# Analysis Method for Laterally Loaded Pile Groups Using an Advanced Modeling of Reinforced Concrete Sections

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Proposed ‘BEM-Based’ Method

#### 2.1. Key Features of the Proposed Method

- a)
- pile-soil, pile-pile interactions are considered using the Mindlin’s solution;
- b)
- horizontally layered elastic soil;
- c)
- non-linear behavior for the reinforced concrete pile section;
- d)
- non-linear soil behavior (incremental analysis);
- e)
- the so-called shadowing effect, has been implemented in the code using an approach similar to that described in [35];
- f)

#### 2.2. Pile Modelling

- 20 blocks with a thickness Δ = D/8, starting from the ground level up to a depth of 2.5D;
- 10 blocks with a thickness Δ = D/4, starting from a depth of 2.5D up to a depth of 5D;
- 10 blocks with a thickness Δ = D/2, starting from a depth of 5D up to a depth of 10D;
- 10 blocks with a thickness Δ = D, starting from a depth of 10D up to a depth of 20D;
- 10 blocks with a thickness Δ = (L − 20D)/10, starting from a depth of 20D up to the pile base depth.

_{j}represents the load applied at the generic pile-block j (located at depth z

_{j}), and y

_{0}and θ

_{0}are the unknown displacement and rotation at the pile-head. Obviously if the pile-head is fixed, the rotation becomes a known term. Each pile-point displacement is a function of n + 2 (or n + 1, for fixed condition) unknowns, n pile-soil interface pressures, y

_{0}and θ

_{0}.

_{p}I

_{p}is assumed to be constant (which means hypothesizing a linear-elastic behavior of the section until the ultimate bending moment occurs). For reinforced concrete sections, the development of cracks, even at low values of the bending moment, requires a different modeling for the pile response. In this case, the “moment-curvature-axial load” relationship is obtained by a model that has the additional feature of taking the influence of tension stiffening into account [43].

_{p}I

_{p}, along the pile shaft. In Equation (3), the variation of both E

_{p}and I

_{p}along the shaft is fully considered by changing I

_{p}of the section, while E

_{p}is kept constant. Consequently, in an incremental analysis, the pile flexibility matrix needs to be updated at each load increment.

_{i}and z

_{j}represent respectively the distance between the fixed node in Figure 2 and the point along the beam in which the displacement is considered and the distance between the same fixed node and the point where the load is applied. On the other hand, l

_{k}, represents the distance between the fixed node and the lower part of block k, and E

_{p}I

_{k}is the flexural rigidity of block k.

#### 2.3. Soil Modelling

_{i}+ E

_{j})/2.

_{ij}at a point i belonging to the half space by a horizontal load P

_{j}applied at point j can be expressed as in Equation (4) (Figure 3). Where the term b

_{ij}represents the general expression for each soil flexibility matrix coefficient.

#### Soil Non-Linear Behavior

_{max}at zero strain and where the tangent is asymptotic to τ

_{max}at infinite strain.

_{ref}= τ

_{max}/G

_{max}) it was possible to rewrite the equation of a hyperbola as a normalized secant shear modulus (G

_{sec}/G

_{max}) that is reduced with a normalized shear strain (γ/γ

_{ref}), Equation (5).

_{f}is equal to 1, while the parameter g ranges between 0.25 and 1. The appropriate value for g, to perform the analysis, can be easily estimated by trying to obtain the best fit with the load-deflection curve of a lateral load test on a single pile or with the load-deflection curve obtained with other available codes [12,14,28,30,34,42].

_{max}and the Poisson ratio. While the input data to define the soil resistance are: the angle of friction or the undrained shear strength for cohesionless or cohesive, respectively.

#### 2.4. Influence of Suction on Pile Group Response to Horizontal Loading

_{c}

_{0}in the porous medium. The role of this parameter is the same as the average capillary rise in the original model developed by Kovacs and is calculated using the expression for the rise of water in a capillary tube (h

_{c}) with a diameter d.

_{10}(in cm) is the diameter corresponding to 10% passing on the grain-size distribution curve, C

_{U}is the coefficient of uniformity (=D

_{60}/D

_{10}), and e is the void ratio.

