Evaluation of Single Disc Springs

The equations provided in DIN 2092 are sufficiently accurate for evaluating disc springs. The equations in DIN 2092 are essentially derived from calculations developed by ALMEN and LÁSZLÓ.

According to these equations, the deformation behaviour of the disc spring is treated as a one-dimensional inversion of a circular ring of rectangular cross-section about an inversion centre point S. The resulting inverted stress condition is overlaid by a bending stress condition caused by the change in the cone angle resulting from the deflection. The cross-section of the disc spring remains rectangular so that force is always applied at the edges I and III.

The behaviour of the material is regarded as linear-elastic without limit. Residual stresses are not taken into account. The calculated stresses are nominal stresses. Mubea has computer-aided calculation programs at its disposal for the design of disc springs.

Disc springs without contact surfaces with force application per DIN

Specifications

The typical characteristic load curve for a disc spring is defined by the ho/t ratio. Assuming unrestricted spring deformation and adherence to permissible load limits, the characteristic load curves shown in the Fig. the page before are obtained. The characteristic curves shown in the Fig. above are specifically for Series A, B and C springs standardised according to DIN 2093.

 

Fig.: Calculated characteristic curves for disc springs to DIN 2093 Series A, B and C as well as the recommended deflection limit. The upper section of the graph shows the average spring parameters for the standard spring series.

Disc springs without contact surfaces with force application per DIN

Load stresses

The fatigue life of the disc spring is dependent upon the stresses thatrun tangentially, while the stresses running radially are negligibly small. As a matter of principle, compressive stresses act on the upper side of the disc and tensile stresses on the lower side.

The calculated stresses do not correspond to the actual stresses in the spring. This is due to residual stresses caused by shot-peening and by the pre-setting of the disc springs in production. The actual stresses are a combination of the already existing residual stresses and the load stresses (see Fig.). To calculate the dynamic strength of the disc springs, the tensile stress at the underside of the disc are decisive. Due to our production methods there are residual p stresses that result in calculated stress levels that are higher than the actual stress levels.

Depending on the ho/t-ratio, the maximum tensile stress will occur at cross-section point II (inner diameter bottom) or III (outer diameter bottom).

On the upper side of the spring, the maximum compressive stress occurs on inside edge of the spring at cross-section point I. This stress is decisive for the setting loss of the spring. Setting is caused by plastic deformation of the disc spring due to high deflections that exceed the elastic limit of the material. This results in a reduction in the free height of the spring.

 

Fig.: Overlaying of load stresses and residual stresses to give total stresses

Disc springs without contact surfaces with force application through shortened lever arms

The moment required for compression is, on the deformation model usually used as a basis, as great as for the load applied via cross-sectional corners I and III. Because of the shortened lever arma greater load F' > F is required to achieve the same spring deflection. The result is a steeper characteristic load curve when compared with a spring that does not have a shortened lever arm.

The calculated load stress is not influenced by the location of load application. It depends only on the variation of the cone angle (see Fig.).

Fig.: Dimensional ratios when load is applied via shortened lever arms

Disc springs with contact surfaces

Contact surfaces are used for disc springs with a thickness greater than 6 mm (Group 3 of DIN 2093). This provides a larger area for load application thereby reducing the friction on the guide elements. The location of the applied force is shifted from de to de' on the outside and from di to di' on the inside. This results in a shortening of the lever arm and an increase in the characteristic load curve (see Fig.).

Disc springs with contact surfaces have the same design force F (at a deflection s = 0.75 x ho) as disc springs without contact surfaces with the same dimensions De, Di and lo.

Fig.: Comparison of the calculated characteristic curves of disc springs with (right) and without (left) seating faces.

The resulting increase in force is counteracted by a reduction in the disc thickness. Due to the requirement for identical overall height lo, the disc spring with contact surfaces will have a greater contact angle o' > o. This results in a characteristic curve which, with the exception of the common design point F' (s=0.75 ho) = F (s=0.75 ho), deviates slightly from that for the standard disc spring (see Fig.).

Comparison of the calculated characteristic curves of disc springs with and without seating faces

The reduction of the disc thickness from t to t' is specified in DIN 2093. The average ratio of disc thickness t' and t is 0.938 for springs of Series A and B and 0.995 for springs of Series C. The load stresses are calculated for the corner points I … IV of the cross section, which no longer exist because of edge rounding. Therefore, the calculated stresses are somewhat higher than. Since these are only nominal values, the error is insignificant.

Special cases

Conversions when using special materials

The characteristic equations applicable to a sharp-edged rectangular cross section yield forces which are 8 % to 9 % too high for spring steel where E = 206000 N/mm2 and µ = 0.3. This is compensated for by the shortening of the lever arm due to the radii at points I and III. Therefore the calculated and measured loads for steel springs are in good agreement. This is no longer true if special materials, especially ones with higher POISSON numbers µ, are used.

Extremely thin disc springs

In the case of disc springs where De/t >> 40 the characteristic equation yields a force which is too high. In this case, the cross section of the spring is no longer rectangular and bowing of the lateral surface line must be taken into consideration (especially if a finite element analysis is made).

Extremely small diameter ratio

In the case of disc springs with De/Di < 1.8, the shortening of the lever arm must be taken into consideration when calculating the characteristic load curve. Otherwise, the calculated load will be too low.