*Corr. Author’s Address: Shandong University of Technology, Zibo, China, fuhongxun615@163.com 143
Strojniški vestnik - Journal of Mechanical Engineering 68(2022)3, 143-154 Received for review: 2021-09-13
© 2022 The Authors. CC BY 4.0 Int. Licensee: SV-JME Received revised form: 2022-01-13
DOI:10.5545/sv-jme.2021.7401 Original Scientific Paper Accepted for publication: 2022-02-02
Thermal Mechanical Coupling Analysis  
of a Flexible Spoke Non-pneumatic Tire
Fu, H. – Liang, X. – Wang, Y . – Ku, L. – Xiao, Z.
Hongxun Fu – Xuemeng Liang – Yan Wang – Laiyun Ku – Zhen Xiao
Shandong University of Technology, School of Transportation and Vehicle Engineering, China
A new type of non-pneumatic tire with a flexible spoke structure is proposed to solve traditional pneumatic tire defects, such as easy bursting, 
the requirement of real-time maintenance to keep tire pressure, and a complex manufacturing process. The thermal mechanical coupling 
characteristics of the flexible spoke non-pneumatic tire are studied in detail to obtain the steady-state temperature field distribution pattern 
under different operating conditions. A unit configuration method is proposed to build the 3D model of the flexible spoke non-pneumatic tire. 
Then, the thermal-mechanical sequential coupling method is used to analyse the deformation, energy loss and heat conduction of the flexible 
spoke non-pneumatic tire. Finally, the steady-state temperature field distribution pattern under thermal-mechanical coupling is obtained. The 
results show that the high temperature region of the non-pneumatic tire is mainly distributed in the middle bending part of the flexible spoke 
unit, and the temperature gradually decreases from the centre of the flexible spoke to the surrounding parts. The influence of load on the 
temperature change of non-pneumatic tire is greater than that of the driving speed. This study provides a theoretical basis and method 
guidance for solving the failure problem of the non-pneumatic tires under the action of thermal-mechanical coupling.
Keywords: non-pneumatic tire, flexible spoke structure, unit configuration method, thermal mechanical coupling, temperature field
Highlights
• 	 A unit configuration method for the structural design of the non-pneumatic tire is proposed.
• 	 The thermal-mechanical sequential coupling method is used to analyse the deformation, energy loss, and heat conduction of 
the flexible spoke non-pneumatic tire.
• 	 The temperature field distribution pattern of the flexible spoke non-pneumatic tire under the influence of thermal mechanical 
coupling is obtained.
• 	 The influence of load on the temperature change of the non-pneumatic tire is greater than that of driving speed.
0  INTRODUCTION
A non-pneumatic tire does not rely on air to adjust the 
tire elasticity and support the weight of the vehicle. 
It solves the problems of the traditional pneumatic 
tire, such as easy bursting, the requirement of real-
time maintenance to keep tire pressure, and a complex 
manufacturing process. Therefore, non-pneumatic tire 
technology has gradually become a research topic of 
great interest in the field of tire safety, and new types 
of non-pneumatic tire structures have been put forward 
[1] to [3]. Compared with pneumatic tires, non-
pneumatic tires have the advantages of anti-puncture, 
explosion-proof, shock absorption and good adhesion 
performance. However, they have poor performance 
in reliability and durability as they are easily affected 
by stress concentrations, alternating stresses, high 
temperatures and other factors [4] and [5]. The above 
problems seriously restrict the further development 
and application of non-pneumatic tire technology. 
Based on the study of the basic mechanical properties 
of a non-pneumatic tire, thermal analysis of a non-
pneumatic tire has also been conducted. Currently, 
the performance of a non-pneumatic tire under the 
action of thermo-mechanical couplings has been 
analysed. This has a high theoretical significance and 
an application value for the development of tire safety 
technology [6] to [8].
Researchers have conducted studies on the 
basic mechanical properties, fatigue, failure of non-
pneumatic tires. Gasmi et al. [9] and Kucewicz et al. 
[10] conducted vertical static deflection simulations 
with the Tweel™ and honeycomb tires, analysing the 
influence of the vertical displacement of the hub, tread 
deformation and grounding pressure on the geometry 
of the flexible spoke structure. Xu et al. [11] studied 
the influence of the ratio of the mechanical elastic 
wheel (MEW) hinge group length to the elastic wheel 
thickness on the radial stiffness of the wheel based 
on the finite element analysis, realizing the multi-
objective optimization of the radial stiffness of the 
MEW based on the artificial fish swarm algorithm. 
Mazur et al. [12], Ju et al. [13] and Jackowski et al. [14] 
obtained the rolling resistance of a non-pneumatic tire 
based on the prototype test method; they studied the 
influence of tire structure and the polyurethane (PU) 
material on the rolling resistance of a non-pneumatic 
tire, and compared pneumatic and non-pneumatic tires 
in terms of the rolling resistance. Fu et al. [15] and [16] 
conducted a preliminary study on the lateral deflection 

