# The Use of Polyurethane Injection as a Geotechnical Seismic Isolation Method in Large-Scale Applications: A Numerical Study

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Soil Improvement Using Polyurethane

#### 2.1. Master Builders Solutions’ Technologies for Soil Improvement through Grouting

_{f}/V

_{i}= ρ

_{i}/ρ

_{f}

_{i}and V

_{f}respectively being the initial and final volume of the polyurethane. By assuming a radial expansion of the material, the configuration of the injected layers is shown in Figure 1b, where the diameter of the expanded polyurethane will depend on CE. Thanks to the mass conservation, the volume ratio of Equation (1) corresponds to a density ratio of initial ρ

_{i}versus final ρ

_{f}.

## 3. Dynamic Characterisation of Polyurethane by Means of Laboratory Tests

#### 3.1. Sample Preparation

^{3}) and B (isocyanate, density 1250 kg/m

^{3}) are mixed in a volumetric ratio of 1:1, as suggested by the Master Builders Solutions’ technical sheet [42], and 20 mL syringes are used for the volumetric control (Figure 2a). The two components are withdrawn separately, injected into a plastic cup and mixed using a propeller-equipped drill (Figure 2b) for an adequate mixing time to obtain a homogeneous foam (about 40 s, compatible with the setting time, after which the foam is not workable anymore). The mixture is then poured inside a cylindrical mould of height 0.10 m and diameter 0.045 m (volume 1.59 × 10

^{−4}m

^{3}).

_{f}) and changing only the initial volume, V

_{i}(see Equation (1)). This is done thanks to the hand press shown in Figure 2c. Specifically, V

_{i},

_{A}= V

_{i},

_{B}= 15 mL for CE = 6, V

_{i},

_{A}= V

_{i},

_{B}= 11.25 mL for CE = 8 and V

_{i},

_{A}= V

_{i},

_{B}= 10 mL for CE = 9, with V

_{i},

_{A}and V

_{i},

_{B}being the initial volume of each component.

_{A}

_{+ B}, depends on the densities and the initial volumes of A- and B-components, which allow to compute the initial volume of the polyurethane as:

_{i}= 2 m

_{A}

_{ + B}/(ρ

_{A}+ ρ

_{B})

_{i}is computed for each specimen, and then the CEs are determined. CE values are shown in Table 1, and the specimen density is also reported.

#### 3.2. Impact Tests

_{1}(f), computed by means of the Fourier transform of the measured force and acceleration, respectively S

_{f}(in N) and S

_{a}(in m/s

^{2}):

_{f}; and H

_{1}is therefore expressed in m/s

^{2}/N. In the following, the experimental results’ elaboration is described, for the evaluation of the elastic modulus and the damping coefficient of the specimens. In both cases, the first natural frequency identified, f

_{1}(fundamental frequency), is considered, depending on the axial modal behaviour of the specimens for the test performed.

_{1}, is equal to 1/(2π)$\sqrt{\hat{k}/m}$, with m the specimen mass and $\hat{k}$ a stiffness, taking into account not only the longitudinal stiffness of the sample (EA/L, with E the elastic modulus, A the cross-section area and L the length of the specimen) but also the ring binding the specimen; because of the impact, the boundary exerts an elastic reaction depending on a stiffness k

_{str}. $\hat{k}$ is therefore evaluated as the difference, EA/L−k

_{str}; from f

_{1}expression, the elastic modulus, E, is derived, simplified as:

^{3}), f is the fundamental frequency (in Hz), while $C$ is a calibration factor including both k

_{str}and the physical and geometrical characteristics of the specimen in the examination (ρ, A and L). The length, L, is expressed in metres. To define an expression for C and calibrate the procedure of experimental data elaboration, the impact tests are initially conducted on steel and aluminium specimens, i.e., materials of known characteristics, of dimensions similar to those of polyurethane specimens.

_{1}, it is possible to evaluate the damping coefficient associated to each system’s mode of vibration through the half-power bandwidth method, whose validity is proven for frequency response functions symmetric around the natural frequencies [36,44,45]. Specifically, for the evaluation of the damping coefficient associated to the first mode of vibration, after having detected the fundamental frequency, f

_{1}, the method consists of determining the two frequencies, f

_{a}and f

_{b}, where the value assumed by H

_{1}is $1/\sqrt{2}{H}_{1}$(f

_{1}), and the damping coefficient is:

## 4. The Finite Element Numerical Model

_{R}, was considered:

- Loose sand (LS), with D
_{R}= 15–35% - Medium sand (MS), with D
_{R}= 35–65% - Medium-dense sand (MDS), with D
_{R}= 65–85% - Dense sand (DS), with D
_{R}= 85–100%

#### 4.1. Model Description

_{b}= 1 m

^{2}, H = 30 m high, discretised with 3D 9-node Standard Brick elements, each of them with a dimension 0.5 × 0.5 × 1 m

^{3}(Figure 4a). The finite element dimensions respect the Lysmer’s relationship [46], so that Δz < v

_{s}/10 f

_{max}(for the cases in exam, ${v}_{s}$ = 80 ÷ 250 m/s and f

_{max}= 10 Hz). Two analysis phases are considered: a gravity analysis first, followed by a dynamic time-history analysis, in which a seismic event is simulated.