_{L}is the liquid limit, and ρ

_{s}is the solid grain density (kg/m

^{3}).

_{r}(or volumetric water content θ) and the matric suction ψ. The model considers that water is held by capillary forces responsible for capillary saturation S

_{c}, and by adhesive forces, causing saturation by adhesion S

_{a}. The S

_{c}component is more important at relatively low suction values, while the S

_{a}component becomes dominant at a higher suction when most capillary water has been withdrawn. The relationship proposed in the MK-Model is written as in Equation (11) for the degree of saturation:

_{a}* is introduced in place of S

_{a}used in the original model. The contribution of the capillary and the adhesion components to the total degree of saturation is defined as a function of h

_{c}

_{0}and ψ using the equations reported in Aubertin et al. [48].

#### 2.5. Group Effects Modelling

_{g}) due to passive wedge interference is evaluated based on an empirical relationship Equation (12), which provides good agreement with field test results [35].

_{j}is determined from all the neighboring piles (sides and front piles) of the pile in question (Figure 5).

_{i}value on the right-hand side of Equation (12), which represents the SL of the single isolated pile, for cohesionless soils in the Strain Wedge model, is defined in Equation (13).

_{hf}(or the deviatoric stress at failure in the triaxial test) is $\Delta {\sigma}_{hf}={\sigma}_{v0}\left[{\mathrm{tan}}^{2}\left({45}^{\xb0}+\phi /2\right)-1\right]$. However, in the proposed method it is assumed that $S{L}_{i}\cong p/{p}_{ult}$, and thus:

_{m}, can be easily obtained if SL

_{i}is known, which is assumed to be approximately equal to the ratio $p/{p}_{ult}$.

_{g}vary with depth and level of loading. They can be used to evaluate the increased value of the pressure at each pile-soil interface (${p}_{g}$) (where this increase is caused by the interferences of the passive wedges) Equations (15) and (16).

_{g}.

_{m}is always 0°. This means that the base angle of the passive wedge, for cohesive soils, is constantly equal to 45° and only the dimension in depth (and thus on the plain) of the passive wedge changes when the load increases. However, in this way only the interaction between the wedge of a pile positioned in a row different from the front row with the wedge of the pile located in front of it can be considered, and thus the interactions between the wedges of piles belonging to the same row are neglected (Figure 6).

_{a}, for shallow depths, could be negative and this means that the soil is in tension. Assuming reasonably that the soil cannot support tension all the values of p

_{a}< 0 are corrected considering directly p

_{a}= 0. The difference between p

_{p}and p

_{a}thus represents the ultimate soil pressure profile, p

_{r}, (in terms of force per unit length) acting along a pile shaft in a row of piles with a spacing of 1D (or on a retaining wall). For example, Figure 7 shows all these steps to define the resulting lateral pressure profile, p

_{r}, considering a homogenous cohesive soil with a constant c

_{u}equal to 50 kPa, a soil unit weight γ equal to 20 kN/m

^{3}and a pile diameter D = 1 m.

_{r}(for a spacing of 1D) and p

_{ult}(for a single isolated pile).

_{r}(assumed for spacing ratio s/D = 1) and the profile p

_{ult}(assumed for spacing ratio s/D ≥ 6).

_{ult,def}), for spacing ratio between 1 and 6, it is assumed that p

_{ult,def}can be expressed as a function of the actual spacing ratio s/D and the depth, z, using this relationship in Equation (20).

_{ult,def}values.

#### 2.6. Solution System

_{0}is the pile-group displacement, θ

_{m}are the m pile-heads rotations, H

_{m}are the m horizontal loads at the pile-heads and [P] is the known-terms vector (with the same dimension as for the vector [X]). [F] is a (km + 2m + 1) × (km + 2m + 1) or (km + m + 1) × (km + m + 1) matrix, obtained by summing:

- the km × km pile flexibility matrix [F
_{P}], composed of the a_{ij}coefficients; - the km × km flexibility matrix [F
_{S}], composed of the b_{ij}coefficients that represent the displacements induced by a load acting at the pile-soil interface j to the pile-soil interface i.

