


Rolling load in a cylindrical roller bearing (friction free Hertzian contact) 



Torque transmission in a shaft-hub connection with shrink fit 





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 



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. 



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 







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 



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 












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