^{3}and 1000 m/s. A uniform shaking was simulated tying the force application node with the remaining base nodes, with master-slave conditions for x- and y-translations (Figure 4b). Periodic boundary conditions were also introduced by tying the nodes at the same vertical coordinate for x- and y-translation.

_{s}= 2 s and ${u}_{0}$ = 0.5 m. The velocity, $\dot{u}\left(t\right)$, for the viscous input force is the first derivative of ${u}_{R}$.

_{1}ω

_{2})/(ω

_{1}+ ω

_{2}) and β = (2$\xi $)/(ω

_{1}+ ω

_{2}), with ω

_{1}and ω

_{2}, two target frequencies evaluated as ω

_{1}= πv

_{s}/2H and ω

_{2}= 5πv

_{s}/2H, i.e., the first and the third natural frequency of a theoretical amplification function for a visco-elastic deposit [52].

#### 4.2. The Pressure-Dependent Multi-Yield Model (PDMY)

^{2}), at a distance of 0.75 m from one another and 0.125 m from the edge. Both two and three injection levels were chosen, resulting in injected surficial depths, h, equal to 2 and 3 metres. The polyurethane mass injected at each level, identified as ${m}_{\mathit{PUR}}$, was set equal to 10 and 15 kg.

^{3}. Note that ${m}_{\mathit{PUR}}$ is considered as the mass injected every metre of depth, which is why the final volume introduced in the formula is 1 metre high. Since the relationship between CE and the soil confinement is not known for the soil under examination, CE is varied parametrically assuming CE = 5, 8 and 10, corresponding to rigid foam density ρ

_{PUR}= 240, 150 and 120 kg/m

^{3}(see Equation (1)).

_{PUR}calculated through Equation (10) are reported in Figure 5, together with the area of the expanded polyurethane A

_{PUR}contained in the 1 m

^{2}column’s cross-section area.

_{PUR}and A

_{PUR}. The PDMY model was also used for this material. To evaluate the parameters of the composite material, a homogenisation procedure is introduced; specifically, for density and damping, a linear homogenisation is considered, in the form:

#### 4.3. Calibration of the PDMY Constitutive Model for the Sand–Polyurethane Composite Material

^{3}) and sand–polyurethane, with the polyurethane introduced at layers of 15, 25 and 45 mm thick, giving rise to different volumetric percentages of the polyurethane in the specimen.

_{0}-γ, the latter obtained from Equation (7) for the PDMY material. The numerical curve is shown to well-interpret the experimental results.

_{0,sand}) and pure polyurethane (G

_{0,PUR}) according to the polyurethane volumetric percentage (%PUR) in the specimen. This is necessary for the evaluation of the G

_{ref}associated to each case of Figure 5.

_{0}/G

_{0,sand}was computed for each confining pressure (p’

_{c}= 100, 200 and 300 kPa), with G

_{0}the small-strain shear modulus for the generic specimen. The experimental points G

_{0}/G

_{0,sand}-%PUR are plotted in Figure 7a, showing an exponential trend. The equation of a negative exponential distribution was derived for G

_{0}-%PUR, so that G

_{0}is equal to G

_{0,sand}and G

_{0,PUR}respectively, for %PUR = 0 and %PUR = 100.

_{0}, G

_{0,sand}and G

_{0,PUR}are expressed in MPa, and it is represented by the lines in Figure 7a, showing an acceptable comparison with the experimental points.

_{0}depends on the confining pressure through G

_{0,sand}, since G

_{0,PUR}is quite independent from the confining pressure [35]. For the numerical model, a homogenised shear modulus will be computed at a reference pressure; then, it is updated according to the specific model pressure through Equation (9).

