Rolling load in a cylindrical roller bearing (friction free Hertzian contact) 
Torque transmission in a shaft-hub connection with shrink fit 


 2009 PTC 23 

FAG-Cylindrical Roller Bearing 
NU314E, Load Rating C0=220 kN 


Shaft diameter 70 mm 
Housing diameter 150 mm 
Bearing width 35 mm 
Bearing inner ring outside diameter 89 mm 
Bearing inner ring outside diameter 89 mm 
Bearing outer ring inside diameter 133 mm 
Roller length 24 mm, load carrying 22 mm 
Roller diameter: 22 mm 
(13 rollers) 



Bearing and shaft material: Steel 
E=210000 MPa; .=0,3 
Housing material: Alu 
E=70000 MPa; .=0,3 
(alternatively also in Steel) 



Contact without friction 


Dr. Hans-Jrgen Bhmer 
(Schaeffler KG) must be thanked 
very much for the information and 
the discussions 

 2009 PTC 24 

The FAG roller bearing catalogue states that the contact pressure at the 
maximum stress position between rolling elements and race reaches 4000 MPa 
on reaching the static load rating C0 (for this bearing 220000 N). This is a 
notional value, calculated through the application of Hertzian contact theory 
assuming linear-elastic material. 


In reality when the bearing is subject to a load C a permanent plastic 

In reality when the bearing is subject to a load C0 a permanent plastic 
deformation would occur in the middle of the contact surfaces of the highest 
loaded roller and race of approximately 1/10000 the roller diameter. Due to high 
demands for positional accuracy required of the bearing, it should not be loaded 
as high as C0, for dynamic loading the bearing load must be much lower. 


There is no catalogue information advising what material the housing and shaft 
should be or what fit and bearing play were used as a basis for the 4000 MPa 
value. For the following studies, these values only serve as guidance to what 
stresses are to be expected in the rolling elements and the bearing races at 
various adopted extreme tolerances. 


 2009 PTC 25 

To calculate the influence from fitting tolerances to the bearing loading, the 
model will be analyzed with different extreme clearances: 
 A variant with minimum clearance: 
 Clearance to housing and shaft: 10 m 
 Bearing clearance also 10 m; this means, each rolling element is 5 m smaller than the half 
 
Bearing clearance also 10 m; this means, each rolling element is 5 m smaller than the half 
diameter difference between inner-and outer race ring of 22 mm (for this bearing size, this is 
equivalent to the minimum clearance of a high precise C1NA-clearance group bearing) 




 A variant with maximum clearance: 
 Clearance to housing and shaft : 100 m 
 
Bearing clearance 160 m, this means, each rolling element is 80 m smaller than the half 
diameter difference between inner-and outer race ring of 22 mm (for this bearing size, this is 
equivalent to the maximum clearance of a C5-clearance group bearing with increased play) 
Hint: Normal group C0-bearings of this size have 40-75 m clearance 
 
In addition, for the latter variant the soft Aluminum housing will be replaced by a stiffer steel 
housing, which should lead to higher contact pressures because of a more worse osculation 











 2009 PTC 26 

For all variants the load vector is applied in a way, that six or seven rolling 
elements are within the loaded half of the bearing 



 2009 PTC 27 

The idealization of the bearing assembly is 
difficult, since the Hertzian stress state 
within a rolling body  having the maximum 
comparative (shear) stress below the 
contact surface  is created by preventing of 
the axial transverse strain. Therefore, here 
the plane stress state cannot be used.
the plane stress state cannot be used. 



In opposite, the housing plate  outside the 
bearing load introduction  is just loaded in 
its plane, so here the plane stress condition 
would be fine for idealization 
Since here just the bearing loads are of 
interest, the plane strain condition will be 
selected 



 2009 PTC 28 

Ensuring the result quality through improved (refined) meshing and creation of 
contact specific measures 

 2009 PTC 29 