Strojniški vestnik - Journal of Mechanical Engineering 68(2022)3, 143-154
144 Fu, H. – Liang, X. – Wang, Y. – Ku, L. – Xiao, Z.
properties of a mechanical elastic wheel and its 
influencing factors and established a theoretical model 
of the lateral loading of the wheel. Zang et al. [17] and 
Du et al. [18] studied the influences of load and roll 
angle on the distribution of the contact pressure of 
a MEW and made a comparison with the pneumatic 
tire. Sriwijaya and Hamzah [19] studied the influence 
of the road inclination angle on the stress distribution 
of the flexible spoke structure of a non-pneumatic tire; 
their results show that a larger road inclination angle 
increases the equivalent stress value of a honeycomb 
flexible spoke structure, thus accelerating the failure 
of the flexible spoke structure. Zhang et al. [20] 
carried out a static grounding performance analysis 
of a staggered non-pneumatic tire, obtaining the S-N 
curve of the flexible spoke PU material by using the 
demosia fatigue test, and entering it into the FE-SAFE 
software to predict the tire fatigue life. The above 
studies have preliminarily explored the structural 
performance and fatigue failure of a non-pneumatic 
tire, but the influence of temperature has not been 
considered in the research process.
During the rolling motion, the non-pneumatic 
tire deforms under the action of external loads, which 
causes a temperature rise of the tire, resulting in a 
possible decline of physical and chemical properties 
of the tire materials, a loss of strength, and eventually 
tire failure [21] and [22]. Many studies have been 
carried out to evaluate both the performance and the 
durability of pneumatic tires with temperature, but 
there are few studies on the influence of temperature of 
non-pneumatic tires. In particular, Chen et al. [23] and 
Zhu et al. [24] analysed the effects of vehicle speed, 
loss factor, and thermal conductivity of the material 
and wheel section width on the maximum temperature 
of a MEW. The authors proposed a theoretical analysis 
method to predict the surface temperature of a MEW 
under different driving conditions.
In this paper, the three dimensional (3D) model 
and numerical analysis model of a 195/50N16 flexible 
spoke non-pneumatic tire were established based on 
the unit configuration method. The thermo-mechanical 
sequential coupling method was used to investigate 
the thermo-mechanical coupling characteristics of 
the flexible spoke non-pneumatic tire under different 
operating conditions, which provides a method 
guidance for the subsequent fatigue failure research 
and structural optimization design of non-pneumatic 
tires.
1  ESTABLISHMENT OF THE NUMERICAL ANALYSIS MODEL 
OF A FLEXIBLE SPOKE NON-PNEUMATIC TIRE
1.1  Establishment of the 3D Model of the Flexible Spoke 
Non-pneumatic Tire using the Unit Configuration Method
Our research group proposed a unit configuration 
method for modelling the structural design of the non-
pneumatic tire. It creates the non-pneumatic tire model 
of different structures by combining the unit structure 
with an arbitrary shape according to a different array 
mode and array number.
In this paper, the flexible spoke support body 
of the flexible spoke non-pneumatic tire consists of 
a circular array of 45 groups of flexible spoke units. 
Every group occupies a construction angle of 8°. 
Each flexible spoke unit consists of an inner ring, a 
circumferential spoke, a side spoke, and an outer 
Fig. 1.  3D model of the flexible spoke non-pneumatic tire

