A New Calculation Method for Belleville Disc Springs with Contact Flats and Reduced Thickness

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Introduction

In 1867 Julien François Belleville of Dunkirk was granted French Patent Number 52399 for the design of a new kind of spring.

That spring was actually a conical shell which could be loaded along its axis either statically or dynamically with the possibility to use not only a single spring but also a stack of springs.

During the last several decades, thanks to their advantageous properties if compared with other types of springs, coned disc springs (also called “Belleville springs”) have been adopted in nearly all areas of technology from the creation of vertical seismic isolation systems (Kitamura, 2003) to the developing of tester for bolt torques in the wind tower field (Abasolo, 2011).

Compared with helical springs, using the same dimensions, disc springs can support a larger load and spring characteristics can be designed to be linear, regressive or, with a suitable arrangement, also progressive.

Furthermore, in comparison with helical springs, if the spring is properly dimensioned, it is possible to obtain a higher service life under dynamic loads

The most known paper about disc springs is unquestionably the one by Almen and László (1939) and its results are still used as standard method in norms as DIN 2092 (2006) or UNI 8736 (1985) to calculate stresses and displacements in coned disc springs.

Figure 1 shows the cross-section points I, II, III, and IV of a uniform thickness coned disc spring.

Section of a disc spring with uniform thickness

Almen and László based their theory on experimental observations according to which the cross-section of the spring merely rotates about a neutral point S0 (assumed to be on the middle line of the cross-section) without undergoing an appreciable deflection.

They agreed with Timoshenko in regarding the radial stresses as negligible and succeeded in calculating tangential stresses and displacements of a coned disc spring subjected to an axial load F uniformly distributed on inner and outer edges (Timoshenko, 1934).

The Curti and Orlando based their hypothesis on the resemblance of fleebly coned springs with circular flat thin plates and they succeeded in calculating not only displacements and tangential stresses, but also radial stresses. However, they concluded with a finding of very low radial stresses and suggested further theoretical and experimental analyses for a more complete evaluation of coned disc springs (Curti & Orlando, 1979).

In standard norms as DIN 2092 (DIN, 2006) and DIN 2093 (DIN, 2006), it is also described a particular kind of disc springs: disc springs with contact flats and reduced thickness.

For disc springs with a thickness of more than 6.00 mm, DIN 2093 specifies small contact surfaces at points I and III in addition to the rounded corners. Figure 2 shows a schematic cross-section of a spring in group 3 as per DIN 2093:

Section of a disc spring with contact flats and reduced thickness, t'

The aim of contact flats is to improve the definition of the point where the load is applied and, particularly for spring stacks, to reduce the friction at the guide rod. The result is a considerable reduction in the lever arm length and a corresponding increase in the spring load, this is in turn compensated by a reduction in the spring thickness from t to t'.

In order to calculate loads and deflections for this kind of coned disc springs, standard norms propose the method of Almen and László with a new parameter, K4, in addition but, as will be shown in this paper, the results are not as accurate as for springs without contact flats.

In this paper it will be shown, indeed, how the calculation of disc springs with contact flats and reduced thickness cannot be made using equations reported in standard norms which are, in fact, contradictory. A method to obtain more realistic results and remove contradictions is finally proposed in order to correct the standard norms.