# Orthogonal Set of Indicators for the Assessment of Flexible Pavement Stiffness from Deflection Monitoring: Theoretical Formalism and Numerical Study

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## Abstract

**:**

## 1. Introduction

#### 1.1. Use of Deflection Measurements

#### 1.2. Recall of the Various Means for Conducting Deflection Measurements

## 2. Construction of Indicators to Assess the Individual Stiffness of Pavement Layers

#### 2.1. Pavement Model for the Determination of Indicators

#### 2.2. Proposed Indicators and Constraints for Their Determination

- $l\in Inde{x}_{u}$
- ${p}_{l}\left(x\right)$ = “Weighting functions” (or distributions) defined on $\mathcal{M}$
- ${\int}_{\mathcal{M}}^{}f\left(x\right)dx$ = linear form for functions $f$from $\mathcal{M}$ to $\mathbb{R}$, defined as either:
`o`- ${\int}_{\mathcal{M}}^{}f\left(x\right)dx={\int}^{}{1}_{\mathcal{M}}\left(x\right)f\left(x\right)dx$,$$\approx {\displaystyle \sum}_{{x}_{i},{x}_{i+1}\in \mathcal{M}}^{}\frac{\left(f\left({x}_{i}\right)+f\left({x}_{i+1}\right)\right)}{2}\left({x}_{i+1}-{x}_{i}\right)$$
`o`- Or: ${\int}_{\mathcal{M}}^{}f\left(x\right)dx=\sum _{i\in \mathcal{M}}f\left({x}_{i}\right)$in the case of discrete measurements

- $f\cdot g={\int}_{\mathcal{M}}^{}f\left(x\right)g\left(x\right)dx$ (=$\sum _{i\in \mathcal{M}}f\left({x}_{i}\right)g\left({x}_{i}\right)$ in the discrete case) = scalar product of functions $f,g$ defined on $\mathcal{M}$ and related to the norm assumed to be finite: $\Vert f\Vert =\sqrt{f.f}=\sqrt{{\int}_{\mathcal{M}}^{}{f}^{2}\left(x\right)dx}(={(\sum _{i\in \mathcal{M}}f{\left({x}_{i}\right)}^{2})}^{\frac{1}{2}}$ in the discrete case)

- Indicator maximizes the sensitivity of the deflection measurements to the stiffness of layer #$l$ (condition #1).
- Indicator is “weakly” sensitive to the stiffness of the other layers #$j$ for $j\in \backslash $ (condition #2). The best case would be for indicators to be independent of the stiffness of the other layers #$j\left(j\ne l\right)$ (orthogonal indicator).
- The functions are imposed to have a finite norm $=<+\infty $, in avoiding infinite values for (condition #3).
- The values that is the magnitude of functions are chosen to give a direct physical meaning to the indicators (condition #4).

#### 2.3. Determination of the Weighting Functions

#### 2.4. Variations of Indicators ${I}_{l}$ along a Given Route

## 3. Numerical Applications of the Method (Theoretical Examples)

_{1}, BM

_{2}), lying on an unbound granular material subgrade layer (UGM) and rigid bedrock 6 m deep. Presence of such bedrock is a common hypothesis in pavement design. Indeed experiments using anchored sensors during the 1960s confirmed that deflection could be assumed null at a depth of 6 m. Large variations in pavement stiffness are considered along the simulated route. Variations lie within a realistic range of stiffnesses, as observed namely by:

- Variations in the Young’s modulus of the upper base layer between 3000 and 18,000 MPa.
- Variations in the Young’s modulus of the subgrade layer between 20 and 200 MPa.

_{1}(${E}_{ref}=9000\mathrm{MPa})$ and UGM (${E}_{ref}=50\mathrm{MPa})$layers. We also confirmed that the subgrade layer indicator remains nearly constant in this interval with respect to large variations of the base layer stiffness (see orthogonality). This property has also been verified for the base layer indicator, but only for small variations in subgrade stiffness. This limitation is due to the high sensitivity of the deflection bowl relative to subgrade stiffness, as shown in Figure 2.

