*Corr. Author’s Address: National Aviation University, Lubomyr Huzar 1, 03680 Kyiv, Ukraine, anatoliykor80@gmail.com 363
Strojniški vestnik - Journal of Mechanical Engineering 67(2021)7-8, 363-368 Received for review: 2021-02-21
© 2021 Journal of Mechanical Engineering. All rights reserved.  Received revised form: 2021-05-11
DOI:10.5545/sv-jme.2021.7139 Original Scientific Paper Accepted for publication: 2021-06-21
Predictive Estimation of Sliding Bearing Load-Carrying Capacity 
and Tribological Durability
Chernets, M. – Opielak, M. – Kornienko, A. – Radko, O.
Myron Chernets
1
 – Marek Opielak
2
 – Anatolii Kornienko
1,*
 – Oleg Radko
3
1
Aerospace Faculty, National Aviation University, Ukraine 
2
Lublin University of Technology, Poland 
3
National Defence University of Ukraine named after Ivan Cherniakhovskyi, Ukraine
A computational method is presented as a method for solving a plane contact problem of the theory of elasticity to determine the contact 
strength and tribological durability of sliding bearings. The effect of load and radial clearance on the initial contact pressures and their 
reduction due to wear is studied. The durability of the bearing is estimated. Qualitative and quantitative regularities of changes in contact 
parameters and durability from the factors under study are established. In particular, it has been shown that both contact angles and 
maximum contact pressures are approximately linearly dependent on the load, and the durability decreases nonlinearly with increasing load.
Keywords: sliding bearing, wear-contact problem, contact and tribocontact parameters, wear, durability
Highlights
• 	 A computational method for sliding bearings is presented.
• 	 A tribokinetic wear model for sliding friction has been developed.
• 	 Evaluation of contact pressures and durability has been carried out.
• 	 Regularities of the influence of wear on contact characteristics and resources have been established.
0  INTRODUCTION
The use of sliding bearings as one of the common 
friction units (Fig. 1) currently remains quite 
significant, where the use of rolling bearings is 
impossible or impractical. The main recommendations 
for their use are high load capacity, use at miniature or 
large shaft diameters, at significant speeds, at shock 
loads, small radial dimensions, low noise, damping 
capacity, etc. The range of their application in practice 
is very diverse [1]. They work in various conditions, 
not only at liquid conditions but also at boundary 
and dry friction in separate cases. Under certain 
operating conditions, special alloys can be used to 
make the shaft, for example [2]. It should be noted that 
the number of new types of composite materials for 
metal bearings is growing rapidly [3]. Therefore, the 
estimated assessment of their bearing capacity, wear, 
and durability at the design stage is an urgent task.
The solutions of the corresponding wear-contact 
problems for such sliding tribosystem are known 
in the literature [4] to [16]). In particular, in [7] and 
[8], a model of the sliding bearing wear in conditions 
of boundary friction was obtained in the form of 
dependence of the wear rate on the dimensionless 
complexes of contact pressure and sliding velocity. 
The parameters of wear resistance in the model were 
determined by the calculation-experimental method 
on the basis of wear tests with the “cone – three 
balls” structure under variable contact conditions. 
Such a non-standard friction structure (ISO 7148-
2) significantly limits the use of this wear model. 
Paper [9] presents the results of an experimental 
and numerical study of fibre-reinforced polymer 
bearings. The authors developed a two-dimensional 
finite-element model to study the stresses in the 
bearing and researched three-dimensional quasi-
static and two-dimensional dynamic models. A study 
on the effect of the clearance on the contact stresses 
and kinematics of large-scale composite bearings in 
[10] was conducted experimentally, using the finite 
element method. The results of the wear study are not 
given. Paper [11] presents an adaptive wear-modelling 
method in plain bearings. Validation was done for a 
laminated polymeric composite bearing. A study of 
the effect of clearance on the wear and the evolution 
of contact pressure due to wear was performed. In 
[12], the method of triboelements and the modelling of 
the behaviour of sliding tribosystems on the basis of 
Archard’s law of abrasive wear with use of ANSYS are 
presented. Paper [13] aimed to study the wear of a fine 
elastic layer with the rigid bearing and shaft with the 
same method. Numerical analysis of the effect of the 
external load scattering and the initial radial clearance 
in the bearing on its wear was carried out. Paper [14] 
presents the results of a numerical simulation with 
the triboelement method to determine the wear of a 
thin elastic layer on a hard bushing of a cylindrical 
linear plain bearing. These works use Arсhard’s law 
of wear. Methods for estimating the parameters of the 

