# Structural FEA-Based Design and Functionality Verification Methodology of Energy-Storing-and-Releasing Prosthetic Feet

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## Abstract

**:**

## 1. Introduction

## 2. Requirements

#### 2.1. The Ankle Rotation on the Sagittal Plane

#### 2.2. The Vertical Ground Reaction Forces during Normal Ground Walking

#### 2.3. The ISO 10328 Standard Static Test

#### 2.4. Biomechanical Requirements

#### 2.5. Foot Prosthesis Configuration

## 3. Materials and Method

#### 3.1. Design Phase

#### 3.1.1. Geometry Optimization: 2D FE Model

**G**and

**M**in Figure 8). The positions of the virtual markers were chosen considering the marker-set protocol [21,25]. Since the shank was fixed, the

**J**marker was considered as (0,0).

#### 3.1.2. Material Properties Optimization: 3D FE Model

**Table 6.**Contacts’ properties. See also Figure 15. AF = ankle frame; UB = upper blade; MB = middle blade; LB = lower blade; SH = spring holder; TC = tube connector.

Surface 1 | Surface 2 | Type | Formulation | Frict. Coeff. | Norm. Stiff. Fact. |
---|---|---|---|---|---|

AF top | UB bottom | bonded | augm.Lagrange | - | 1.00 |

UB bottom | MB top | frictional | pure penalty | 0.20 | 0.01 |

MB bottom | LB top | frictional | pure penalty | 0.20 | 0.01 |

MB bottom | SH top | frictional | pure penalty | 0.20 | 0.01 |

#### 3.2. Mechanical Test Phase

#### 3.3. Functionality Verification Phase

#### 3.3.1. Two-Dimensional FE Model Adjustment

#### 3.3.2. ISO 10328 Cyclic Tests

#### 3.3.3. ISO 22675–ISO/TS 16955 Dynamic Test

## 4. Results and Discussion

#### 4.1. Design Phase Results

#### 4.1.1. Geometry Optimization: 2D FE Model Results

#### 4.1.2. Material Properties Optimization: 3D FE Model

#### 4.2. ISO 10328 Equivalent Static Test Results

#### 4.3. Functionality Verification Results

#### 4.3.1. ISO 10328 Cyclic Tests

#### 4.3.2. ISO 22675–ISO/TS 16955 Roll-over Test

#### 4.3.3. Elapsed Time of Calculation Comparison: 2D FE Model vs. 3D FE Model

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Deflection of the blades in respect to each other when the foot prosthesis is loaded at the heel and at the forefoot.

**Figure 3.**Flowchart of the design methodology: design phase, validation phase and functional verification phase.

**Figure 4.**Schematic image of the boundary conditions used for the 2D FE model of the Design step, which was used to simulate the static tests according to ISO 10328. The heel was loaded with an inclined platform (−15°) to which a displacement was imposed (10 mm); the forefoot was compressed with a 20° inclined platform which was moved until 50 mm. In both cases, the platform was free to move along its longitudinal direction.

**Figure 5.**Geometric parameters of MyFlex-$\gamma $ varied in the 2D FE model of the design phase—see Table 3.

**Figure 6.**Mesh modeling and width assignment in the transversal direction. The 2D FE model was meshed with PLANE183 elements, for a total of 10,000 nodes without the platform and 20,000 with the platform.

**Figure 7.**Joint and contact modeling in the 2D FE model. See also Table 4 for detailed information.

**Figure 8.**The foot rotation $\alpha $ is calculated as the variation of the angle between the shank axis and the

**GM**line. The

**G**and

**M**markers taken into account in (

**A**) are referred to the reflective markers (grey circles in (

**B**)) considered by Leardini et al. [25].

**Figure 9.**Virtual markers displacement during static dorsiflexion test. Platform displacement at 0 (black and white image) mm vs. platform displacement at 50 mm.

**Figure 10.**The ISO 10328 dorsiflexion static test compared to the equivalent test with vertical actuator.

**Figure 11.**Isotropic parts were provided as solid, while composite components were surfaces. The Össur lower blade was assumed as isotropic with approximated properties. The foot shell was simplified considering the areas underneath the lower blade only.

**Figure 12.**Mesh modeling of the components considered flexible in the analysis. The elastic elements, including the lower blade and the foot shell, were mainly modeled with SOLID186 elements (quadratic behavior). Ankle frame, spring holder and platform were modeled with SOLID185 elements (linear behavior).