_{0}values experimentally obtained for the generic confining pressure, p’, were normalised with respect to the shear modulus evaluated at p’

_{ref}= 100 kPa, and the results are shown in Figure 7b, together with curves derived from Equation (9), with d values giving the best fit. d was observed to diminish by increasing the polyurethane volumetric percentage, going towards d = 0 for pure polyurethane specimen, whose stiffness is independent from the confining pressure. A linear expression describes the variation d (dimensionless) with the polyurethane percentage:

## 5. Results

#### 5.1. Elastic Modulus and Damping Coefficient for Polyurethane MP355 at Different Densities, Evaluated through Impact Tests

_{1}, evaluated through Equation (3) from the impact tests performed on steel and aluminium specimens: the fundamental frequencies are f

_{1,s}= 157.88 Hz and f

_{1,al}= 205.08 Hz. These results are used to define the k

_{str}value, by $\left[\left(\frac{E}{L}{-\text{}4\pi}^{2}{\rho \mathit{Lf}}^{2}\right)A\right]$. Knowing the elastic moduli for steel (E

_{s}= 210 GPa) and aluminium (E

_{al}= 70 GPa), values of k

_{str,s}= 3.34 × 10

^{10}N/m and k

_{str,a}= 1.11 × 10

^{10}N/m were obtained. k

_{str}is therefore not constant, by changing the material specimen, that which is constant is the ratio k

_{str}/E. An empirical expression was derived for C (see Equation (4)), applicable knowing only the specimen geometry (area A and height L) and physical characteristics (density $\rho $).

_{s}is the steel density. The elastic modulus predicted for the aluminium specimen is 69.9 MPa, comparable with its typical value.

_{1}computed from the experimental results of impact tests conducted on the polyurethane specimens are shown in Figure 8b; through Equations (4) and (14), the fundamental frequency and the density of each specimen (Table 1) allow us to evaluate the elastic modulus, E

_{PUR}.

_{PUR}in MPa and ρ

_{PUR}in kg/m

^{3}. On the other hand, pairs ${\xi}_{\mathit{PUR}}$−ρ

_{PUR}stay on a horizontal line (${\xi}_{\mathit{PUR}}\cong 0.05\forall {\rho}_{\mathit{PUR}}$).

#### 5.2. Seismic Response of Composite Material

_{ref}= 80 kPa were used (Table 2).

_{PUR}, is evaluated through Equation (15), being E

_{PUR}= 300 MPa for CE = 5, E

_{PUR}= 187.5 MPa for CE = 8 and E

_{PUR}= 150 MPa for CE = 10. ${G}_{0,\mathit{PUR}}$ is then derived through the elastic theory (${G}_{0,\mathit{PUR}}$ = E

_{PUR}/2/(1 + ν

_{PUR})), with ν

_{PUR}equal to 0.38.

^{2}respectively, for pure soil, and the injected soil with h = 2 m and h = 3 m. Minor acceleration reductions were observed for dense sand and CE = 5 (maximum accelerations are 8.71, 8.67 and 8.65 m/s

^{2}, for pure soil, h = 2 m and h = 3 m).

^{2}for m

_{PUR}= 10 kg and m

_{PUR}= 15 kg. It was observed that injecting a greater amount of polyurethane per level provides bigger reductions.

_{f}, were also evaluated, to analyse the effects of polyurethane injection on the frequency response of the deposit; particularly, A

_{f}is computed by dividing the Fourier spectrum of surficial and base accelerations. Figure 13 shows A

_{f}of the models having h = 3 m, with m

_{PUR}equal to 10 and 15 kg. A slight increase of the fundamental frequency was observed in the composite models.

## 6. Discussion

_{F}, and it is equal to:

^{2}) are the total height and the cross area of the homogeneous soil deposit and ${m}_{\mathit{PUR}}$ is expressed in kg. Each case numerically analysed can be described by a value of I

_{F}. By representing the acceleration reduction with the injection factor, Figure 14 was obtained.

_{soil}needs to be derived.

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Polyurethane injection scheme underneath foundation slabs: (

**a**) Application example realised by Bequadro S.r.l., and (

**b**) schematisation of polyurethane injections as cylinders (green) in the soil matrix (red).

**Figure 2.**Sample preparation: (

**a**) Sampling of the two components with syringes (Volume ratio 1:1), (

**b**) injection of both components into a plastic cup being mixed through a propeller-equipped drill, (

**c**) outpour of the mixed foam inside a mould, closed by a hand press, and (

**d**) specimens for tests.

**Figure 4.**FE numerical modelling: (

**a**) 3D model and finite element adopted, (

**b**) input application at an end node of the base, with master-slave node for uniform excitation introduction, and (

**c**) modelling of a viscous-elastic bedrock.

**Figure 5.**Summary of the injection cases analysed. The polyurethane diameter and area are shown, corresponding to each considered combination injected mass per level, ${m}_{\mathit{PUR}}$—expansion coefficients CE.

**Figure 6.**Numerical (Num) vs. experimental (Exp) comparison of the shear modulus decay for composite sand–polyurethane samples at three thicknesses (15, 25 and 45 mm) and for three confining pressures.