Strojniški vestnik - Journal of Mechanical Engineering 67(2021)7-8, 363-368
364 Chernets, M. – Opielak, M. – Kornienko, A. – Radko, O.
durability model under the mechanism of high-cycle 
fatigue under sliding friction conditions are proposed 
in [15]. The results are recommended for analysing the 
reliability and durability of friction units of machines 
under fatigue wear conditions. For the numerical 
modelling of the wear kinetics of tribosystems, an 
iterative approach is presented in [16], which takes 
into account the discrete states and operational factors 
affecting wear. The wear of the tribosystem elements 
is calculated.
The abovementioned methods have not yet 
found proper practical application due to the use of a 
simplified Archard’s law of abrasive wear assuming 
the wear intensity linear dependence on the contact 
pressure and the friction path, although this type of 
wear is unacceptable in sliding bearings. Today, in 
engineering practice and in the design calculations 
of sliding bearings, it is customary to use two main 
parameters: the average pressure p and the parameter 
pv. This simplification is very approximate, because 
the contact area characteristics depend not only on 
the load and the diameter of the shaft journal but also 
significantly on the radial clearance in the bearing and 
the elastic characteristics of the element materials. The 
last of the specified significant factors of influence are 
not considered in any way in the specified criteria, and 
the problem of predictive evaluation of sliding bearings 
durability at the design stage is not considered here 
at all. Therefore, reasonable methods for calculating 
bearings should be based on the contact problems of 
the theory of elasticity for cylindrical bodies of close 
radii. This study aims to use the author’s generalized 
computational method [4] and [17] to [23] to estimate 
tribocontact pressures and durability when the 
bearing wears. This method is based on the concept 
[4], [17] and [18] of near-surface layers frictional-
fatigue destruction of tribosystem elements in the 
process of sliding friction. In particular, in [17], the 
author presented a method of approximate solution of 
cylindrical sliding tribosystem consisting of elements 
with small non-circularity of its contours for the first 
time. The model of the triboprocess and the method of 
calculating the contact pressures were considered; in 
[18], a set of different contact problems is considered, 
taking into account wear for cylindrical tribosystems 
consisting of elements with non-circular contours; 
in [19], a cumulative analytical model of wear and 
durability of plain bearings is presented and schemes 
of plain bearings with different faceting of the shaft 
and the bushing are investigated; in [20], a generalized 
method for solving the contact problem for a 
cylindrical joint with complex faceting is presented. 
The parameters of one- and two-region contact 
are determined; in [21], according to the author’s 
cumulative model of wear of plain bearings with 
technological ovality of adjacent parts, the accuracy 
of calculations of their service life was evaluated. 
The developed express calculation method of the 
tribocontact interaction of the shaft and the bushing 
is given; in [22], the generalized cumulative model 
of research of wear kinetics for the plain bearings in 
the case of one- and two-region contact is given. The 
results of solving the nonclassical contact problem 
and the wear contact problem are presented; in [23], 
the solution of the wear-contact problem for a bearing 
with different faceting of the shaft is given. Using the 
cumulative model, the influence of the faceting on the 
service life of the bearing at the complete one-region 
and mixed-region contact was investigated.
Based on the above methods of solving a complex 
wear-contact problem of the theory of elasticity, an 
easier-to-implement engineering method is presented 
below.
1  WEAR TRIBOKINETIC MODEL OF SLIDING FRICTION
According to [4], the materials’ wear kinetics in sliding 
tribosystem is described by a system of ordinary 
differential equation:
 
k
k
k
v
h
t

1
1
d
d
() ,  (1)
where h
k
 is the linear wear function of tribosystem 
elements; γ
k 
is the wear rate of their materials; v is 
the sliding speed; t is the triboprocess time; Φ(τ) is 
the basic parameter of the model as the characteristic 
function of wear resistance of tribocouple materials;  
k = 1, 2 is the numbering of tribosystem elements.
The specific force of friction in mechanics and 
tribology is determined by the Amonton-Coulomb 
formula:
   f
r
, (2)
where f is the sliding friction coefficient;  
σ
r
 = –p(α) is the contact stress calculated according 
to the methods of the elasticity theory; p(α) are the 
contact pressures.
The characteristic function Φ
i
(τ
i
) of wear 
resistance of materials for discrete values of specific 
friction forces τ
i
 is established by the results of tribo-
experimental studies according to the method [18] and 
[19]:
 
ii
i
i
L
h
   , (3)

Strojniški vestnik - Journal of Mechanical Engineering 67(2021)7-8, 363-368
365 Predictive Estimation of Sliding Bearing Load-Carrying Capacity and Tribological Durability
where L = vt is the friction path; і = 1, 2, 3, ... are the 
levels of load in the tribo-experiment. 
Approximation of experimental values of wear 
resistance functions Φ
i
(τ
i
) is carried out by the relation 
[18] and [19]:
 
kk
k
m
k
m
B
k
k



 
 
