Zhao, M.H.; Shi, C.J.; Zhao, H. Settlement Calculation of Composite Foundation Reinforced with Stone Columns. Int. J. Geomech.
**2013**, 13, 248–256. [Google Scholar] [CrossRef] - Chang, D.W.; Lu, C.W.; Tu, Y.J.; Cheng, S.H. Settlements and Subgrade Reactions of Surface Raft Foundations Subjected to Vertically Uniform Load. Appl. Sci.
**2022**, 12, 5484. [Google Scholar] [CrossRef] - Pantelidis, L. Empirical Relationships Between the Elastic Settlement of Rigid Rectangular Foundations and the Settlement of the Respective Flexible Foundations. Geotech. Geol. Eng.
**2021**, 39, 3959–3971. [Google Scholar] [CrossRef] - Wang, J.; Ji, H.G.; Wang, C.Q.; Bai, Y. Analysis the Settlement Calculation Methods of Combined Piles in Composite Foundation. Appl. Mech. Mater.
**2011**, 71–78, 28–31. [Google Scholar] - Yang, G.H. Nonlinear settlement computation of the soil foundation with the undisturbed soil tangent modulus method. Chin. J. Geotech. Eng.
**2006**, 28, 1927–1931. (In Chinese) [Google Scholar] - Duncan, J.M.; Chang, C.Y. Nonlinear Analysis of Stress and Strain in Soils. J. Soil Mech. Found. Div.
**1970**, 96, 1629–1653. [Google Scholar] [CrossRef] - Yang, G.H.; Wang, P.; Qiao, Y. An undisturbed-soil secant modulus method for calculation of nonlinear settlement of soil foundations. China Civ. Eng. J.
**2007**, 40, 49–52. (In Chinese) [Google Scholar] - Yang, G.H. New computation method for soil foundation settlements. Chin. J. Rock Mech. Eng.
**2008**, 27, 679–686. (In Chinese) [Google Scholar] - Peng, C.X.; Yang, G.H. A simplified method for determining e-p curve of soft soil and its application to analyzing nonlinear settlement of foundation. Rock Soil Mech.
**2008**, 29, 1706–1710. (In Chinese) [Google Scholar] - GB50007-2011; Code for Design of Building Foundation. China Architecture & Building Press: Beijing, China, 2011.
- JGJ79-2012; Technical Code for Ground Treatment of Buildings. China Architecture & Building Press: Beijing, China, 2012.
- Vakili, K.N.; Barciaga, T.; Lavasan, A.A.; Schanz, T. A practical approach to constitutive models for the analysis of geotechnical problems. Third Int. Symp. Comput. Geomech.
**2013**, 1, 738–749. [Google Scholar] - Mohsan, M.; Vardon, P.J.; Vossepoel, F.C. On the use of different constitutive models in data assimilation for slope stability. Comput. Geotech.
**2021**, 138, 104332. [Google Scholar] [CrossRef] - Wu, J.T.H.; Tung, S.C.-Y. Determination of Model Parameters for the Hardening Soil Model. Transp. Infrastruct. Geotechnol.
**2019**, 7, 55–68. [Google Scholar] [CrossRef] - JGJ8-2007; Code for Deformation Measurement of Building and Structure. China Architecture & Building Press: Beijing, China, 2007.
- Hua, J.X.; Zheng, J.G. Geological Engineering Handbook, 5th ed.; China Architecture & Building Press: Beijing, China, 2018; pp. 206–223. [Google Scholar]
- Zhu, M.; Chen, X.S.; Zhang, G.T.; Pang, X.C.; Su, D.; Liu, J.Q. Parameter back-analysis of hardening soil model for granite residual soil and its engineering applications. Rock Soil Mech.
**2022**, 43, 1061–1072. (In Chinese) [Google Scholar] - Luo, M.M.; Chen, Y.; Zhou, J. Research status and prospect of parameter selection for the HS-small model. Ind. Constr.
**2021**, 51, 172–180. (In Chinese) [Google Scholar] - Liu, W.H.; Zhu, H.; He, S.J.; Yan, J.B.; Xu, C.J. Experimental study on parameters of hardening soil model for soils and its application in foundation pit engineering in Nanchang. J. Civ. Environ. Eng.
**2021**, 43, 38–47. (In Chinese) [Google Scholar]

**Figure 7.**Comparison of calculated and measured p–s curves: (

**a**) midpoint of the boundary line; and (

**b**) corner point.

**Figure 10.**Settlement comparison of p–s curve: (

**a**) the midpoint of boundary line; and (

**b**) the corner point.

**Figure 14.**Finite element model of rigid pile composite foundation of Building 15: (

**a**) raft; (

**b**) raft and pile; and (

**c**) FEM model.

**Figure 16.**Comparison of finite element and measured p–s curves: (

**a**) the midpoint of the boundary line; and (

**b**) the corner point.

$\overline{{E}_{s}}$ (MPa) | 4 | 7 | 15 | 20 | 35 |

${\psi}_{s}$ | 1 | 0.7 | 0.4 | 0.25 | 0.2 |

Soil Layer | Poisson’s Ratio $\mathit{\mu}$ | Gravity (kN/m ^{3}) | Characteristic Value ${\mathit{f}}_{\mathit{a}\mathit{k}}$ (kPa) | Modulus of Compression ${\mathit{E}}_{\mathit{s}}$ (MPa) | C (kPa) | $\mathit{\phi}$ (Degrees) | SPT Blow Count N |
---|---|---|---|---|---|---|---|

Silty clay | 0.3 | 18.8 | 130 | 5.91 | 17 | 15.61 | 6 |

Gravel layer | 0.24 | 20.5 | 340 | 27 | 0 | 34 | (12) |

Completely weathered calcareous conglomerate | 0.26 | 19.5 | 210 | 15 | 32 | 20 | 16 |