**Figure 14.**Modeling of the composite parts. The lower blade was considered to be isotropic with elastic properties close to Össur Pro-Flex Pivot’s sole blade.

**Figure 15.**Joint and contact modeling in the 3D FE model. See also Table 6.

**Figure 16.**ISO 1328 equivalent static tests: actual plantarflexion and dorsiflexion tests and 3D Cad model of the test set up, in plantarflexion test configuration.

**Figure 17.**The width of each part in the 2D FE model was constant and the holes were not considered.

**Figure 18.**ISO 10328 cyclic test configuration: the heel platform was inclined at −15°, while the forefoot platform had an inclination of 20°. The heel and the forefoot forces followed the paths given in Figure 19.

**Figure 19.**M-shaped force: the qualitative behavior of the heel and the forefoot forces are based on ISO 10328 dynamic test, while the maximum values (130% of the body weight for the heel force and 108% for the forefoot force) are from literature—see the Biomechanical Requirements in Section 2.4. In the present work, a 60 kg body weight was used; therefore the peaks were 764 N and 635 N. The forces were imposed as shown in Figure 18.

**Figure 20.**ISO–22675: the tilting table simulates the relative rotation between the ground and the thigh. The force F depends on the weight category and the maximum value was 130% of the body weight (BW). For the 60 kg weight category, the maximum value of the force F was 764 N. The tilting table ranged from −20° to 40°.

**Figure 21.**Results from the first stage of the design phase. The results from the pre-optimized configuration (Table 3) are compared to the results of a random configuration. The vertical shadow is the rotation range that the foot must have (values defined and specified in Section 2.4, “Biomechanical Requirements”, i.e., between −5 degrees and −8 degrees in the plantarflexion and between 14 degrees and 18 degrees in the dorsiflexion). The horizontal shadow is the range of the ground reaction force (between 95% and 130% in plantarflexion and between 95% and 108% in dorsiflexion of the body weight of the user). The intersection area between the two shadows is the optimal area in which the curve of stiffness of the foot must fall, so that the foot rotates up to the degrees desired when subjected to ground reaction forces.

**Figure 22.**The stiffness curves presented as foot rotation–reaction force in Figure 21 can be also presented as platform displacement–reaction force.

**Figure 24.**Comparisons among the static plantarflexion and dorsiflexion 2D FEAs and 3D FEAs and the mechanical tests on the physical prototype with stiffness curves givens as reaction force–rotation. Again, the vertical shadow is the rotation range that the foot must have (values defined and specified in Section 2.4, “Biomechanical Requirements”, i.e., between −5 degrees and −8 degrees in the plantarflexion and between 14 degrees and 18 degrees in the dorsiflexion). The horizontal shadow is the range of the ground reaction force (between 95% and 130% in plantarflexion and between 95% and 108% in dorsiflexion of the body weight of the user). The intersection area between the two shadows is the optimal area in which the curve of stiffness of the foot must fall, so that the foot rotates up to the degrees desired when subjected to ground reaction forces.

**Figure 25.**Comparisons among the static plantarflexion and dorsiflexion 2D FEAs and 3D FEAs and the mechanical tests on the physical prototype with stiffness curves given as reaction force–displacement.

**Figure 26.**Upper blade and middle blade evaluated with the Tsai–Wu criterion; the most critical areas, when the foot was loaded with the 220% of body weight of intended users, presented an inverse reserve factor of 0.5, which means a safety factor of 2.

**Figure 27.**Plantarflexion test: the heel of the foot shell was not completely covered by the platform.

**Figure 28.**The deflection sequence of the 2D FE model of the prosthesis during ISO 10328 cyclic test.

Manufacturer | Model | Website (Access Date) | Country |
---|---|---|---|

Blatchford | Elan | www.blatchford.co.uk (1 November 2021) | UK |

Blatchford | Elan${}^{IC}$ | www.blatchford.co.uk (1 November 2021) | UK |

Fillauer | Raize | www.fillauer.com (1 November 2021) | USA |

Freedom-Innovations | Kinnex 2.0 | www.freedom-innovations.com (1 November 2021) | USA |

Össur | Proprio Foot | www.ossur.com (1 November 2021) | Iceland |

Ottobock | Empower | www.ottobock.com (1 November 2021) | Germany |

**Table 2.**Aim and type of simulations and material properties used in previous works where finite element analysis is applied to study foot prostheses.