0
0
, (4)
where B
k
, m
k
, τ
k0
 are the wear resistance characteristics 
of materials in the tribocouple.
2  FORMULATION OF A TRIBOCONTACT PROBLEM
The arrangement of the sliding bearing is presented in 
Fig. 1. Shaft 2 rotates at a constant angular velocity 
ω
2
. Under the influence of the reduced external 
load N = F / l, the contact pressures p(α) unknown in 
distribution and magnitude arise in the contact area. 
There is the radial clearance ε = R
1
 – R
2
 > 0 in the 
bearing. The materials of the shaft and the bushing 
usually have different elastic properties and different 
wear resistance. The bearing elements have different 
wear areas: the bushing 1 in the area 2R
2
α
0
 and shaft 
2 along the contour. The problem is solved as a plain 
problem of the elasticity theory, where the external 
load F on the shaft is related to the length of the 
journal l.
Fig. 1.  Scheme of the sliding bearing
When solving the problem, it is necessary to 
determine: initial contact angle 2α
0
; maximum 
initial contact pressures p(0); tribocontact angle 
2α
0h
; maximum tribocontact pressures at wear  
p(0, t, h); bearing durability t at the accepted elements 
permissible wear h
k*
; bearing elements wear h
k
 during 
the accepted service life t
*
.
According to the methods in [4], [18], and [20], 
the equilibrium Eq. (5) is used to determine the initial 
contact semi-angle α
0
 under the action of the reduced 
load N = F/l, the journal radius R
2
 and the radial 
clearance ε.
    NR pd RE 







 22
2 0
0
0
4
4
() coss in ,  



 (5)
where E
e
R
e
EE
Z



cos
,,
2
0
2
12
4 4  /
 09 0
0
   ,   
 
ZE E    () () () () 11 11
11 22 21
  , 
kk
 34,
E
k
, μ
k
 are Young’s modules and Poisson’s ratios of 
shaft 1 and bearing bushing 2 materials. Eq. (5) is 
solved using the method of successive approximations, 
by ensuring the equality of its left and right parts with 
the accepted accuracy. Accordingly, to determine the 
maximum initial contact pressures p(0), which 
characterize the bearing load-carrying capacity, the 
developed method uses the formula:
 pE 0
2
0
 








tan . (6)
Determining the tribocontact semi-angle α
0h
 
while shaft and bearing bushing wear is carried out by 
a similar dependence as for the contact semi-angle α
0
:
 NR EC
hh
h
 






4
4
2
2 0
 

sin, (7)
where 
hk t
k
k
hK h   

max
()
,   hh h
12 1
, K
t
(1)
=1,  K
t
(2)

0
 are the overlap coefficients; C
h
 > 0 is the 
wear rate indicator; h
1*
 is the permissible bushing 
wear:
  



 
 

h
h
h
B
B
K
m
m
m
m t 1
2
1
1
2
11 02 0
22 01 0
2
1
2
2
1




 
 
,
  



 
 

h
h
h
B
B
K
m
m
m
m t 2
1
2
2
1
22 01 0
11 02 0
1
2
1
1
2




 
 
,
where     fp fE () tan 02
0
 is the maximum 
specific friction force acting at the wear process 
beginning when t = 0; h
1
, h
2
 are the linear wear of 
bushing and shaft, respectively.
Maximum contact pressure p(0, t, h) in the 
bearing at elements wear:

Strojniški vestnik - Journal of Mechanical Engineering 67(2021)7-8, 363-368
366 Chernets, M. – Opielak, M. – Kornienko, A. – Radko, O.
 pt hp ph () () () 00 0 ,, ,.  (8)
The change in the maximum initial contact 
pressure p(0, h) due to bearing elements wear is 
calculated as follows:
 ph EC
hh
h
() tan 0
2
0
,. 








 (9)
Since the bushing material is less wear-resistant 
than the shaft material and the bushing wears at 
a limited contact area, the bearing service life is 
determined by the bushing durability when it reaches 
the maximum permissible wear. According to [4], [22], 
and [23], taking into account the dependences, Eqs. 
(1), (2), (4), (6) and (9), the bearing service life t
* 
is 
calculated by the formula:
 
t
B
vC ShmK
SS CS
m
hh t
m
hhh
*
'




 

  

10 1
11
1
1
1
11

 
(1)
()
1 1
1


m
, (10)
where S = f · p(0) / ε, S
h
 = f · p(0, h) / (ε
h
 · C
h
), v = ω · R
2
.
If it is necessary to estimate the shaft wear 
h
2 
along its entire contour over time t
*
, the wear 
is calculated as follows (after the corresponding 
transformation of Eq. (10)):
 
h
CS Kh K
MS t
M
S
hh tt
m
m
2
2
2
2
2
1
2
1
1
2
2

 