Monitoring Point | Cumulative Settlement (mm) | |||
---|---|---|---|---|

Construction to the 7th Floor (84 kN/m ^{2}) | Construction to the 14th Floor (168 kN/m ^{2}) | Construction to the 21st Floor (252 kN/m ^{2}) | Construction to the 28th Floor (336 kN/m ^{2}) | |

CJ-1 | 1.69 | 3.21 | 5.01 | 10.35 |

CJ-2 | 1.74 | 3.33 | 6.20 | 12.18 |

CJ-3 | 1.72 | 3.39 | 7.96 | 14.85 |

CJ-4 | 1.52 | 3.22 | 7.35 | 14.79 |

CJ-5 | 1.76 | 3.31 | 7.31 | 15.80 |

CJ-6 | 1.68 | 3.42 | 6.66 | 15.30 |

CJ-7 | 1.61 | 3.32 | 7.00 | 15.42 |

CJ-8 | 1.74 | 3.25 | 5.05 | 10.87 |

CJ-9 | 1.65 | 3.39 | 4.91 | 10.30 |

Soil | Ultimate Bearing Capacity ${\mathit{p}}_{\mathit{u}}$ (kPa) | Depth of Soil Layer (m) | Tangent Modulus of Soil Layer under Different Load Levels (MPa) | ||||||
---|---|---|---|---|---|---|---|---|---|

105 | 210 | 315 | 420 | 525 | 630 | 735 | |||

gravel layer | 814 | 0.566 | 343.5 | 249.3 | 170.1 | 106.0 | 57.0 | 23.1 | 4.3 |

1264 | 1.131 | 394.6 | 340.4 | 290.2 | 244.0 | 201.8 | 163.6 | 129.5 | |

1714 | 1.697 | 428.3 | 404.4 | 381.3 | 358.8 | 337.0 | 315.9 | 295.5 | |

2163 | 2.262 | 441.6 | 430.6 | 419.7 | 408.9 | 398.3 | 387.8 | 377.5 | |

2613 | 2.828 | 447.0 | 441.2 | 435.5 | 429.8 | 424.2 | 418.5 | 413.0 | |

3063 | 3.394 | 449.4 | 446.1 | 442.8 | 439.5 | 436.2 | 432.9 | 429.6 | |

3513 | 3.959 | 450.7 | 448.7 | 446.6 | 444.6 | 442.6 | 440.5 | 438.5 | |

3963 | 4.525 | 451.5 | 450.1 | 448.8 | 447.4 | 446.1 | 444.8 | 443.4 | |

4413 | 5.090 | 451.8 | 450.7 | 449.7 | 448.7 | 447.6 | 446.6 | 445.6 | |

4863 | 5.656 | 452.2 | 451.5 | 450.9 | 450.3 | 449.7 | 449.0 | 448.4 | |

5313 | 6.222 | 452.3 | 451.8 | 451.3 | 450.8 | 450.3 | 449.8 | 449.3 | |

5762 | 6.787 | 452.5 | 452.1 | 451.8 | 451.5 | 451.1 | 450.8 | 450.5 | |

6212 | 7.353 | 452.5 | 452.3 | 452.1 | 451.8 | 451.6 | 451.3 | 451.1 | |

6662 | 7.918 | 452.6 | 452.5 | 452.3 | 452.1 | 451.9 | 451.8 | 451.6 | |

7112 | 8.484 | 452.7 | 452.6 | 452.5 | 452.4 | 452.3 | 452.2 | 452.0 |

Monitoring Point | Cumulative Settlement (mm) | |||
---|---|---|---|---|

Construction to the 7th Floor (84 kN/m ^{2}) | Construction to the 14th Floor (168 kN/m ^{2}) | Construction to the 21st Floor (252 kN/m ^{2}) | Construction to the 28th Floor (336 kN/m ^{2}) | |

CJ-1 | 2.04 | 3.15 | 6.19 | 12.30 |

CJ-2 | 2.13 | 3.19 | 6.82 | 13.79 |

CJ-3 | 2.11 | 3.13 | 6.47 | 13.31 |

CJ-4 | 2.16 | 3.18 | 6.71 | 13.96 |

CJ-5 | 2.09 | 3.16 | 6.64 | 13.62 |

CJ-6 | 2.12 | 3.28 | 6.65 | 14.04 |

CJ-7 | 2.18 | 3.26 | 6.69 | 13.77 |

CJ-8 | 2.16 | 3.26 | 6.3 | 14.25 |

CJ-9 | 2.13 | 3.25 | 6.54 | 14.74 |

CJ-10 | 2.06 | 3.12 | 6.48 | 13.93 |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Li, Y.; Yao, L.; Chen, G.; Zhao, W.; Liu, X. Nonlinear Settlement Calculation of Composite Foundation Based on Tangent Modulus Method: Two Case Studies. *Buildings* **2023**, *13*, 892.
https://doi.org/10.3390/buildings13040892

**AMA Style**

Li Y, Yao L, Chen G, Zhao W, Liu X. Nonlinear Settlement Calculation of Composite Foundation Based on Tangent Modulus Method: Two Case Studies. *Buildings*. 2023; 13(4):892.
https://doi.org/10.3390/buildings13040892

**Chicago/Turabian Style**

Li, Yonghua, Lei Yao, Gaoxiang Chen, Weijian Zhao, and Xiangang Liu. 2023. "Nonlinear Settlement Calculation of Composite Foundation Based on Tangent Modulus Method: Two Case Studies" *Buildings* 13, no. 4: 892.
https://doi.org/10.3390/buildings13040892