_{P}], is updated only in the case of a non-linear “moment-curvature” relationship for the pile section. [F

_{P}] is updated using the tangent flexural rigidities, according to the bending-moments reached at each pile-node in the previous load increment.

_{k}, an iterative process is performed where two solutions are obtained, the first using h

_{k}as the load increment, the second using two load steps equal to h

_{k}/2. The iterative scheme is described in Figure 9, which, for the sake of simplicity refers to the explicit Euler method with step-doubling and adaptive step-size control. However, a fourth order Runge-Kutta method can also be used to obtain some improvement in the accuracy of the solution. The adaptive step-size control numerical technique is fully described in Press et al. [49].

_{1}and Δu

_{2}are the incremental displacement at the pile-head evaluated using one and two steps, respectively. The ε value is compared with a predefined tolerance taken as equal to 0.001 (Figure 10).

_{k}

^{new}which should be able to achieve the desired accuracy and can be estimated using Equation (24) [49].

## 3. Validation of the Proposed Method

#### 3.1. Analysis Results with the Proposed BEM Method for a Specific Lateral Load Test on a Bored Pile Group

#### 3.1.1. Soil and Pile Properties Description

_{max}data profiles.

^{2}) were realized using bentonite-mud with reverse circulation. Two of the 13 bored piles were realized by means of a drilling device with hydraulic oscillator at full length. The measurement instruments (strain gauges and inclinometers) were attached to the longitudinal reinforcement bars, inserted into the hole before casting the concrete. Bored pile properties are summarized in Table 3.

#### 3.1.2. Single Bored Pile B7 (Free-Head) and Pile Group (Fixed-Head): Analysis Results

_{max}profile was that provided in [4]. This profile was simplified and assumed linearly increasing from 15 to 150 MPa. The Poisson ratio was set equal to 0.35.

_{group}/(n H

_{single}); where, H

_{group}= the total horizontal load in the pile group, H

_{single}= the horizontal load in the isolated single pile (at the same displacement-level) and n = the number of piles in the group.

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 4.**Mobilized soil passive wedges and pile-group interaction scheme (similarly as described in the context of the so-called Strain Wedge Model [35]).

**Figure 5.**Lateral interaction for a specific pile in the group (similarly as described in the context of the so-called Strain Wedge Model [35]).

**Figure 11.**Measured vs. computed horizontal loads (H) for a given normalized displacement (y/D): (

**a**) y/D = 0–0.5%; (

**b**) y/D = 1.0–2.0%; (

**c**) y/D = 4.0–5.0%; (

**d**) y/D = 8.0–10.0%.

**Figure 16.**Computed vs. measured deflection profiles at H = 10948 kN of piles B2, B5, B6, B8, B9, B10.

Case Reference | Pile Material | Pile Diameter (m) | Pile Length (m) | Soil Type | H_{max} (kN) |
---|---|---|---|---|---|