Ref. | Aim | Type | Mat. Prop. |
---|---|---|---|

Omasta et al. [6] | analysis | 3D Static | linear, isotropic |

Bonnet et al. [7] | analysis | 3D static | linear, isotropic |

Naveed et al. [8] | design | 3D dynamic | linear, isotropic |

Santana et al. [9] | design | 3D static | non linear, orthotropic |

Prost et al. [10] | design | 3D static | linear, isotropic |

Shepherd et al. [11] | design | 3D static | linear, isotropic |

Mahmoodi et al. [12] | design | 3D dynamic | linear, isotropic |

Ke et al. [13] | design | 3D static | linear, isotropic |

Dao et al. [14] | design | 3D static/dynamic | linear, isotropic |

Rigney et al. [15] | analysis | 3D static/dynamic | linear, isotropic |

Tryggvason et al. [16] | design | 3D dynamic | non linear, orthotropic |

**Table 3.**Geometric parameters of MyFlex-$\gamma $ varied in the 2D FE model of the design phase—see Figure 5.

Parameter | Min Value | Max Value | |
---|---|---|---|

upper blade thickness UB${}_{t}$ (mm) | 6.00 | 8.50 | defined in step 2 |

upper blade curvature 1 UB c${}_{1}$ (deg) | 1.00 | 3.00 | |

upper blade curvature 2 UB c${}_{2}$ (deg) | 1.00 | 3.00 | |

upper blade curvature 3 UB c${}_{3}$ (deg) | 1.00 | 3.00 | |

upper blade curvature 4 UB c${}_{4}$ (deg) | 1.00 | 3.00 | |

upper blade curvature 5 UB c${}_{5}$ (deg) | 3.00 | 5.00 | |

middle blade length MB${}_{L}$ (mm) | 150 | 175 | |

middle blade thickness MB${}_{t}$ (mm) | 7.00 | 10.00 | defined in step 2 |

**Table 4.**Contacts’ properties. See also Figure 7. AF = ankle frame; UB = upper blade; MB = middle blade; LB = lower blade; SH = spring holder; TC = tube connector.

Surface 1 | Surface 2 | Type | Formulation | Frict. Coeff. | Norm. Stiff. Fact. |
---|---|---|---|---|---|

AF top | UB bottom | bonded | augm.Lagrange | - | 1.00 |

UB bottom | MB top | frictional | pure penalty | 0.20 | 0.01 |

MB bottom | LB top | frictional | pure penalty | 0.20 | 0.01 |

MB bottom | SH top | frictional | pure penalty | 0.20 | 0.01 |

AF ankle | TC ankle | no separation | augm.Lagrange | - | 1.00 |

**Table 5.**Orthotropic elasticity of CFRP prepregs used to manufacture the upper blade, middle blade and lower blade.

Type | Gramm. | Thick. | ${\mathit{E}}_{1}$ | ${\mathit{E}}_{2}$ | ${\mathit{E}}_{3}$ | ${\mathit{G}}_{12}$ | ${\mathit{G}}_{23}$ | ${\mathit{G}}_{13}$ | ${\mathit{\u03f5}}_{12}$ | ${\mathit{\u03f5}}_{23}$ | ${\mathit{\u03f5}}_{13}$ |
---|---|---|---|---|---|---|---|---|---|---|---|

g/m${}^{\mathbf{2}}$ | mm | GPa | GPa | GPa | GPa | GPa | GPa | - | - | - | |

UD | 150 | 0.151 | 112.5 | 7.4 | 7.4 | 4.3 | 2.6 | 4.3 | 0.33 | 0.44 | 0.33 |

UD | 250 | 0.251 | 112.5 | 7.4 | 7.4 | 4.3 | 2.6 | 4.3 | 0.33 | 0.44 | 0.33 |

W | 200 | 0.234 | 61.3 | 61.3 | 6.9 | 3.3 | 3.3 | 2.7 | 0.04 | 0.30 | 0.30 |

**Table 7.**Final geometric parameters that define the profile shape of MyFlex-$\gamma $ elastic elements, for the 60 kg weight category. See Figure 5.