Strojniški vestnik - Journal of Mechanical Engineering 68(2022)3, 143-154
145 Thermal Mechanical Coupling Analysis of a Flexible Spoke Non-pneumatic Tire 
ring. The circumferential spokes of the flexible 
spokes are arranged in a layered structure along the 
radial direction of the tire, and the side spokes of 
the flexible spokes are combined as a wave line by 
“curved surface” treatment, with a curvature of 1/15. 
The flexible spoke non-pneumatic tire comprises 
a rigid hub, a flexible spoke support body made of 
PU material and a rubber tread. The flexible spoke 
support body connects the hub and the tread as a 
whole. In this case, the rigid hub is composed of the 
rim, the hub bulge and the hub adapter assembly, 
and the rubber tread contains two layers of wire belt. 
The flexible spoke non-pneumatic tire based on the 
unit configuration method is shown in Fig. 1, and the 
detailed structural parameters are shown in Table 1.
Table 1.  Detailed structural parameters of the tire
Structure Parameters
Tire model 195/50N16
Tread width [mm] 195
Outside diameter [mm] 602
Rim diameter [mm] 406.4
Spoke thickness [mm] 5
Curvature of spoke 1/15
Construction angle [°] 8
Number of unit groups 45
1.2  Construction of Numerical Analysis Model
The flexible spoke support body is the main structure 
of the flexible spoke non-pneumatic tire. The heat 
generation of the entire tire mainly comes from the 
flexible spoke support body. In the steady-state rolling 
process, the frictional heat generated by the contact 
between the tread and the ground is small and quickly 
carried away by the air, which has little influence on 
the overall temperature distribution of the flexible 
spoke non-pneumatic tire. 
Therefore, the influence of the tread on the 
temperature distribution of the flexible spoke non-
pneumatic tire can be ignored in the numerical 
analysis. To obtain the basic material properties of the 
PU material forming the flexible spoke support body, 
uniaxial tensile tests of the PU material were carried 
out by using an electronic universal testing machine to 
obtain the stress-strain relationship, as shown in Fig. 
2.
For nonlinear materials, both the fitting accuracy 
and the stability are high by using Yeoh material 
constitutive model. The strain energy function of the 
hyper-elastic Yeoh model is as follows:
 
WC I
CI CI CI
i
i
i

  

 01
1
3
10 12 01
2
30 1
3
3
33 3
()
()() () , (1)
where W is strain energy density; I
1
 is the first 
invariant of principal elongation ratio; C
10
, C
20
, C
30
 
are material constants. In the first step of our study, 
the PU material is modelled as nonlinear elastic for 
simplification. Based on this approach, the mechanical 
loadings in the spokes of the non-pneumatic tire can 
be estimated.
Fig. 2.  Stress-strain relationship of PU material
The mesh of the flexible spoke non-pneumatic 
tire was divided. Since the hexahedral element has a 
higher accuracy in the calculations and is convenient 
for extracting the stress and the strain of the flexible 
spoke unit, the mesh of the flexible spoke unit consists 
of C3D8RH elements. The element number of a 
flexible spoke unit was 6,909, and that of the entire 
flexible spoke non-pneumatic tire was 193,050. The 
analysis steps and interaction settings of the flexible 
spoke non-pneumatic tire were carried out, and the 
friction coefficient was set to 0.5.
1.3  Verification of the Effectiveness of the Numerical 
Analysis Model
The static analysis of the flexible spoke non-
pneumatic tire was carried out in ABAQUS, and the 
vertical load was evenly increased to the maximum 
level of 5000 N with intervals of 500 N. The loading 
test of the flexible spoke non-pneumatic tire under the 
same working condition was carried out based on the 
tire testing machine. After the completion of the test, 
the axial sinking under different vertical loads was 

