# Soil Consolidation Analysis in the Context of Intermediate Foundation as a New Material Perspective in the Calibration of Numerical–Material Models

^{*}

## Abstract

**:**

## 1. Introduction

- Foundations founded directly (footings or slabs) on load-bearing and low-deformation soils;
- Indirect (deep) foundations based on lower layers of load-bearing soils, in the form of a certain number of piles or piles connected to the structure by a cap (grate, footing, slab) to transfer loads.

## 2. Materials and Methods—Background Information

## 3. Effect of Soil Consolidation on the Total Settlement of a Single Pile

_{c}= s∞, as in the case of estimating the settlement of a direct foundation, consisted of selecting the parameters of the Meyer function (De, p, α, and s∞) to match the results of the settlements measured during the test loading of the column in the field.

_{e}, as part of the settlement that may occur during the operational stage of the structure, to the total settlement s

_{c}, can be considered constant and independent of the value of stress in the soil medium of the same consistency. Under this assumption, the ratio of the settlement of the column at the operational stage s

_{e}to the total settlement of the column s

_{c}was determined by the increments of settlement caused by the change in the load on the column head from 500 (kN) to 600 (kN) (Δσ = const = 100 (kN)). The settlement of the column at the s

_{e}operation stage was calculated as the difference between the total settlement s

_{c}and the settlement observed after the contractual settlement stabilization time, i.e., after about 30 min for most columns. All data was presented in Table 1, Table 2 and Table 3.

## 4. Consolidation Settlements

_{total}= 1 to 1.25 × s

_{w_time_realisation}.

_{v}denotes the consolidation coefficient ${c}_{v}=\frac{k\cdot M}{{\gamma}_{w}}$, was derived based on the assumption that an equilibrium condition is maintained between the difference in the amount of water flowing in and out of the elementary volume of soil and the change in this volume. The water flow was defined according to Darcy’s law, in which the flow velocity is proportionally dependent on the induced hydraulic gradient, through a constant filtration coefficient k.

_{t}mobilized after time t, related to total settlements s

_{c}after the consolidation process, defined as the averaged degree of consolidation U

_{av}= s

_{t}/s

_{c}. Sivaram and Swamee [21] proposed an empirical formula for U

_{av}[%]:

_{dr}—filtration path.

_{v}for a given load increment. The Taylor method (square root method) is most often used:

_{90}= 0.848;

_{90}—time after which U

_{av}= 90% (90% of total consolidation settlement);

_{dr}

_{90}—filtration path at t

_{90}.

_{50}= 0.197;

_{50}—time after which U

_{av}= 50% (50% of total consolidation settlement);

_{dr}

_{50}—filtration path at t

_{50}.

_{v}is calculated as the product of the filtration coefficient and oedometric modulus, determined for a given load range, related to the volume weight of water $\frac{k{E}_{\mathrm{oedo}}}{{\gamma}_{w}}$.

_{v}determined by one of the methods given above form the basis for calculating the dimensionless time index T

_{v}, needed to estimate the consolidation settlement st after time t, at a known (assumed) value of the total settlement mobilized in the consolidation process and with the assumption that the actual filtration path H

_{dr}is known. Analogous to oedometric tests, in specially prepared samplers, tests were performed to determine t

_{50}according to the Casagrand method for the subsoil without and with reinforcement (using piles). Figure 5, Figure 6, Figure 7 and Figure 8 show the labeled c

_{v}values determined by the logarithmic method, assuming H

_{dr}= 0.10 m as equal to the width of the slab, and H

_{dr}= 0.15 m as the total thickness of the soil calculated from the base of the piles to the bottom of the test tank (test tank). The curves in Figure 5, Figure 6, Figure 7 and Figure 8 showing the variation in settlement as a function of the logarithm of time allowed easy determination of t

_{50}, and ultimately the determination of the coefficient of consolidation c

_{v}.

_{v}as a function of load determined assuming H

_{dr}= 0.10 m.

_{v}on load for slab and slab–pile foundations (Figure 9), supported by the averaged c

_{v}results of Figure 5, Figure 6, Figure 7 and Figure 8, indicate two different types of behavior: a stand-alone slab and slab with 4 piles, and a slab supported by 9 and 16 piles. The first group is characterized by c

_{v}consolidation coefficients with average values of 5.71 × 10

^{−8}(slab) and 7.04 × 10

^{−8}m

^{2}/s (slab + four piles). The average c

_{v}values for the second group are an order of magnitude lower: 7.73 × 10

^{−9}(slab + 9 piles) and 8.02 × 10

^{−9}m

^{2}/s (slab + 16 piles). Such observations lead to the conclusion that the consolidation of the subsoil of slab-and-pile foundations, in which the piles are spaced widely (about r/D = 6), proceeds similarly to the consolidation of the subsoil of direct foundations. The CPRF foundation, with piles in close proximity, behaves differently, where consolidation proceeds more slowly.

