# Hybrid Shell-Beam Inverse Finite Element Method for the Shape Sensing of Stiffened Thin-Walled Structures: Formulation and Experimental Validation on a Composite Wing-Shaped Panel

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## Abstract

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## 1. Introduction

## 2. Hybrid Inverse Finite Element Method

#### 2.1. 1D Inverse Finite Element Method

#### 2.1.1. Kinematic Relations

#### 2.1.2. Experimental Sectional Strains

#### 2.1.3. Least-Squares Error Functional

#### 2.2. 2D Inverse Finite Element Method

#### 2.2.1. Kinematic Relations

#### 2.2.2. Experimental Strain Measures

#### 2.2.3. Least-Squares Error Functional

#### 2.3. Hybrid Formulation

## 3. Experimental Test Specimen

#### 3.1. Composite Wing Panel

#### 3.2. Inverse Element Models

#### 3.3. Test Configuration

#### 3.4. Strain Sensors

## 4. Experimental Setup and Results

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Definition of the variables used to describe Timoshenko beam kinematics; the location and component of strains measured by a sensor placed on the beam surface are also shown.

**Figure 2.**Illustration of the plate structure: (

**a**) kinematic variables used to describe the plate deformations, and (

**b**) sensors mounted on the top and bottom plate surfaces.

**Figure 5.**Transverse section of the stringers. The figure shows the stacking sequence of the stringer’s web and how it is folded to obtain the two caps. Moreover, the strain sensor configuration for the web is illustrated. All dimensions are expressed in mm.

**Figure 8.**Testing configuration of the panel with the boundary conditions indicated. Moreover, the transverse loads ${F}_{1}$ and ${F}_{2}$, relative to the first and second loading conditions, are presented. The locations of the six transverse displacement transducers (${w}_{1-6}$) are also shown. All dimensions are expressed in mm.

**Figure 9.**Torsional load case: contour plot of the transverse deformation (along z) for the first load case. All dimensions and displacements are expressed in mm.

**Figure 10.**Primarily bending load case: contour plot of the transverse deformation (along z) for the second load case. All dimensions and displacements are expressed in mm.

**Figure 14.**The load application system for the first load case. The sphere that transmits the load can be moved along the bar to obtain the second load case.

${\mathit{E}}_{11}$ [GPa] | ${\mathit{E}}_{22}$ [GPa] | ${\mathit{\nu}}_{12}$ | ${\mathit{G}}_{12}={\mathit{G}}_{23}={\mathit{G}}_{13}$ [GPa] | Thickness [mm] |
---|---|---|---|---|

59.7 | 59.7 | 0.09 | 3.8 | 0.25 |

**Table 2.**Shape sensing results: experimentally measured and reconstructed transverse displacements are reported for the two load cases. In parentheses, the percentage errors with respect to the experimental values are reported. Moreover, the mean of the absolute value of the percentage error is also reported ($\mu \left(\right|\%Err\left|\right)$).

Torsional Load Case | Primarily Bending Load Case | |||||
---|---|---|---|---|---|---|

Experimental | Hybrid iFEM | Shell-Only iFEM | Experimental | Hybrid iFEM | Shell-Only iFEM | |

F [N] | 200 | 200 | ||||

${w}_{1}$ [mm] | 2.42 | 2.52 | 2.50 | 2.44 | 2.56 | 2.55 |

($\%Er{r}_{{w}_{1}}$) | (+4.3%) | (+3.5%) | (+5.0%) | (+4.2%) | ||

${w}_{2}$ [mm] | 2.98 | 3.14 | 3.14 | 3.72 | 3.89 | 3.89 |

($\%Er{r}_{{w}_{2}}$) | (+5.2%) | (+5.4%) | (+4.5%) | (+4.5%) | ||

${w}_{3}$ [mm] | 2.98 | 3.13 | 3.14 | 3.72 | 3.89 | 3.90 |

($\%Er{r}_{{w}_{3}}$) | (+5.3%) | (+5.4%) | (+4.7%) | (+4.8%) | ||

${w}_{4}$ [mm] | 5.18 | 5.07 | 4.84 | 3.77 | 3.70 | 3.51 |

($\%Er{r}_{{w}_{4}}$) | (−2.0%) | (−6.5%) | (−1.9%) | (−7.0%) | ||

${w}_{5}$ [mm] | 4.72 | 4.54 | 4.35 | 3.42 | 3.35 | 3.19 |

($\%Er{r}_{{w}_{5}}$) | (−3.9%) | (−8.0%) | (−2.2%) | (−6.8%) | ||

${w}_{6}$ [mm] | 3.75 | 3.90 | 3.82 | 3.29 | 3.42 | 3.36 |

($\%Er{r}_{{w}_{6}}$) | (+3.9%) | (+1.8%) | (+3.9%) | (+1.9%) | ||

$\mu \left(\right|\%Err\left|\right)$ | 4.1% | 5.1% | 3.7% | 4.9% |

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## Share and Cite

**MDPI and ACS Style**

Esposito, M.; Roy, R.; Surace, C.; Gherlone, M.
Hybrid Shell-Beam Inverse Finite Element Method for the Shape Sensing of Stiffened Thin-Walled Structures: Formulation and Experimental Validation on a Composite Wing-Shaped Panel. *Sensors* **2023**, *23*, 5962.
https://doi.org/10.3390/s23135962

**AMA Style**

Esposito M, Roy R, Surace C, Gherlone M.
Hybrid Shell-Beam Inverse Finite Element Method for the Shape Sensing of Stiffened Thin-Walled Structures: Formulation and Experimental Validation on a Composite Wing-Shaped Panel. *Sensors*. 2023; 23(13):5962.
https://doi.org/10.3390/s23135962

**Chicago/Turabian Style**

Esposito, Marco, Rinto Roy, Cecilia Surace, and Marco Gherlone.
2023. "Hybrid Shell-Beam Inverse Finite Element Method for the Shape Sensing of Stiffened Thin-Walled Structures: Formulation and Experimental Validation on a Composite Wing-Shaped Panel" *Sensors* 23, no. 13: 5962.
https://doi.org/10.3390/s23135962