[51] 3 × 3; s = 3D | Steel with Grout-fill | 0.273 | 13.11 | OC Clay | 695 |

[5] 3 × 3; s = 3D | Steel with Grout-fill | 0.273 | 13.11 | Sand | 808.5 |

[4] 3 × 2; s = 3D | Bored RC | 1.5 | 34.9 | Silty Sand | 11043 |

[8] ϕ’ = 34°; 3 × 3; s =3D | Aluminum | 0.43 | 13.3 | Sand | 761.2 |

[8] ϕ’ = 39°; 3 × 3; s = 3D | Aluminum | 0.43 | 13.3 | Sand | 1508.2 |

[8] ϕ’ = 34°; 3 × 3; s = 5D | Aluminum | 0.43 | 13.3 | Sand | 1110.5 |

[8] ϕ’ = 39°; 3 × 3; s = 5D | Aluminum | 0.43 | 13.3 | Sand | 1424 |

[52] 2 × 1; s = 2D | Aluminum | 0.72 | 12 | Sand | 1183 |

[52] 2 × 1; s = 4D | Aluminum | 0.72 | 12 | Sand | 1220.1 |

[52] 2 × 1; s = 6D | Aluminum | 0.72 | 12 | Sand | 1030.72 |

[53] 3 × 3; s = 3D | Steel with Grout-fill | 0.305 | 8.7 | Clay | 927.05 |

[11] 3 × 3; s = 3D | Steel pipe | 0.324 | 11.5 | Sand | 488.6 |

[54] 3 × 3; s = 5.65D | Steel pipe | 0.324 | 11.9 | Clay | 1407 |

[54] 3 × 4; s = 4.4D | Steel pipe | 0.324 | 11.9 | Clay | 1353.8 |

[54] 3 × 5; s = 3.3D | Steel pipe | 0.324 | 11.9 | Clay | 1942.5 |

Case | E_{p}I_{p} (MNm^{2}) | γ (kN/m^{3}) | ϕ (°) | D_{R} (%) | c_{u} (kPa) | E_{max} (Linear Increasing with Depth) (MPa) | G (-) | F (m) | W.T. (m) | Head B.C. |
---|---|---|---|---|---|---|---|---|---|---|

[51] | 16.0 | 19.0 | - | - | 58–145 (0–5.5 m) | 70–200 (0–5.5 m) | 0.25 | 0.305 | 0.0 | Free |

[5] | 16.0 | 19.5 | 47 | 90 | - | 35–100 (0–2.0 m) | 1.0 | 0.305 | 0.0 | Free |

[4] | variable | 18.5 | 34 | 50 | - | 40–400 (0–34.9 m) | 0.5 | 1.0 | 1.0 | Fixed |

[8] | 72.1 | 14.51 | 34 | 33 | - | 60–300 (0–13.3 m) | 0.25 | 1.68 | - | Free |

[8] | 72.1 | 15.18 | 39 | 55 | - | 50–260 (0–13.3 m) | 0.5 | 1.68 | - | Free |

[52] | 514.0 | 16.3 | 40 | 89 | - | 40–200 (0–12.0 m) | 1.0 | 1.6 | - | Free |

[53] | 26.91 | 19.0 | - | - | 50–75 (0–2.9 m) | 60–170 (0–8.7 m) | 0.25 | 0.4 | 0.0 | Free |

[11] | 30.03 | 19.5 | 40 | 44 | - | 20–150 (0–11.5 m) | 0.25 | 0.86 | 0.0 | Free |

[54] | 30.03 | 19.0 | - | - | 60 (0–1 m)120 (1.0–4.0 m) | 50–60 (0–4.0 m) | 0.25 | 0.48 | 1.0 | Free |

_{p}I

_{p}= pile flexural rigidity; γ = soil unit weight; ϕ = peak friction angle; D

_{R}= relative density; c

_{u}= undrained shear strength; E

_{max}= soil elastic modulus at small strain level; g = parameter of the modulus reduction curve; f = load eccentricity; W.T. = water table depth below the ground surface; Head B.C. = pile-head boundary conditions (free-head or fixed head).

Pile Diameter D (mm) | Pile Length (m) | Cross Sectional Area (cm^{2}) | Concrete Compressive Strength f’_{c} (MPa) | Reinforcement Yield Stress f_{y} (MPa) | Steel Ratio ρ_{s} | Intact Flexural Rigidity EI (GNm^{2}) |
---|---|---|---|---|---|---|

1500 | 34.9 | 17672 | 27.5 | 471 | 0.025 | 6.86 |

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**MDPI and ACS Style**

Stacul, S.; Squeglia, N.
Analysis Method for Laterally Loaded Pile Groups Using an Advanced Modeling of Reinforced Concrete Sections. *Materials* **2018**, *11*, 300.
https://doi.org/10.3390/ma11020300

**AMA Style**

Stacul S, Squeglia N.
Analysis Method for Laterally Loaded Pile Groups Using an Advanced Modeling of Reinforced Concrete Sections. *Materials*. 2018; 11(2):300.
https://doi.org/10.3390/ma11020300

**Chicago/Turabian Style**

Stacul, Stefano, and Nunziante Squeglia.
2018. "Analysis Method for Laterally Loaded Pile Groups Using an Advanced Modeling of Reinforced Concrete Sections" *Materials* 11, no. 2: 300.
https://doi.org/10.3390/ma11020300