_{1}(3000 MPa) and UGM (20 MPa). For example

#### 3.1. Local Variations of E-Moduli (Theoretical Application Example)

_{1}) or subgrade layer (UGM), but the variations here have the shape of a descending staircase as shown in Figure 6. Only small changes in the deflection bowls can be observed among the various stiffness conditions. The maximum deflection varies by just 10 µm in the first case and 50 µm in the second. Therefore, for conventional deflection indicators (Table 1), the pavement structure appears to be nearly homogeneous. Regardless of the conventional parameter chosen (e.g., Dmax, RoC, BLI, etc.), the variations are less than 2% for Young’s modulus variations in the base layer and 5% for variations in the subgrade layer.

#### 3.2. Sensitivity of the Indicators to Measurement Errors

## 4. Possible Extensions to the Method

#### 4.1. Model with Interface Shear Stiffness

#### 4.2. Visco-Dynamic Models for FWD or HWD Measurements

#### 4.3. Application to Structural Health Monitoring with Embedded Sensors

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Sensitivity of the Optimized Indicators to Deflection Measurement Uncertainties

Weighting Function | Configuration with 2 Geophones Position of Geophones (cm) | Norm of Indicators | ||||||
---|---|---|---|---|---|---|---|---|

G1 | G2 | |||||||

0 | 30 | |||||||

Weighting coefficients | ||||||||

${p}_{\mathrm{BM}1}^{\left(\mathrm{FWD}2\right)}$ | −550 | 566 | 789 | |||||

Weighting coefficients | ||||||||

${p}_{\mathrm{UGM}}^{\left(\mathrm{FWD}2\right)}$ | 0.1568 | −0.2946 | 0.33 | |||||

Weighting function | Configuration with 7 geophones Position of geophones (cm) | Norm of indicators | ||||||

G1 | G2 | G3 | G4 | G5 | G6 | G7 | ||

0 | 20 | 30 | 45 | 60 | 90 | 120 | ||

Weighting coefficients | ||||||||

${p}_{\mathrm{BM}1}^{\left(\mathrm{FWD}7\right)}$ | −249 | −88 | −17 | 49 | 95 | 145 | 159 | 357 |

Weighting coefficients | ||||||||

${p}_{\mathrm{UGM}}^{\left(\mathrm{FWD}7\right)}$ | 0.0438 | 0.0010 | −0.0176 | −0.0345 | −0.0457 | −0.0567 | −0.0578 | 0.11 |

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**Figure 1.**Typical elastic multi-layer structure used as a direct model for calculating the indicators. The load simulates that used for deflection measurements. The number of layers${n}_{L}$ can differ from that of the pavements. Some model layers have a predetermined and fixed stiffness ($j\in Inde{x}_{f})$, while others ($j\in Inde{x}_{u})$ are considered with unknown values of Young’s modulus, which must be specified from the deflection measurements (${n}_{f}$ + ${n}_{u}={n}_{L})$.

**Figure 2.**Theoretical curviameter deflection basins;

**left**: stiffness variations in the bituminous base layer;

**right**: stiffness variations in the subgrade layer.

**Figure 3.**Weighting functions for the reference structure of Table 2; blue: function relative to the upper base layer (

**left**scale); red: function relative to the subgrade layer (

**right**scale).

**Figure 4.**Evolution of indicators ($\Delta {I}_{\mathrm{BM}1})$, ($\Delta {I}_{\mathrm{UGM}})$ vs. the Young’s modulus of the base and subgrade layers. The $y$ coordinate provides an estimate of the difference between the actual and reference stiffness moduli of the layers. These calculations have been performed for the theoretical deflection bowl of the curviameter. The left (resp. right) curve has been obtained for the reference modulus of the subgrade layer (resp. base layer).