_{v}takes similar values, close to 1 × 10

^{−7}m

^{2}/s. This may mean that after exceeding a certain conventional load value, greater than, for example, the value of preconsolidation stress, characteristic of the soil, i.e., when the soil is considered normally consolidated, direct and slab–pile foundations, regardless of the number of piles, settle over time in a similar manner.

_{v}for a specific engineering task is difficult. It should be remembered that the actual settling velocity of a structure’s foundation is often two to four times higher than the velocities predicted from the c

_{v}measured using intact samples [23]. At loads of 94, 142, and 189 kPa, the reduction in consolidation settlement using 16 instead of 9 piles is identical or very similar, which means that using more than 9 piles for the studied foundations in the analyzed load range is ineffective.

_{av}consolidation, the consolidation settlement after time t, for each of the five selected loading steps, was related to the total consolidation settlement estimated by the Meyer method. For the first three loading steps, the ground consolidations of slab–pile systems with 9 and 16 piles are very similar. Slab and slab with four piles behave differently. At a load of 95 kPa, we can clearly distinguish between the two types of behavior. The last two loading steps (142 and 189 kPa), discarding the results for the slab with nine piles, indicate that consolidation proceeds in a similar manner regardless of the type of slab–pile foundation. In the case of the nine-pile foundation, analysis of the load–settlement relationship indicated that a lack of smooth entry into the original load path was observed after the reapplication of the load. Interpreting this as a possible measurement error, the results for the slab with nine piles were removed from the above analysis. Then, the selected relations U

_{av}

_{(t)}, in Figure 10, were compared with the consolidation path created from Equation (3). The empirical formula proposed by Sivaram and Swamee [21] indicates a faster completion of the process of dispersion of excess pore water pressure. Hence, for two loading steps (142 and 189 kPa) and three types of CPRF (slab, slab + 4 piles, and slab + 16 piles), the approximation of U

_{av}changes as a function of time was carried out using Equation (5). The values of the potentiometric coefficients A, B, and C were determined using nonlinear numerical optimization of in-house results (Table 4). The medians of the potentiometric coefficients for the six selected U

_{av}

_{(t)}relationships allowed us to write the empirical formula, which is the solution of Terzaghi’s one-dimensional consolidation equation, in the form:

_{v}determined by oedometric tests, as for direct foundations.

## 5. Preconsolidation Stress σ’c

_{u}= 57 kPa—rotary shear, s

_{u}= 64 kPa—piston penetrometer. The preconsolidation stress interpreted from the stress–settlement graphs increases as the number of piles increases (for 4, 9, and 16 piles, σ’c = 87, 94, and 123 kPa, respectively), which can be analogously transposed to an increase in the equivalent shear strength without drainage of the pile-reinforced foundation.

_{FI,d}of the subsoil can be calculated using the standard formula in Annex D3 of [2] PN-EN 1997-1:2008 Eurocode 7:

_{u}—wall strength without drain;

_{c,E}

_{7}—slope factor of the foundation base;

_{c,E}

_{7}—shape factor of the foundation;

_{c,E}

_{7}—load slope factor.

## 6. Numerical Analysis of Large-Dimensional Test Loads of Slab–Pile Foundations

#### 6.1. Geometric Systems

#### 6.2. Comparison of Model Test Results with FEM Analysis Results

#### 6.3. Analysis of Column Behavior Based on FEM Analysis

- The greater the settlement (increase in column head settlement from 4 mm to 30 mm), the greater the value of friction mobilized along the column (Figure 14, Figure 15, Figure 16 and Figure 17, figure on the right, a noticeable increase in friction—smallest values for blue, higher for orange, highest for black lines);
- By comparing the mobilization of friction on the shaft of the column (Figure 14, Figure 15, Figure 16 and Figure 17, figure on the right) for example for the central columns marked with a continuous blue line, we observe under the slab in pile–raft foundations a reduction in the mobilization of friction on the pile shaft; for example, for a depth of 2 m, the friction (kPa) decreases from 17.1 kPa—single column to 15.8 kPa—column in 4 pile–raft foundation, to 6.4 kPa—column in 9 pile–raft foundation, and finally to 5.88 kPa—column in 16 pile–raft foundation); we observe the so-called formation of a “dead zone” to a depth that depends on the spacing and mutual location of the piles, as well as the amount of settlement of the foundation;
- For the range of very small column head settlements (0.01D, blue line), for relative column spacing r/D = 3.3 (Figure 17, figure on the left) to r/D = 5 (Figure 16, figure on the left), the center columns (marked with a continuous line) mobilize less resistance than the edge columns (indicated by a dashed line), or the corner columns (indicated by a dotted line) working most effectively;
- For a range of small column head settlements (0.03D, orange line), for relative column spacing r/D = 3.3 (Figure 17, figure on the left), center columns (marked with a continuous line) mobilize less resistance than edge columns (indicated by a dashed line) or corner columns (indicated by a dotted line);
- For the range of small column head settlements (0.03D, orange line), for relative column spacings r/D = 5 (Figure 16, figure on the left), regardless of their location in the group, the columns mobilize similar resistance;
- For the range of large settlements of the column head (0.08D, black line), for the relative spacing of the columns r/D = 3.3 (Figure 17, figure on the left) to r/D=5 (Figure 16, figure on the left), the center columns (continuous line), due to the significant pressure of the slab, mobilize the greatest resistance; the corner columns (dotted line) work least effectively.