Strojniški vestnik - Journal of Mechanical Engineering 68(2022)3, 143-154
146 Fu, H. – Liang, X. – Wang, Y. – Ku, L. – Xiao, Z.
extracted, and the simulated and measured data were 
presented as vertical stiffness curves. The comparison 
of vertical stiffness curves of the flexible spoke non-
pneumatic tire simulation and the test is shown in Fig. 
3. RP represents the centre of the flexible spoke non-
pneumatic tire.
It can be seen from Fig. 3 that the vertical 
stiffness of the flexible spoke non-pneumatic tire in 
the simulation and test tended to be consistent. After 
calculation, the relative error between the simulation 
and the test verification was only 4.7 %, which 
demonstrates the accuracy of the numerical analysis 
model.
Fig. 3.  Comparison of vertical stiffness curves of simulation and 
test
2 THERMAL-MECHANICAL COUPLING ANALYSIS  
OF THE FLEXIBLE SPOKE NON-PNEUMATIC TIRE
2.1  Thermal-mechanical Sequential Coupling Method
In this study, the thermo-mechanical sequential 
coupling method was used to analyse the mechanical 
fields of the flexible spoke non-pneumatic tire firstly, 
and then the analysis results of mechanical fields 
were used as input conditions for the thermal field 
analysis. Finally, the thermo-mechanical coupling 
characteristics of the flexible spoke non-pneumatic 
tire were obtained. Fig. 4 shows the flow chart of the 
thermo-mechanical coupling analysis of the flexible 
spoke non-pneumatic tire, which mainly consists of 
deformation analysis, energy loss analysis and heat 
conduction analysis. In the deformation analysis, the 
static and dynamic mechanical characteristics were 
analysed using ABAQUS software. The accuracy of 
the numerical analysis model was verified in the static 
analysis. The stress and strain of each mesh element 
during the steady-state rolling process of a flexible 
spoke unit were extracted in the dynamic analysis. 
In the energy loss analysis, a Fourier transformation 
of the equivalent stress and equivalent strain was 
carried out using MATLAB software, and the lagging 
energy loss and heat generation rate were calculated. 
In the heat conduction analysis, a user subroutine was 
written to define the internal heat source, to endow 
the material with thermal properties and to set the 
thermal boundary conditions. Finally, the steady-state 
temperature field distribution pattern of the flexible 
spoke non-pneumatic tire under thermal-mechanical 
coupling was obtained.
2.2  Deformation Analysis
In the deformation analysis, the steady-state rolling 
of the flexible spoke non-pneumatic tire was carried 
out under a vertical load of 5000 N and driving speeds 
of 45 km/h, 65 km/h and 85 km/h, respectively. 
When the driving speed was 85 km/h, vertical loads 
of 3000 N, 4000 N, and 5000 N were applied to the 
non-pneumatic tire. The stress and strain of each 
mesh element in the flexible spoke unit under a 
rolling period (one cycle) were extracted. The data  
contained a total of 90 stress and strain data sets in 
one rolling period. Each set contains six components: 
S11 (LE11), S22 (LE22), S33 (LE33), S12 (LE12), 
S13 (LE13) and S23 (LE23). It was extracted once 
per mesh element at an interval of 4°. The stress and 
strain curves of a mesh element under a load of 5000 
N and a driving speed of 85 km/h are shown in Fig. 
5. This work provides a source of data for subsequent 
calculations of equivalent stress and equivalent strain.
2.3  Energy Loss Analysis
Fig. 6 shows the stress-strain phase diagram of the 
viscoelastic PU material. It can be seen that the stress 
and the strain of PU are not synchronized during 