**Figure 5.**Theoretical comparison between the sensitivity of the conventional and orthogonal normalized indicators, relative to deflection measurements;

**left**: sensitivity to the subbase stiffness modulus;

**right**: sensitivity to the subgrade stiffness modulus.

**Figure 6.**Theoretical curviameter bowls for the pavement structure presented above the graphs. The Young’s modulus value is assumed to decrease every 2 m;

**left**: variations in the base layer;

**right**: variations in the subgrade layer. The deflection bowls have been computed for the geophone placed at the center of the areas with constant stiffness.

**Figure 7.**Simulation of the response of indicators ${I}_{\mathrm{BM}1}$ (

**left**) and ${I}_{\mathrm{UGM}}$ (

**right**) for the pavement structure presented above the graphs and for the deflection bowls in Figure 6.

**Table 1.**Conventional indicators used to interpret deflection measurements (as synthesized from Le Boursicault [7]).

Index | Definition | Comments | References |
---|---|---|---|

D_{0}: Maximum Deflection | D_{0} = D_{max} | Affected by all layers | [2,3,4,5,6,7,10,11,12,13,14,15,16] |

D_{i}: Deflections | Deflection measurement recorded by sensor #i or at “i” millimeters from the center of the plate | [5,6,7,10,11] | |

R_{o}C: Radius of Curvature | Second derivative of the deflection basin at the maximum deflection Calculation method depending on the device | Sensitive to both the base layer and interface | [4,7,12,13] |

Rd: | $\mathrm{Rd}={\mathrm{R}}_{\mathrm{o}}\mathrm{C}\times $D_{0} | Sensitive to platform variations for flexible pavements | [4,7] |

BLI: Base Layer Index or SCI: Surface Curvature Index | BLI = D_{0} − D_{300} | More sensitive to surface layers | [10,11] |

MLI: Middle Layer Index or BDI: Base Damage Index | MLI = D_{300} − D_{600} | More sensitive to base layers | [10,11] |

LLI: Lower Layer Index or BCI: Base Damage Index | LLI = D_{900} − D_{600} | More sensitive to both base and foundation layers | [10,11] |

**Table 2.**Theoretical example for constructing orthogonal deflection indicators. Case of a flexible pavement structure with fictitious large variations in layer stiffness.

Material Type | Thickness (m) | Reference Structure Young’s Modulus (MPa) | Variations (MPa) |
---|---|---|---|

BBSG | 0.06 | 7000 | |

BM1 | 0.08 | 9000 | 3000 to 18,000 |

BM2 | 0.08 | 9000 | |

UGM | 6 | 50 | 20 to 200 |

Rigid bedrock | Infinite | 55,000 |

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**MDPI and ACS Style**

Simonin, J.-M.; Piau, J.-M.; Le-Boursicault, V.; Freitas, M.
Orthogonal Set of Indicators for the Assessment of Flexible Pavement Stiffness from Deflection Monitoring: Theoretical Formalism and Numerical Study. *Remote Sens.* **2022**, *14*, 500.
https://doi.org/10.3390/rs14030500

**AMA Style**

Simonin J-M, Piau J-M, Le-Boursicault V, Freitas M.
Orthogonal Set of Indicators for the Assessment of Flexible Pavement Stiffness from Deflection Monitoring: Theoretical Formalism and Numerical Study. *Remote Sensing*. 2022; 14(3):500.
https://doi.org/10.3390/rs14030500

**Chicago/Turabian Style**

Simonin, Jean-Michel, Jean-Michel Piau, Vinciane Le-Boursicault, and Murilo Freitas.
2022. "Orthogonal Set of Indicators for the Assessment of Flexible Pavement Stiffness from Deflection Monitoring: Theoretical Formalism and Numerical Study" *Remote Sensing* 14, no. 3: 500.
https://doi.org/10.3390/rs14030500