## 7. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**The most common types of foundations. Adapted from [16].

**Figure 2.**Slab–pile foundation (CPRF) as a complex geotechnical structure. Interaction of the elements of a slab–pile foundation according to Hanisch et al. [13].

**Figure 3.**The course of the dependence of the coefficient αCPRF as a function of the ratio of the settlement of the CPRF foundation consisting of a direct foundation (e.g., slab) and a group of piles sCPRF to the settlement of the direct foundation sFI (sCPRF/sFI means the degree of reduction in the settlement of the direct foundation after the introduction of piles into the foundation).

**Figure 5.**Plate without piles. Approximation of selected compressibility curves with Meyer curve to determine total consolidation settlement. Contribution of total settlement to settlement after 24 h. Values of c

_{v}for H

_{dr}= 10 and 15 cm.

**Figure 6.**Slab with 4 piles. Approximation of selected compressibility curves with Meyer curve to determine total consolidation settlement. Contribution of total settlement to settlement after 24 h. Values of c

_{v}for H

_{dr}= 10 and 15 cm.

**Figure 7.**Slab with 9 piles. Approximation of selected compressibility curves with Meyer curve to determine total consolidation settlement. Contribution of total settlement to settlement after 24 h.

**Figure 8.**Slab with 16 piles. Approximation of selected compressibility curves with Meyer curve to determine total consolidation settlement. Contribution of total settlement to settlement after 24 h.

**Figure 9.**Coefficient c

_{v}values for H

_{dr}= 10 cm as a function of load. Designations: mP10—median for slab, m4p10—median for 4 piles, m9p10—median for 9 piles, and m16p10—median for 16 piles.

**Figure 10.**Dependence of Uav as a function of time. Comparison of the empirical formula of Sivaram and Swamee (Equation (3)) [21] with test results for 142 kPa and 189 kPa loading. Power coefficients A, B, and C matched to the authors’ own test results (5).

**Figure 11.**Interpretation of preconsolidation stress σ’c on compressibility curve according to the LCPC method.

**Figure 12.**Diagram of pile placement under the foundation slab. Slab (

**a**) and slab–pile models with 4 (

**b**), 9 (

**c**), and 16 (

**d**) piles.

**Figure 13.**Comparison of model test results with FEM analysis results using the Hs model. Stress on a logarithmic scale.

**Figure 14.**Distribution of axial force and friction on the side along the column working alone. Settlement of the column head s = 0.01D, 0.03D, and 0.08D.

**Figure 15.**Distribution of axial force and friction on the side along the corner column. Plate and 4 columns. Axial spacing of columns r/D = 10. Settlement of column head s = 0.01 D, 0.03 D, and 0.08 D.

**Figure 16.**Distribution of axial force and friction on the side along the corner, edge, and center columns. Plate and 9 columns. Axial spacing of columns r/D = 5. Column head settlement s = 0.01D, 0.03D, and 0.08D.

**Figure 17.**Distribution of axial force and friction on the side along the corner, edge, and center columns. Plate and 16 columns. Axial spacing of columns r/D = 3.3. Column head settlement s = 0.01 D, 0.03 D, and 0.08D.

Column | 1 | 3 | 4 | 10 | 12 | 13 | 14 | 17 |
---|---|---|---|---|---|---|---|---|

s_{∞} | 2.73 | 1.23 | 1.99 | 1.05 | 1.90 | 1.84 | 1.10 | 1.00 |

D_{e} | 3.01 | 5.00 | 1.39 | 1.00 × ^{–11} | 4.99E | 3.80 | 3.65 | 2.85 |

p | 7.39 × 10^{−1} | 4.19 × 10^{–1} | 1.00 × 10^{–3} | 1.05 × 10^{1} | 6.19 × 10^{–1} | 2.34 × 10^{–1} | 3.72 × 10^{–1} | 3.10 × 10^{–1} |

α | 1.00 × 10^{–6} | 1.34 | 1.41 | 4.82 | 4.03 | 8.19 × 10^{–1} | 2.10 | 1.00 × 10^{–6} |