Strojniški vestnik - Journal of Mechanical Engineering 68(2022)3, 143-154
147 Thermal Mechanical Coupling Analysis of a Flexible Spoke Non-pneumatic Tire 
Fig. 4.  Flow chart of thermal mechanical coupling characteristics analysis
a)              b) 
Fig. 5. a) Stress and b) strain curve of a mesh element
deformation, and the lagging phase angle δ appeared, 
as shown in Fig. 6a. Under the action of a cyclic 
load, the non-synchronization of stress and strain 
caused a hysteresis loop in the stress-strain curve 
of the material. The hysteresis area is basically the 
energy loss of the viscoelastic material, as shown in 
Fig. 6b. Assuming that the PU material is subjected 
to sinusoidal stress and strain, both variables can be 
expressed as:
 
 
 





0
0
sin
sin
()
,
t
t
 (2)
where σ is the stress; ε is the strain; σ
0
 is the stress 
amplitude; ε
0
 is the strain amplitude; ω is the angular 
frequency; δ is the lagging phase angle; t is the time.
According to the stress-strain relationship, the 
energy loss of the non-pneumatic tire in a rolling 
period T can be expressed as:

Strojniški vestnik - Journal of Mechanical Engineering 68(2022)3, 143-154
148 Fu, H. – Liang, X. – Wang, Y. – Ku, L. – Xiao, Z.
   

d
T
0
. (3)
Substituting Eq. (2) into Eq. (3) yields:
    
00
sin, (4)
where, sin δ is the energy loss coefficient.
a) 
b) 
Fig. 6.  Stress-strain phase diagram;  
a) stress-strain phase relationship, and b) lagging loop
The energy loss of nonlinear viscoelastic PU 
materials mainly comes from the deformation of 
the materials. According to the strength theory of 
materials, the state of nonlinear materials can be 
expressed by an equivalent stress and an equivalent 
strain, which can be expressed as follows:
 
  
 

 
  

3
22
6
3
22
22
22 22
1
2
[( )( )
() () ]
xy yz
zx xy yz zx
[ [( )( )
() () ]
 
 
xy yz
zx xy yz zx
 
  









22
22 22
1
2
6
 

. (5)
According to Eq. (5), the equivalent stress σ and 
the equivalent strain ε of each mesh element were 
calculated in MATLAB, and fitted by the Fourier 
series fitting method based on the least square method. 
The related equation is as follows:
 
  
 
 
 




01
1
2
01
2
2
2
1
[c os() sin( )]
sin(
n
n
m
n
nn
n
m
nt nt
n 
  
 

t
nt nt
n
n
n
m
n
nn
n

 
 



)
[c os() sin( )]
01
1
2
01
2
2
2
1 1
m
n
nt















sin( )
.


 (6)
The constant m represents the fitting order. For 
most of the data, when m = 8, a better fitting value 
can be obtained, but a small part of the data loses its 
meaning. Therefore, we set five different orders of 
fitting from m = 8 to m = 4; when m = 8, the data lose 
its meaning, m = 7 will be used for fitting, and so on.
a) 
b) 
Fig. 7.  Fitting curve of equivalent stress and strain of a mesh 
element; a) equivalent stress; b) equivalent strain
Fig. 7 shows the fitting curves of the equivalent 
stress and the equivalent strain of a randomly selected 
mesh element. It is clear that the equivalent stress and 

Strojniški vestnik - Journal of Mechanical Engineering 68(2022)3, 143-154
149 Thermal Mechanical Coupling Analysis of a Flexible Spoke Non-pneumatic Tire 
equivalent strain cycle are anharmonic under steady-
state rolling conditions. For the equivalent stress and 
equivalent strain cycle of anharmonic variation, it 
is necessary to express them as the superposition of 
a group of harmonics by the Fourier transformation, 
so as to obtain the equivalent stress amplitude 
and equivalent strain amplitude of each harmonic 
component at its corresponding frequency.
According to Eq. (6), the corresponding 
equivalent stress amplitude σ
n
 and equivalent strain 
amplitude ε
n
 can be calculated as follows:
 
