# Modeling and Validation of a Passenger Car Tire Using Finite Element Analysis

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Finite Element Tire Modeling

^{2}in the Pam-Crash acceleration.

#### 2.1. Meshing Techniques

#### 2.2. Tire Material Definition

#### 2.3. Tire–Road Contact Algorithm

## 3. FEA Tire Validation

#### 3.1. Static Tire Validation

#### 3.1.1. Static Footprint

#### 3.1.2. Tire Static Vertical Stiffness

#### 3.1.3. Tire Static Lateral Stiffness

#### 3.1.4. Tire Static Longitudinal Stiffness

#### 3.2. Tire Dynamic Validation

#### 3.2.1. Frequency Analysis

#### 3.2.2. Tire Cornering Stiffness

#### 3.2.3. Tire Rolling Resistance Coefficient

## 4. Results and Discussion

#### 4.1. Effect of Load and Inflation Pressure on the Footprint and Vertical Stiffness

^{2}at an inflation pressure of 172 kPa, which is similar to the static footprint at 228 kPa. On the other hand, the vertical load has a negligible effect on the variation of vertical stiffness. By increasing the vertical load and inflation pressure, the tire’s sidewall deformation increases, which leads to higher vertical stiffness. Furthermore, tire vertical stiffness is directly affected by the inflation pressure as it can be seen that a higher inflation pressure makes the tire stiffer.