Column | 7 | 8 | 9 | 11 | 16 |
---|---|---|---|---|---|

s_{∞} | 5.32 × 10^{−1} | 7.10 × 10^{−1} | 1.01 | 6.17 × 10^{−1} | 7.77 × 10^{−1} |

D_{e} | 4.10 × 10^{−1} | 2.68 | 5.56 × 10^{−1} | 5.00 | 6.92 × 10^{−1} |

p | 1.00 × 10^{−3} | 3.11 × 10^{−1} | 8.72 × 10^{−2} | 4.99 × 10^{−1} | 6.61 × 10^{−2} |

α | 1.82 | 1.00 × 10^{−6} | 2.90 × 10^{−1} | 5.00 | 4.80 × 10^{−5} |

**Table 3.**Contribution of column settlement during the operation stage to the total settlement according to Meyer’s formula.

Column | 1 | 3 | 4 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 16 | 17 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

s_{e} | 0.08 | 0.03 | 0.24 | 0.13 | 0.10 | 0.54 | 0.04 | 0.01 | 0.00 | 0.06 | 0.04 | 0.42 | 0.12 |

s_{c} | 2.73 | 1.23 | 1.99 | 0.53 | 0.71 | 1.01 | 1.05 | 0.62 | 1.90 | 1.84 | 1.10 | 0.78 | 1.00 |

s_{e}/s_{c} | 0.03 | 0.02 | 0.12 | 0.25 | 0.14 | 0.53 | 0.04 | 0.01 | 0.00 | 0.03 | 0.03 | 0.54 | 0.12 |

Slab | Slab + 4 Piles | Slab + 16 Piles | |||
---|---|---|---|---|---|

142 kPa | 142 kPa | 142 kPa | |||

A | 0.434 | A | 0.396 | A | 0.387 |

B | 0.793 | B | 0.649 | B | 0.970 |

C | 0.543 | C | 0.601 | C | 0.366 |

189 kPa | 189 kPa | 189 kPa | |||

A | 0.456 | A | 0.494 | A | 0.441 |

B | 1.117 | B | 1.014 | B | 1.320 |

C | 0.399 | C | 0.487 | C | 0.325 |

γ (kN/m^{3}) | Material | E (kN/m^{2}) | ν | Thickness (m) |
---|---|---|---|---|

0 | Linear, isotropic | 30 × 10^{6} | 0.2 | 0.3 |

γ (kN/m^{3})
| Conditions | e_{initial} | E_{50}^{ref}(kN/m^{2}) | E_{edo}^{ref}(kN/m^{2}) | E_{ur}^{ref}(kN/m^{2}) | p_{ref} |

15 | drained | 0.5 | 67.2 × 10^{3} | 67.2 × 10^{3} | 134.4 × 10^{3} | 100 |

ν_{ur} | c
(kN/m^{2}) | ϕ
(^{ο}) | ψ
(^{ο}) | K_{0}^{NC} | Interface | R_{f} |

0.3 | 4 | 18.5 | 0 | 0.68 | 0.67 | 0.9 |

γ (kN/m^{3}) | Material | Type | E (kN/m^{2}) | ν | Interface | Diameter (m) |
---|---|---|---|---|---|---|

24 | Linear isotropic | Poreless | 30 × 10^{6} | 0.2 | 1.0 | 0.4 |

Type CPRF | Approximate Load Range (kPa) | Approximating Function | Coefficient of Determination R ^{2} |
---|---|---|---|

slab | 119–154 | y = 43.795ln(x) − 191.43 | 0.985 |

slab + 4 piles | 180–250 | y = 44.83ln(x) − 217.92 | 0.962 |

slab + 9 piles | 254–351 | y = 44.975ln(x) − 234.26 | 0.992 |

slab + 16 piles | 343–481 | y = 44.435ln(x) − 245.27 | 0.990 |

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**MDPI and ACS Style**

Kacprzak, G.; Frydrych, M.
Soil Consolidation Analysis in the Context of Intermediate Foundation as a New Material Perspective in the Calibration of Numerical–Material Models. *Constr. Mater.* **2023**, *3*, 414-433.
https://doi.org/10.3390/constrmater3040027

**AMA Style**

Kacprzak G, Frydrych M.
Soil Consolidation Analysis in the Context of Intermediate Foundation as a New Material Perspective in the Calibration of Numerical–Material Models. *Construction Materials*. 2023; 3(4):414-433.
https://doi.org/10.3390/constrmater3040027

**Chicago/Turabian Style**

Kacprzak, Grzegorz, and Mateusz Frydrych.
2023. "Soil Consolidation Analysis in the Context of Intermediate Foundation as a New Material Perspective in the Calibration of Numerical–Material Models" *Construction Materials* 3, no. 4: 414-433.
https://doi.org/10.3390/constrmater3040027