Editorial Type:
Article Category: Other
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Online Publication Date: 09 Jun 2025

A Look Back at the First Two Decades of Tire Finite Element Analysis—Laying the Foundation

Page Range: 78 – 123
DOI: 10.2346/TST-24-001
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ABSTRACT

Finite element analysis has become a standard tool in the engineer’s toolbox for tire and vehicle development work, with the goal of simulations replacing early prototypes and tests becoming a reality. However, 50 years ago this was just a glimmer on the horizon, with most of the fundamental model features and solution methodologies used in today’s tire simulations, which are now taken for granted as routine, not existing in the finite element codes. Hence, tire researchers in the early 1970s began developing and incorporating technologies into the codes so that basic tire mechanics could be represented, allowing increasing levels of tire performance simulations to be performed. The current model and simulation capabilities have been built upon this foundation of research and applications to give engineers a wide range of tire performance solutions that can be entrusted to virtually develop tires and vehicles for production. This article reviews the early years of the development and application of tire finite element analyses, giving current users of the method an understanding of the foundational work that was done and the main people involved in its development.

Keywords: tire; finite element analysis; history
FIG. 1
FIG. 1

(a) Axisymmetric shell finite element model of a bias tire from a report by Zorowski and Dunn, November 1970 [2]. (b) FEA results of the tire half-cross section shapes at 1200 rpm at inflation pressures of 138 kPa and 220 kPa (20 psi and 32 psi).

FIG. 2
FIG. 2

Finite element results for the deformed shape of a bias tire subjected to vertical and lateral loading, from Bendix Corporation report to Firestone, June 1970 [4].

FIG. 3
FIG. 3

The complex behaviors of a tire that needed to be understood and represented to give a proper representation for simulations [6].

FIG. 4
FIG. 4

Results of crown radial growth (left plot) and max section lateral growth (right plot) vs. inflation pressure (vertical axis), showing less growth with inclusion of updated “follower” force, from a 1979 Stafford and Tabaddor paper [7].

FIG. 5
FIG. 5

Typical computer hardware during the 1970s at Firestone.

FIG. 6
FIG. 6

Axisymmetric model and linear finite element results for a radial tire presented by Durand and Jankovich [8].

FIG. 7
FIG. 7

(a) Axisymmetric tire model from Kaga et al. paper [14]. (b) Deformed shapes for inflation and at center of footprint when loaded against flat and drum surfaces.

FIG. 8
FIG. 8

(a) Shell finite element 3D tire model by Deak and Atluri [15]. (b) Cross section (lower) and circumferential (upper) deformed shape due to vertical deflection. (c) Nodal contact forces at tire–ground interface.

FIG. 9
FIG. 9

(a) Flat plate finite element model of a toroid from DeEskinazi et al. 1975 paper [16]. (b) Discretization of a radial tire using flat plate elements from DeEskinazi et al. 1978 paper [17]. (c) Load-deflection response for the deflected tire showing good agreement with experiment.

FIG. 10
FIG. 11
FIG. 11

May 1980 ASTM F9 Committee Symposium on tire finite element modeling.

FIG. 12
FIG. 12

Extension-shear coupling requiring a circumferential displacement component in axisymmetric elements, from a paper by Stalnaker et al. [24].

FIG. 13
FIG. 13

(a) Statically deflected radial tire from a paper by Yamagishi [25]. (b) Predicted interply shear deformations at the belt edge around the tire circumference (left plot) and across the belt width for all belt plies (right plot). (c) Temperature profile based on the predicted strain energy density.

FIG. 14
FIG. 14

(a) Laminated shell finite element model for tire natural frequency prediction from 1983 Hunkler et al. paper [28]. (b) Predicted natural frequencies showing the importance of including the bending-membrane coupling in the element properties.

FIG. 15
FIG. 15

(a) Tire modal model developed from FEA predicted natural frequencies and mode shapes [30]. (b) Predicted vs. measured vibration transmission from road to steering column using the tire and vehicle modal model [31].

FIG. 16
FIG. 16

(a) Flow chart of the rolling resistance prediction procedure by Whicker et al. from their 1981 paper [32]. (b) Rolling resistance predictions for the effects of load (upper left plot), inflation pressure (upper right plot), and speed (lower plot).

FIG. 17
FIG. 17

(a) Finite element model used for air diffusion prediction by Coddington, 1983 [33]. (b) Predicted vs. measured inflation pressure vs. time for three inner liner material compositions. (c) Predicted internal pressure distribution due to air diffusion.

FIG. 18
FIG. 18

(a) Axisymmetric shell element model used by Padovan for standing wave prediction in his 1977 paper [34]. (b) Effect of rotational speed (Ω) on the damped tire response, showing the formation of the standing wave around the tire circumference (θ) as the speed increased.

FIG. 19
FIG. 19

(a) 2D ring-on-elastic-foundation finite element model from Zeid and Padovan’s 1980 paper [35]. (b) Predicted tire deformation and spring rate response at standing wave speed.

FIG. 20
FIG. 20

(a) 3D tire model for steady-state rolling response prediction from Kennedy and Padovan’s 1987 papers [37, 38]. (b) Predicted steady-state, damped rolling response at standing wave speed.

FIG. 21
FIG. 21

(a) Predicted deformation and hub force response of steady-state rolling tire interacting with a bump at 128 kph (80 mph), from 1987 paper by Nakajima and Padovan [40]. (b) Predicted horizontal hub force vs. time due to the tire rolling over a bump at 32, 64, 96, and 128 kph (20, 40, 60, and 80 mph).

FIG. 22
FIG. 22

(a) Quarter-tire 3D finite element model in exploded view to show the carcass, tread, and sidewall element groups, from Trinko’s 1983 paper [41]. (b) Initial shape and deflected shape at the center of contact area of a tire loaded against the road surface. (c) Measured (left) and predicted (right) contact pressure profiles through the length of the footprint for locations from shoulder (lower) to tire centerline (upper).

FIG. 23
FIG. 23

(a) Solid and truss elements model of a motorcycle tire from Watanabe and Kaldjian’s 1983 paper [42]. (b) Predicted force and moment values for the tire at a 20° camber angle at increasing vertical deflection.

FIG. 24
FIG. 24

(a) 3D tire model used in the simulations of Rothert and his students in their 1985 papers [44, 45]. (b) Predicted contact pressure distribution from the quarter-tire model. (c) Predicted stick and sliding zones due to the presence of friction at three vertical load conditions. Two mesh refinements were used: mesh No. 1, 104 elements, mesh No. 2, 448 elements.

FIG. 25
FIG. 25

Graphics workstation with attached digitizing tablet available around 1980.

FIG. 26
FIG. 26

Flow chart for an FEA system from a 1983 presentation by Stafford [29].

FIG. 27
FIG. 27

Graphical display of predicted tire mode shapes to improve identification, from 1985 paper by Kung et al. [47].

FIG. 28
FIG. 28

(a) 3D wedge model of tire cross section used for inflation analysis, from 1986 paper by Rothert and Gall [46]. (b) Half-section 3D model revolved from the wedge model, used for static deflection analysis.

FIG. 29
FIG. 29

Flow chart for the incorporation of FEA for inflation and static deflection analyses into the tire development process, from a 1988 Chang et al. paper [49].

FIG. 30
FIG. 30

(a) Tire model used with the FEA code created through the NASA Langley National Tire Model consortium, from the 1992 Faria et al. paper [51]. (b) Static road contact results from the model.

Contributor Notes

1Corresponding author. Firestone, Bridgestone/Firestone, Hankook Tire, CenTiRe. Email: ronkenn@vt.edu
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