 
 
nn n
nn n







1
2
2
2
1
2
2
2
. (7)
According to Eqs. (4) and (7), the energy loss 
ξ
i
 of each mesh element of the flexible spoke non-
pneumatic tire in one rolling period T can be calculated 
according to Eq. (8):
   
in
n
m
n
n 


() sin.
1
 (8)
The constant σ
n
 represents the equivalent stress 
amplitude; ε
n
 represents the equivalent strain 
amplitude; sin δ represents the energy loss coefficient; 
m is the fitting order; i represents the element number 
(1 to 4170).
The ratio of the energy loss of each mesh element 
to the rolling period T is equal to the heat generation 
rate Q
i
 of each element, which can be expressed as 
follows:
 QT
ii
 /, (9)
here, ξ
i
 is the energy loss of each mesh element, T 
is the rolling period, T = πd / v in [s], d is the outer 
diameter of the tire [m], and v is the tire rolling speed 
[m/s].
2.4  Heat Conduction Analysis
In the heat conduction analysis, the heat generation 
rate calculated by ABAQUS is assigned to each mesh 
element as the heat load to complete the definition of 
the internal heat source. In this paper, two subroutines 
were used to define the internal heat sources: the 
subroutines USDFLD was used to redefine the field 
variables, and the subroutine HETVAL was used to 
define the internal heat source.
For the numerical simulation of the temperature 
field of the non-pneumatic tire, the following four 
assumptions are generally considered:
(1) The material properties of each part of the non-
pneumatic tire are not influenced by temperature, 
and the entire energy loss in the rolling process is 
converted into heat, without heat loss.
(2) The temperature distribution is steady, and there 
is no temperature gradient in the circumferential 
direction of the non-pneumatic tire, meaning 
that the temperature distribution of each flexible 
spoke unit is exactly the same.
(3) The material of each part of the tire is isotropic.
(4) The non-pneumatic tire will be in thermal 
equilibrium after rolling at a constant speed for 
some time.
According to the above assumptions, the material 
is endowed with thermal properties and thermal 
boundary conditions are set. The thermal properties of 
the PU material mainly include thermal conductivity, 
thermal expansion coefficient, specific heat and 
convective thermal transfer coefficient, as shown in 
Table 2.
Table 2.  Thermal properties of PU material
Thermal properties Parameter
Thermal conductivity k [W/(m·K)] 0.3
Thermal expansion coefficient a [°C
–1
] 0.0002
Specific heat c [J/(kg·K)] 2×10
9
The thermal boundary conditions of the flexible 
spoke non-pneumatic tire include the tread boundary 
in contact with the ground, body surface boundary in 
contact with air and inner circle boundary in contact 
with the rigid hub. In the actual rolling process, the 
heat exchange between the tread and the road, as well 
as that between the flexible spoke inner ring and the 
rigid hub, can be calculated in the form of thermal 
conduction. The heat exchange between the flexible 
spoke support body and the air can be calculated in the 
form of convective thermal transfer. The convective 
heat transfer coefficient is approximately described 
according to the following equation:
 h 22
08 4
.,
.
 (10)
here, h is the convective heat transfer coefficient of 
the surface boundary of the body [W/(m
2
∙K)], and v is 
the tire rolling speed in [m/s].