#### 4.2. Effect of Load and Inflation Pressure on the First Vertical Mode of Frequency

#### 4.3. Effect of Load and Inflation Pressure on the First Longitudinal Mode of Frequency

#### 4.4. Effect of Load, Inflation Pressure, and Speed on the Rolling Resistance Coefficient

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Zorowski, C. Mathematical prediction of dynamic tire behavior. Tire Sci. Technol.
**1973**, 1, 99–117. [Google Scholar] [CrossRef] - Ridha, R. Analysis for tire mold design. Tire Sci. Technol.
**1974**, 2, 195–210. [Google Scholar] [CrossRef] - Kennedy, R.; Padovan, J. Finite Element Analysis of a Steady-State Rotating Tire Subjected to Point Load or Ground Contact. Tire Sci. Technol.
**1987**, 15, 243–260. [Google Scholar] [CrossRef] - Shiraishi, M.; Yoshinaga, H.; Miyori, A.; Takahashi, E. Simulation of dynamically rolling tire. Tire Sci. Technol.
**2000**, 28, 264–276. [Google Scholar] [CrossRef] - Kabe, K.; Koishi, M. Tire cornering simulation using finite element analysis. J. Appl. Polym. Sci.
**2000**, 78, 1566–1572. [Google Scholar] [CrossRef] - Ghoreishy, M. Steady state rolling analysis of a radial tyre: Comparison with experimental results. Proc. Inst. Mech. Eng. Part D J. Automob. Eng.
**2006**, 220, 713–721. [Google Scholar] [CrossRef] - Chae, S. Nonlinear Finite Element Modeling and Analysis of a Truck Tire. Ph.D. Thesis, Pennsylvania State University, State College, PA, USA, 2006. [Google Scholar]
- Krmela, J. The Computational Modelling of Tire. Recent
**2009**, 10, 333–336. [Google Scholar] - Lardner, K.L.; El-GIndy, M.; Oijer, F.; Johansson, I.; Philipps, D. Determining the Vertical and Longitudinal First Mode of Vibration of a Wide Base FEA Truck Tire; SAE Technical Paper 0148-7191; SAE International: Warrendale, PA, USA, 2016. [Google Scholar]
- Nussbaumer, H.J.; Nussbaumer, H.J. The Fast Fourier Transform; Springer: Berlin/Heidelberg, Germany, 1982. [Google Scholar]
- MathWorks. FAst Fourier Transform (FFT). Available online: https://www.mathworks.com/discovery/fft.html (accessed on 25 December 2023).
- Krmela, J. Tire Casings and Their Material Characteristics for Computational Modeling: Scientific Monograph; Oficyna Wydawnicza Stowarzyszenia Menadżerów Jakości i Produkcji: Częstochowa, Poland, 2017. [Google Scholar]
- Rafei, M.; Ghoreishy, M.H.R.; Naderi, G. Thermo-mechanical coupled finite element simulation of tire cornering characteristics—Effect of complex material models and friction law. Math. Comput. Simul.
**2018**, 144, 35–51. [Google Scholar] [CrossRef] - El-Sayegh, Z.; El-Gindy, M. Cornering characteristics of a truck tire on wet surface using finite element analysis and smoothed-particle hydrodynamics. Int. J. Dyn. Control
**2018**, 6, 1567–1576. [Google Scholar] [CrossRef] - Krmela, J. Experiments and Computational Modelling of Tires Textbooks for University Students; Trenčianska Univerzita Alexandra Dubčeka v Trenčíne: Trenčín, Slovakia, 2021; ISBN 978-80-270-9020-4. [Google Scholar]
- El-Sayegh, Z.; El-Gindy, M.; Johansson, I.; Öijer, F. Development and validation of off-road tire-gravelly soil interaction using advanced computational techniques. J. Terramech.
**2020**, 91, 45–51. [Google Scholar] [CrossRef] - Ali, S.N. Rolling Resistance Estimation for PCR Tyre Design Using the Finite Element Method. In Finite Element Methods and Their Applications; IntechOpen: London, UK, 2020. [Google Scholar]
- Gao, X.; Xiong, Y.; Liu, W.; Zhuang, Y. Modeling and experimental study of tire deformation characteristics under high-speed rolling condition. Polym. Test.
**2021**, 99, 107052. [Google Scholar] [CrossRef] - Phromjan, J.; Suvanjumrat, C. Development of solid tire model for finite element analysis of compressive loading. Songklanakarin J. Sci. Technol.
**2021**, 43, 229–236. [Google Scholar] - Király, T.; Primusz, P.; Tóth, C. Simulation of Static Tyre–Pavement Interaction Using Two FE Models of Different Complexity. Appl. Sci.
**2022**, 12, 2388. [Google Scholar] [CrossRef] - Lu, D.; Yang, W.; Wu, H.; Zhou, T. Research on simplified tire finite element modeling and simulation method. Proc. Inst. Mech. Eng. Part D J. Automob. Eng.
**2023**. [Google Scholar] [CrossRef] - Fathi, H.; Khosravi, M.; El-Sayegh, Z.; El-Gindy, M. An Advancement in Truck-Tire–Road Interaction Using the Finite Element Analysis. Mathematics
**2023**, 11, 2462. [Google Scholar] [CrossRef] - Continental. Available online: https://continentaltire.com/tires/crosscontact-lx-sport (accessed on 25 December 2023).
- Haug, E. Finite element analysis of nonlinear membrane structures. In Proceedings of the IASS Pacific Symposium PartII on Tension Structures and Space Frames; University of California: Berkeley, CA, USA, 1972; pp. 165–176. [Google Scholar]
- Group, E. Pam-Crash Theory Notes Manual; PAM System International: Joliet, IL, USA, 2000. [Google Scholar]
- Rivlin, R.S. Large elastic deformations of isotropic materials IV. Further developments of the general theory. Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Sci.
**1948**, 241, 379–397. [Google Scholar] - Rivlin, R.S.; Saunders, D. Large elastic deformations of isotropic materials VII. Experiments on the deformation of rubber. Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Sci.
**1951**, 243, 251–288. [Google Scholar] - Chang, Y.-p.D. Nonlinear FEA Rotating Tire Modeling for Transient Response Simulations; The Pennsylvania State University: State College, PA, USA, 2002. [Google Scholar]
- KuliKowsKi, K.; Szpica, D. Determination of directional stiffnesses of vehicles’ tires under a static load operation. Eksploat. I Niezawodn.
**2014**, 16, 66–72. [Google Scholar] - Sun, P.; Feng, G.; Zhou, S.; Qiu, C.; Fan, J. Experimental analysis of radial tire stiffness and grounding characteristics. IOP Conf. Ser. Mater. Sci. Eng.
**2019**, 677, 022111. [Google Scholar] [CrossRef] - Dudziak, M.; Lewandowski, A.; Waluś, K.J. Static tests the stiffness of car tires. IOP Conf. Ser. Mater. Sci. Eng.
**2020**, 776, 012071. [Google Scholar] [CrossRef] - ISO 20908:2023; Tyre Sound Emission Test—Methods of Drum. ISO: Geneva, Switzerland, 2023. Available online: https://www.iso.org/standard/79096.html (accessed on 25 December 2023).
- ISO 13326:1998(en); Test Methods for Measuring Tyre Uniformity. ISO: Geneva, Switzerland, 1998. Available online: https://www.iso.org/standard/21715.html (accessed on 25 December 2023).
- ISO 28580:2018; Passenger Car, Truck and Bus Tyre Rolling Resistance Measurement Method: Single Point Test and Correlation of Measurement Results. ISO: Geneva, Switzerland, 2018.
- Ejsmont, J.; Taryma, S.; Ronowski, G.; Swieczko-Zurek, B. Influence of load and inflation pressure on the tyre rolling resistance. Int. J. Automot. Technol.
**2016**, 17, 237–244. [Google Scholar] [CrossRef] - Taghavifar, H.; Mardani, A. Investigating the effect of velocity, inflation pressure, and vertical load on rolling resistance of a radial ply tire. J. Terramech.
**2013**, 50, 99–106. [Google Scholar] [CrossRef] - Grappe, F.; Candau, R.; Barbier, B.; Hoffman, M.; Belli, A.; Rouillon, J.-D. Influence of tyre pressure and vertical load on coefficient of rolling resistance and simulated cycling performance. Ergonomics
**1999**, 42, 1361–1371. [Google Scholar] [CrossRef] - Redrouthu, B.M.; Das, S. Tyre Modelling for Rolling Resistance. Master’s Thesis, Chalmers University of Technology, Gothenburg, Sweden, 2014. [Google Scholar]
- Wong, J.Y. Theory of Ground Vehicles; John Wiley & Sons: Hoboken, NJ, USA, 2022. [Google Scholar]