() ()
()
()
'


 
, (11)
where M
B
vC Sm Kh K
m
hh tt
2
20 2
2
2
2
2
2
1

 


()
,
() () '
 and 
   hK Kh
tt 2
21
1
() ()
/.
3  RESULTS OF NUMERICAL SOLUTION
Data for calculation: N = 25 N, 62.5 N, 125 
N, F = Nl; D
2
 = 50 mm; l = D
2
; ω
2 
= 1 s
–1
,  
5 s
–1
; v = 25 mm/s, 125 mm/s; ε = 0.125 
mm, 0.25 mm; f = 0.05 at boundary friction;  
h
1*
 = 0.5 ε;  C
h 
= 0.05.
Bushing material: teen bronze Е
1 
= 1.2·10
5 
МPа, 
μ
1
 = 0.34; B
1 
= 1.9·10
9
, m
1 
= 0.76, τ
01 
= 0.1 MPa; shaft 
material: hardened steel Е
2 
= 2.1·10
5 
МPа, μ
2
 = 0.3;  
B
2 
= 4.9·10
9
, m
2 
= 0.66, τ
02 
= τ
01
.
The calculation was performed according to the 
given flow diagram (Fig. 2).
Fig. 2.  The flow diagram
The results of solving the considered wear-
contact problem are presented in Figs. 3 to 5.
■ ε = 0.25 mm
□ ε = 0.25 mm
● ε = 0.125 mm
○ ε = 0.125 mm
Fig. 3.  Dependences of initial contact angles on loading and their 
changing at wear: 2α
0
 in dashed lines, and 2α
0h
 in solid lines
For the initial contact angle 2α
0 
in the studied 
range of loads, there is an almost linear dependence 
in their increase. Naturally, with smaller radial 
clearances, these angles will be larger. When the 
accepted permissible wear is reached, the tribocontact 
angles 2α
0h
 increase up to 2  times at both values of 
the radial clearances.

Strojniški vestnik - Journal of Mechanical Engineering 67(2021)7-8, 363-368
367 Predictive Estimation of Sliding Bearing Load-Carrying Capacity and Tribological Durability
■ ε = 0.25 mm
□ ε = 0.25 mm
● ε = 0.125 mm
○ ε = 0.125 mm
Fig. 4.  Dependences of initial contact angles on loading and their 
changing at wear: p(0) in dashed lines, p(0, t, h) in solid lines
At relatively low loading, a nonlinear increase 
in the initial maximum pressures p(0) is observed, 
and a further increase in the load leads to their linear 
increase. The wear of the bronze bushing contributes 
to a significant reduction in pressure. Tribocontact 
pressures p(0, t, h) depend to varying degrees on radial 
clearance and wear.
■ ω = 1 rad/s, ε = 0.25 mm
□ ω = 5 rad/s, ε = 0.25 mm
● ω = 1 rad/s, ε = 0.125 mm
○ ω = 5 rad/s, ε = 0.125 mm
Fig. 5.  Dependence of the bearing service life on loading
As the load increases, the bearing service life 
decreases nonlinearly. With a fivefold increase in 
angular velocity, there is a directly proportional 
decrease in service life.
4  CONCLUSIONS
1. The presented method of predictive estimation 
of sliding bearings load-carrying capacity and 
tribological durability allows carrying out 
substantiated and effective research on such basic 
factors of influence as external loading, shaft 
diameter, radial clearances, and wear.
2.  An important feature of the method is the ability 
at the design stage to perform both the calculation 
of bearing service life, and the solution of inverse 
problem: the assessment of the bushing and 
shaft wear accepted service life. The solution 
is presented in a closed form, and this allows 
its implementation using the simplest software, 
starting from Excel (Figs. 3 to 5).
3.  It should also be noted that the method (presented 
in Section 2) can be used without any restrictions 
not only for the calculation of bearings with metal 
elements, as presented above, but also when 
the friction surfaces are coated with different 
composition and purpose (protective, antifriction, 
wear-resistant) coatings. 
4.  On the basis of this method, it is possible to 
carry out the optimization on criteria of contact 
strength, wear resistance and durability, as well 
as an optimum choice of materials at the stage of 
the bearings designing. It is also very promising 
to use the method for hybrid bearings, with 
materials that are significantly different in their 
properties are used.
5.  It is an advanced method for the calculation 
of metal-polymer bearing assemblies, because 
there are no calculation methods for such friction 
assemblies.
6.  Solutions of this type of wear-contact problems 
can also be used to estimate the error of 
calculations obtained by various numerical 
methods (the finite element method, the boundary 
element method, etc.).
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