Strojniški vestnik - Journal of Mechanical Engineering 68(2022)3, 143-154
150 Fu, H. – Liang, X. – Wang, Y. – Ku, L. – Xiao, Z.
axis and lateral axis of flexible spoke unit. According 
to Figs. 9a, b, and c, with the increase in the driving 
speed, the temperature distribution of the flexible 
spoke non-pneumatic tire was almost unchanged. This 
indicates that the driving speed did not influence the 
temperature distribution patterns and characteristics 
of the non-pneumatic tire, but both the overall 
temperature and the maximum temperature slightly 
increase with the increase of the driving speed; in fact, 
the maximum temperature increased from 58.3 ℃ to 
71.9 ℃. According to the energy loss analysis, with 
the increase of driving speed, the rolling period of 
the non-pneumatic tire became gradually shorter. The 
heat generation rate of each mesh element increased 
as well, which implies the overall temperature of the 
non-pneumatic tire increased. It can be seen from 
Figs. 9d and e that under different driving speeds, the 
node temperature on the vertical axis and lateral axis 
of the flexible spoke unit showed a tendency of first 
increasing and then decreasing, and the temperature at 
the centre of the flexible spoke was higher than that 
at the edge of it. With the increase of driving speed, 
the temperature rise in the centre of the flexible 
spoke is greater than that at the edge of it. When the 
driving speed increased from 45 km/h to 65 km/h, the 
maximum temperature of the node was increased by 
4.19 ℃; when the driving speed increased from 65 
km/h to 85 km/h, the maximum temperature of the 
node was increased by 8.44 ℃, indicating that the 
influence of the driving speed on the temperature 
change of the non-pneumatic tire is greater under high 
speed than under the low speed condition.
3  RESULT ANALYSIS
3.1  Analysis of Overall Distribution Trend of Steady-state 
Temperature Field
Fig. 8 shows the comparison diagram of steady-
state temperature field distribution and the stress 
distribution under a vertical load of 5000 N and 
a driving speed of 85 km/h. The vertical load led 
to greater deformation in the middle part of the 
flexible spoke unit, causing greater stress and strain 
and increased energy loss, resulting in more heat 
generation. Therefore, the high-temperature region 
is mainly distributed in the middle bending part of 
the flexible spoke unit, which is consistent with the 
main stress deformation part. The overall temperature 
gradually decreases from the centre of the flexible 
spoke unit to the surrounding area, and the central 
temperature reached 71.9 ℃. The problem of local 
high temperatures can be solved by improving the 
structure of the flexible spoke unit and reducing the 
stress concentration.
3.2  Influence of Driving Speed on Steady-state Temperature 
Field Distribution
Fig. 9 shows the steady-state temperature field 
distribution trend of the flexible spoke non-pneumatic 
tire under a vertical load of 5000 N at different driving 
speeds. Figs. 9a, b, and c show the temperature 
distributions of a flexible spoke unit for different 
driving speeds; Figs. 9d and e show the temperature 
distribution trend maps of each node on the vertical 
Fig. 8.  Comparison diagram of steady-state temperature field distribution and stress distribution

Strojniški vestnik - Journal of Mechanical Engineering 68(2022)3, 143-154
151 Thermal Mechanical Coupling Analysis of a Flexible Spoke Non-pneumatic Tire 
3.3  Influence of Load on Steady-state Temperature Field 
Distribution
Fig. 10 shows the steady-state temperature field 
distribution trend of the flexible spoke non-pneumatic 
tire under different loads at the driving speed of 85 
km/h. Figs. 10a, b, and c show the temperature 
field distribution maps of the flexible spoke unit 
under different loads, and Figs. 10d and e show the 
temperature distribution trend maps of each node on 
the vertical axis and lateral axis of the flexible spoke 
unit. 
According to Figs. 10 a, b, and c, the change of 
load has little influence on the temperature distribution 
of the flexible spoke non-pneumatic tire, and the high-
temperature region always existed in the central part 
of the flexible spoke unit. However, the overall and 
maximum temperature of the non-pneumatic tire 
increased significantly with the increase of the load, 
and, in fact, the maximum temperature increases 
from 31.9 ℃ to 71.9 ℃. According to the energy loss 
analysis, with the increase of the load, the sinking 
amount of non-pneumatic tire gradually increased, 
and the deformation amplitude of each part of the 
tire increased as well, which increased the stress 
amplitude, strain amplitude, and the energy loss of 
each mesh element, resulting in an increase of the 
heat generation rate. It can be seen from Figs. 10 d 
and e that under different loads, the node temperature 
on the vertical axis and lateral axis of the flexible 
spoke unit showed a tendency of first increasing and 
then decreasing, and the temperature at the centre of 
the flexible spoke was higher than that at the edge of 
the flexible spoke. With the increase of the load, the 
temperature rise at the centre of the flexible spoke 
was significantly greater than that at the edge of 
the flexible spoke. In addition, the temperature rise 
amplitude caused by the load increasing from 3000 
N to 4000 N was almost the same as that caused by 
the load increasing from 4000 N to 5000 N. This 
Fig. 9.  Steady-state temperature field distribution under different driving speeds