**Figure 1.**Membrane element concepts in Pam-Crash [15].

**Figure 2.**FEA 235/55R19 tire in Pam-Crash. (

**a**) FEA full rotated tire; (

**b**) cross-section of the tire.

**Figure 4.**Contact definition in Pam-Crash, perforation, and penetration [25].

**Figure 6.**The nodal displacement contour at footprint test for nominal inflation pressure and nominal load.

**Figure 8.**Frequency analysis of a 235/55R19 tire at 228 kPa inflation pressure and 5 kN vertical load.

**Figure 9.**Tire RRC as a function of vertical load at three inflation pressures and longitudinal speeds: (

**a**) 10 km/h longitudinal speed; (

**b**) 40 km/h longitudinal speed.

**Figure 10.**Effect of longitudinal speed on the tire’s rolling resistance coefficient at different vertical loads and a nominal inflation pressure of 228 kPa.

**Figure 11.**RRC as a function of five longitudinal speeds (10 km/h, 30 km/h, 40 km/h, 50 km/h, and 80 km/h) at 228 kPa inflation pressures and an 8 kN vertical load.

**Table 1.**Comparison between the size and weight of a 235/55R19 101H passenger car tire model and measurements.

Tire Information | Manufacturer * Diameter (mm) | Simulation Diameter (mm) | Manufacturer * Weight (kg) | Simulation Weight (kg) |
---|---|---|---|---|

Rim and Wheel | 482.6 (19 inch) | 445.27 | 11.4 (25.13 lbs.) | 11.05 |

Tire | 741.68 (29.2 inch) | 739.69 | 13.65 (30.1 lbs.) | 14.81 |

Tire Components | ${\mathit{C}}_{10}$ (MPa) | ${\mathit{C}}_{01}$ (MPa) | Poisson’s Ratio |
---|---|---|---|

Tread Top | 2.49 | 0.67 | 0.473 |

Under Tread | 0.51 | 1.86 | 0.409 |

Tire Dynamics | Manufacture * Data | Simulation Results | Unit |
---|---|---|---|

Vertical stiffness | 181.1 | 235.86 | N/mm |

Lateral stiffness | 115–140 | 116.66 | N/mm |

Longitudinal stiffness | 200–250 | 373.11 | N/mm |

Tire Dynamics | Manufacture * Data | Simulation Data | Unit |
---|---|---|---|

Critical vertical frequency | 70–90 | 74 | Hz |

Cornering stiffness | 1200–1800 | 1057.26 | N/degree |

Rolling resistance coefficient | 0.015–0.03 | 0.006–0.01 | N/A |

Contact Area (mm^{2}) | |||
---|---|---|---|

Inflation Pressure (kPa) | Vertical Load (kN) | ||

3.5 | 5 | 8 | |

172 | 20,000 | 22,500 | 52,500 |

228 | 17,500 | 22,500 | 32,500 |

352 | 15,000 | 20,000 | 22,500 |

Static Vertical Stiffness (N/mm) | |||
---|---|---|---|

Inflation Pressure (kPa) | Vertical Load (kN) | ||

3.5 | 5 | 8 | |

172 | 196.09 | 200.18 | 206.75 |

228 | 230.74 | 235.86 | 245.82 |

352 | 297.56 | 309.12 | 322.55 |

Vertical First Mode of Vibration (Hz) | |||
---|---|---|---|

Inflation Pressure (kPa) | Vertical Load (kN) | ||

3.5 | 5 | 8 | |

172 | 68 | 69 | 70.5 |

228 | 73.5 | 74 | 74.5 |

352 | 83 | 83 | 83.5 |

**Table 8.**The tire’s 1st longitudinal (horizontal) mode of vibration at several inflation pressures and vertical loads.

Longitudinal First Mode of Vibration (Hz) | |||
---|---|---|---|

Inflation Pressure (kPa) | Vertical Load (kN) | ||

3.5 | 5 | 8 | |

172 | 35.5 | 32 | 32 |

228 | 35.5 | 32 | 33.5 |

352 | 37 | 37 | 37 |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2024 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Fathi, H.; El-Sayegh, Z.; Ren, J.; El-Gindy, M.
Modeling and Validation of a Passenger Car Tire Using Finite Element Analysis. *Vehicles* **2024**, *6*, 384-402.
https://doi.org/10.3390/vehicles6010016

**AMA Style**

Fathi H, El-Sayegh Z, Ren J, El-Gindy M.
Modeling and Validation of a Passenger Car Tire Using Finite Element Analysis. *Vehicles*. 2024; 6(1):384-402.
https://doi.org/10.3390/vehicles6010016

**Chicago/Turabian Style**

Fathi, Haniyeh, Zeinab El-Sayegh, Jing Ren, and Moustafa El-Gindy.
2024. "Modeling and Validation of a Passenger Car Tire Using Finite Element Analysis" *Vehicles* 6, no. 1: 384-402.
https://doi.org/10.3390/vehicles6010016