Parameter | Final Value |
---|---|

upper blade thickness UB${}_{t}$ (mm) | 6.80 |

upper blade curvature 1 UB c${}_{1}$ (deg) | 2.00 |

upper blade curvature 2 UB c${}_{2}$ (deg) | 2.00 |

upper blade curvature 3 UB c${}_{3}$ (deg) | 2.20 |

upper blade curvature 4 UB c${}_{4}$ (deg) | 2.50 |

upper blade curvature 5 UB c${}_{5}$ (deg) | 4.20 |

middle blade length MB${}_{L}$ (mm) | 163 |

middle blade thickness MB${}_{t}$ (mm) | 7.60 |

Part | Type | Orientation (deg) | No. of Layers | Total Thickness (mm) |
---|---|---|---|---|

Woven 200 g/m${}^{2}$ | 0 | 3 | 0.702 | |

Woven 200 g/m${}^{2}$ | 45 | 2 | 0.468 | |

Upper blade | Unidir. 250 g/m${}^{2}$ | 0 | 18 | 4.518 |

Woven 200 g/m${}^{2}$ | 45 | 2 | 0.468 | |

Woven 200 g/m${}^{2}$ | 0 | 3 | 0.702 | |

total = | 6.858 | |||

Woven 200 g/m${}^{2}$ | 0 | 3 | 0.702 | |

Unidir. 250 g/m${}^{2}$ | 0 | 5 | 1.255 | |

Unidir. 150 g/m${}^{2}$ | 0 | 10 | 1.510 | |

Middle blade | Woven 200 g/m${}^{2}$ | 0 | 3 | 0.702 |

Unidir. 150 g/m${}^{2}$ | 0 | 10 | 1.510 | |

Unidir. 250 g/m${}^{2}$ | 0 | 5 | 1.255 | |

Woven 200 g/m${}^{2}$ | 0 | 3 | 0.702 | |

total = | 7.636 |

**Table 9.**Orthotropic strength of CFRP prepregs used to manufacture the upper blade, middle blade and lower blade. T${}_{1}$, T${}_{2}$ and T${}_{3}$ are the tensile strength; C${}_{1}$, C${}_{2}$ and C${}_{3}$ are the compressive strength; S${}_{12}$, S${}_{23}$ and S${}_{13}$ are the shear strength.

Type | Gramm. | Thick. | T${}_{1}$ | T${}_{2}$ | T${}_{3}$ | C${}_{1}$ | C${}_{2}$ | C${}_{3}$ | S${}_{12}$ | S${}_{23}$ | S${}_{13}$ |
---|---|---|---|---|---|---|---|---|---|---|---|

g/m${}^{\mathbf{2}}$ | mm | MPa | MPa | MPa | MPa | MPa | MPa | MPa | MPa | MPa | |

UD | 150 | 0.151 | 2200 | 29 | 29 | −1082 | −100 | −100 | 60 | 30 | 60 |

UD | 250 | 0.251 | 2200 | 29 | 29 | −1082 | −100 | −100 | 60 | 30 | 60 |

W | 200 | 0.234 | 805 | 805 | 50 | −509 | −509 | −170 | 125 | 65 | 65 |

**Table 10.**Number of nodes and number of degrees of freedom from the 2D FE model (Section 3.1.1) and the 3D FE model (Section 3.1.2).

Step | Type of Simulation | Number of Nodes | Number of Deg. of Freedom |
---|---|---|---|

1 | 2D Static | 10,000 | 20,000 |

2 | 3D Static | 600,000 | 1,800,000 |

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**MDPI and ACS Style**

Tabucol, J.; Brugo, T.M.; Povolo, M.; Leopaldi, M.; Oddsson, M.; Carloni, R.; Zucchelli, A.
Structural FEA-Based Design and Functionality Verification Methodology of Energy-Storing-and-Releasing Prosthetic Feet. *Appl. Sci.* **2022**, *12*, 97.
https://doi.org/10.3390/app12010097

**AMA Style**

Tabucol J, Brugo TM, Povolo M, Leopaldi M, Oddsson M, Carloni R, Zucchelli A.
Structural FEA-Based Design and Functionality Verification Methodology of Energy-Storing-and-Releasing Prosthetic Feet. *Applied Sciences*. 2022; 12(1):97.
https://doi.org/10.3390/app12010097

**Chicago/Turabian Style**

Tabucol, Johnnidel, Tommaso Maria Brugo, Marco Povolo, Marco Leopaldi, Magnus Oddsson, Raffaella Carloni, and Andrea Zucchelli.
2022. "Structural FEA-Based Design and Functionality Verification Methodology of Energy-Storing-and-Releasing Prosthetic Feet" *Applied Sciences* 12, no. 1: 97.
https://doi.org/10.3390/app12010097