BOStrab Guidance Regulations (Technishe Regeln Spurführung - TR Sp) Appendix 4 Page 1 of 8

Appendix 4

Minimum wheel support width and/or minimum flange tip width

The investigation of both the minimum tread contact width and the minimum flange tip

width required to support the wheel can be undertaken using the same methodology

because both cases can be approximated to a Hertzian contact pairing, in which a line

contact can be assumed between a cylinder and a plane.

This is generally a good approximation because the curvature of the contact between

wheel and rail in the transverse direction is effectively flat when compared to the curvature

of the wheel in the longitudinal axis of the rails.

The criterion for the contact stress is the Hertzian compression with its maximum value

pmax. This value is determined by the material characteristics as well as of the wheel

diameter, the wheel load and the contact width, i.e. the minimum wheel tread contact width

or the minimum width of the flange tip, such that the stress through the Hertzian pressure

does not exceed the limits of the materials of the wheel and rail.

For the line contact between a cylinder and a flat plane the Hertzian pressure is calculated

from

( )2min

max 1 νπ −⋅⋅⋅⋅

=bd

EFp N

where

E Elastic modulus of the material

v Poisson’s Ratio for the material

pmax Hertzian Pressure

FN Maximum wheel load

d Wheel diameter

bmin Minimum contact width

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For steel, for which the Modulus of Elasticity ( E ) = 2.1 • 105 N/mm2 and the Poissons ratio

( v ) = 0.3, the smallest contact width bmin (in mm) is dependent upon the wheel diameter d

(in mm), the maximum wheel load FN (in N) and the Hertzian compression pmax (in N/mm2),

the material dependent limit plim which may not be exceeded, and is given by

dpF

b N

⋅

⋅= 2

limmin

73456

The determination of the material specific limit for plim results from assumptions for the

stress condition, that from the reference stress the corresponding (shear stress?) can be

determined from hypotheses for the deformation strain. Applying the reference stress to

the Tresca / St. Venant hypotheses, the resulting maximum shear stress for the material

σVSmax is given by

σVSmax = 0.6 plim

This reference stress must, in order to avoid large and cumulative permanent plastic

deformations, lie beneath the elastic limit for the material Rp0.2.

σVSmax < 0.6 Rp0.2

from which

plim < 1.67 Rp0.2

From that, the minimum wheel support width and/or the smallest flange tip width can be

calculated in relation to

• the elastic limit of the material strength with the Hertzian contact stress (pressure) plim

• the wheel load FN, and

• the wheel diameter d

BOStrab Guidance Regulations (Technishe Regeln Spurführung - TR Sp) Appendix 4 Page 3 of 8

The minimum wheel support width to be prepared bmin, that is to be allowed for especially

for in switch installations in narrow curves, and/or the minimal flange tip width required for

running in flat groove rails is significant. Examples of the resulting minimum width in

relation to wheel diameter and wheel load for various qualities of rail materials with varying

strengths are shown in diagrams 4.1, 4.2 and 4.3. The characteristic values for the wheel

material are used to calculate the minimum wheel support width if the rail material has the

greater strength. Examples for two wheel materials of differing strength are shown in

diagrams 4.4 and 4.5.

The calculation quoted here is for a simplified observation based approach to the

calculation of the minimum wheel support width and/or the minimum width of the flange tip

in relation to the various values for material strength, wheel diameter and wheel load.

More detailed evaluation, for example with Finite-element analysis methods, that model

the actual situation more exactly, or by proof under actual operating conditions can allow

for deviation from the values that can be determined from the calculation method

recommended above.

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Diagram 4.1: Calculated minimum wheel support width for an S700 grade rail having a yield strength of 460 MPa

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Diagram 4.2: Calculated minimum wheel support width for an S800 grade rail having a yield strength of 520 MPa

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Diagram 4.3: Calculated minimum wheel support width for an S900A grade rail having a yield strength of 580 MPa

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Diagram 4.4: Calculated minimum wheel support width for a tyre steel of yield strength 416 MPa

BOStrab Guidance Regulations (Technishe Regeln Spurführung - TR Sp) Appendix 4 Page 8 of 8

Diagram 4.5: Calculated minimum wheel support width for a tyre steel of yield strength 580 MPa