**Figure 7.**Homogenisation procedure of the small-strain parameters for the composite specimens. (

**a**) Experimental (points) vs. analytical (lines) G

_{0}/G

_{0,sand}varying with the polyurethane volumetric percentage. (

**b**) Calibration of d exponent for shear modulus variation with the isotropic stress.

**Figure 8.**Frequency response functions, H

_{1}, for (

**a**) steel and aluminium specimens and (

**b**) polyurethane specimens at different densities. In both figures, the marker shows the fundamental frequencies; in Figure (

**a**), the application of the half-power bandwidth method for the damping evaluation is also illustrated.

**Figure 9.**Parameters obtained from the elaboration of impact test results. Variation with polyurethane density of (

**a**) the elastic modulus and (

**b**) the damping coefficient.

**Figure 10.**Time-history accelerations recorded on numerical models of pure cohesionless soil at different relative densities and on “modified” cohesionless soil: evidence of the effects of the thickness of the modified layer (m

_{PUR}= 15 kg injected per level).

**Figure 11.**Time-history accelerations recorded on numerical models of pure cohesionless soil at different relative densities and on “modified” cohesionless soil: evidence of the effects of the mass injected per level (h = 3 m).

**Figure 14.**Reduction of surficial accelerations in relation to the injection factor, representative of the number of injections, mass of injected polyurethane, depth of injected soil, as well as the soil stiffness.

Specimen | Initial CE | Mass, m_{A + B} (g) | V_{i}(mL) | Final CE | Density, ρ_{PUR} (kg/m ^{3}) |
---|---|---|---|---|---|

A | 6 | 25 | 22 | 7.16 | 157 |

B | 6 | 26 | 23 | 6.88 | 164 |

C | 6 | 24 | 21 | 7.45 | 151 |

D | 6 | 25 | 22 | 7.16 | 157 |

E | 8 | 21 | 19 | 8.52 | 132 |

F | 8 | 19 | 17 | 9.41 | 120 |

G | 8 | 22 | 20 | 8.13 | 138 |

H | 8 | 22 | 20 | 8.13 | 138 |

I | 9 | 20 | 18 | 8.94 | 126 |

L | 9 | 19 | 17 | 9.41 | 120 |

M | 9 | 18 | 16 | 9.94 | 113 |

N | 9 | 19 | 17 | 9.41 | 120 |

**Table 2.**Density, ρ, elastic properties (shear and bulk moduli, G

_{ref}and B

_{ref}) at a reference confining pressure (80 kPa) and friction angle, ${\phi}^{\u2019},$ for the PDMY material.

Cohesionless Material | ||||
---|---|---|---|---|

LS | MS | MDS | DS | |

Ρ (t/m3) | 1.7 | 1.9 | 2 | 2.1 |

G_{ref} (MPa) | 55 | 75 | 100 | 130 |

B_{ref} (MPa) | 150 | 200 | 300 | 390 |

${\phi}^{\u2019}$(°) | 29 | 33 | 37 | 40 |

CE = 5 | CE = 8 | CE = 10 | ||||
---|---|---|---|---|---|---|

m_{PUR} = 10 kg | m_{PUR} = 15 kg | m_{PUR} = 10 kg | m_{PUR} = 15 kg | m_{PUR} = 10 kg | m_{PUR} = 15 kg | |

LS | 62 | 65 | 58 | 59 | 55 | 55 |

MS | 80 | 82 | 73 | 73 | 69 | 67 |

MDS | 101 | 102 | 91 | 88 | 84 | 80 |

DS | 126 | 125 | 111 | 106 | 102 | 94 |

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**MDPI and ACS Style**

Gatto, M.P.A.; Lentini, V.; Castelli, F.; Montrasio, L.; Grassi, D. The Use of Polyurethane Injection as a Geotechnical Seismic Isolation Method in Large-Scale Applications: A Numerical Study. *Geosciences* **2021**, *11*, 201.
https://doi.org/10.3390/geosciences11050201

**AMA Style**

Gatto MPA, Lentini V, Castelli F, Montrasio L, Grassi D. The Use of Polyurethane Injection as a Geotechnical Seismic Isolation Method in Large-Scale Applications: A Numerical Study. *Geosciences*. 2021; 11(5):201.
https://doi.org/10.3390/geosciences11050201

**Chicago/Turabian Style**

Gatto, Michele Placido Antonio, Valentina Lentini, Francesco Castelli, Lorella Montrasio, and Davide Grassi. 2021. "The Use of Polyurethane Injection as a Geotechnical Seismic Isolation Method in Large-Scale Applications: A Numerical Study" *Geosciences* 11, no. 5: 201.
https://doi.org/10.3390/geosciences11050201