Strojniški vestnik - Journal of Mechanical Engineering 68(2022)3, 143-154
152 Fu, H. – Liang, X. – Wang, Y. – Ku, L. – Xiao, Z.
indicates that the influence of the load change on the 
temperature of the flexible spoke non-pneumatic tire 
has nothing to do with the loading conditions of the 
tire.
4  CONCLUSIONS
The 3D model and numerical analysis model of 
the 195/50N16 flexible spoke non-pneumatic tire 
were established based on the unit configuration 
method. The thermo-mechanical sequential coupling 
method was used to carry out a deformation analysis, 
an energy loss analysis, and a heat conduction 
analysis on the flexible-spoke non-pneumatic tire. 
The temperature distribution of the flexible spoke 
non-pneumatic tire under the influence of thermo-
mechanical coupling was obtained. After comparative 
analysis, the conclusions are as follows:
1. The high temperature region of the flexible 
spoke non-pneumatic tire is mainly concentrated 
in the middle bending part of the flexible spoke 
unit, which is consistent with the main stress 
deformation part. The overall temperature 
gradually decreases from the centre of the flexible 
spoke to the surrounding parts.
2. Under the condition of constant vertical load, 
the overall temperature and the maximum 
temperature of the non-pneumatic tire increase 
slightly with the increase of the driving speed. 
The centre temperature of the flexible spoke 
unit is higher than the edge temperature of the 
flexible spoke unit, and with the increase of the 
driving speed, the temperature rise of the centre 
of the flexible spoke unit is greater than that of 
the edge of the flexible spoke unit. The influence 
of the driving speed on the temperature change of 
Fig. 10.  Steady-state temperature field distribution under different loads

Strojniški vestnik - Journal of Mechanical Engineering 68(2022)3, 143-154
153 Thermal Mechanical Coupling Analysis of a Flexible Spoke Non-pneumatic Tire 
the non-pneumatic tire is greater at higher speeds 
than that at lower speeds.
3. Under the condition of constant driving speed, 
the overall temperature and the maximum 
temperature of the non-pneumatic tire increase 
significantly with the increase of the load. The 
centre temperature of the flexible spoke unit is 
higher than the edge temperature of the flexible 
spoke unit. With the increase of the vertical load, 
the temperature rise of the centre of the flexible 
spoke unit is significantly greater than that of the 
edge of the flexible spoke unit, and the overall 
temperature rise is larger. The influence of load 
change on the temperature of the flexible spoke 
non-pneumatic tire has nothing to do the loading 
conditions of the tire.
4. Both the driving speed and the vertical load 
affect the overall temperature and the maximum 
temperature of the flexible spoke non-pneumatic 
tire. However, the load has a significantly 
larger influence on the temperature change than 
the driving speed. Therefore, it is possible to 
increase the driving speed appropriately under 
the condition of reducing the load, and thus solve 
the problem of high temperature failure caused by 
high-speed operation for current non-pneumatic 
tires.
5  ACKNOWLEDGEMENTS
This work was supported by the National Natural 
Science Foundation of China [No. 52002231]; 
the China Postdoctoral Science Foundation [No. 
2020M672029]; and the Shandong Provincial Natural 
Science Foundation [No. ZR2018PEE008].
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