1  Calculation Formulas for Flat Springs by Shapes

・1-1 Cantilever Springs with Rectangular Cross-section

The simplest flat spring is a cantilever spring with a rectangular cross-section.
When the fixed end is A and the free end is B, and the load P is applied to point B, the calculation formula is as follows.

Here, I is the second moment of area.

It is expressed by

, and when is large, it is

Thus, the calculation formula for when is large is

ν represents the Poisson’s ratio, and in the case of steel, ν ≈ 0.3. The maximum stress is at the fixed end and is represented by the formula below.

Table 2 shows the longitudinal elasticity modulus (E) value of the main flat spring materials.

・1-2 Calculation Formulas for Flat Springs with Trapezoidal Cantilever

Figure 2

As shown in Figure 2, when the plate thickness of the flat spring is fixed and the plate width changes linearly, the deflection of the free endis as per the following formula.

Formula 4

To calculate B in the formula, use the following two formulas according to the plate thickness.

If the plate is thick, use the formula below.

If the plate is thin, use the following formula:

Also, the value in the formula can be calculated from Figure 3 by β = b1 / b.

Figure 3

・1-3 Calculation Formulas for Flat Springs with Joggled Plate Width

Figure 4

As shown in Figure 4, the deflection at the free end of a flat spring with a fixed plate thickness and a joggled plate width is as per the following formula.

Formula 5

Here,、are the deflection and deflection angle of the joggled part A due to P, and is the deflection at the free end of the cantilever with length l2 and plate width .

・1-4 Calculation Formulas for Flat Springs with Circular Cantilever

Figure 5

When the vertical load (P) acts on the free end with the shape as shown in Figure 5 in which the center of the plate thickness is a straight line and the center line of the plate width is an arc, the deflection (δφ) at any position φ is as per the following formula.

Formula 6

Here, C represents the torsional strength of the plate.

・1-5 Calculation Formulas for Flat Springs with Arc-Shaped Cantilever

Figure 6

Generally, Castigliano’s theorem is used to calculate the deflection when a load is applied to a cantilever spring whose center line of plate thickness is an arc. The calculation results using this theorem are shown below.

When the vertical load (P) and the horizontal load (W) are simply applied to the position of the central angle on the arc-shaped flat spring shown in Figure 6, deflection in the y-direction and deflection in the x-direction at the central angle position are as follows.

Deflection due to P
When :

Formula 7

When :

Formula 8

Deflection due to W:

When :

Formula 9

When :

Formula 10

Figure 7

In Figure 7, δy and δx 、are as follows.

Formula 11

Formula 12

Figure 8

In Figure 8, δy and δx are as follows.

Formula 13

Formula 14

The maximum stress due to P always occurs at the fixed end.

Formula 15

The maximum stress due to W occurs at point A in Figure 8 for and at the fixed end for .

Formula 16

・1-6 Calculation Formula for Ring-shaped Flat Springs

Figure 9

The circular spring shown in Figure 9 is vertically symmetrical, and so the total deflection is twice the deflection of the shape in Figure 8.

Formula 17

・1-7 Calculation Formulas for Flat Springs with Half-circle and Quarter-circle Combination

Figure 10

In the flat spring with the combination of the half-circle and the quarter-circle shown in Figure 10, the deflection is as per the following formula.

Formula 18

The maximum stress occurs at the fixed end can be calculate with the following formula:

Formula 19

・1-8 Calculation Formulas for Arced Flat Springs

Figure 11

The deflection of the free end of the shape shown on the left side of Figure 11 is as per the following formula.

Formula 20

In the case of an arc that is constrained in the horizontal direction as shown on the right side of Figure 11, the deflection can be calculated with the following formula.

Formula 21

In both cases, the maximum stress is calculated with the following formula.

Formula 22

・1-9 Calculation Formulas for Flat Springs with Arc and Straight Line Part

  ・1-9-1  Calculation Formula for Flat Spring with Arc and Straight Line Part (No.1)

Figure 12

As shown in Figure 12, when the straight-line part (AB) and the circular arc part (BD) are combined, one end (D) is fixed, and the vertical load (P) or the horizontal load (W) acts on the other end (A), and will be as follows.

Formula 23

Formula 24

When ,

Formula C25

If W acts, the formula will be as below.

Formula 26

Formula 27

Here, in the formula represents

The maximum stress occurs at the fixed end when , and at the point C when .

  ・1-9-2 Calculation Formulas for Flat Springs with Arc and Straight Line Part (No.2)

Figure 13

The spring in Figure 13 is a combination of the two springs in Figure 12, and the deflection is times the deflection obtained in formula 23.

Formula 28

Figure 14

As shown in Figure 14, the deflection of the A end of the spring with a straight line part and an arced part is as per the following formula.

Formula 29

Here, and .

The maximum bending stress occurs at point C can be calcumated with the following formula.

Formula 30

In the case of , if , the maximum stress occurs at the fixed end, and when and , the formula is as follows.

Formula 31

  ・1-9-3 Calculation Formulas for Flat Springs with Arc and Straight Line Part (No.3)

Figure 15

In the case of the shape shown in Figure 15, it is possible to calculate the deflection of part A by dividing the AC part and the CD part, doubling the deflection of formula 25 and calculating the deflection of each formula before combining them.

Formula 32

  ・1-9-4 Calculation Formulas for Flat Springs with Arc and Straight Line Part (No.4)

Figure 16

As shown in Figure 16, the straight part is fixed, and so when a load is applied to the A end of the arc part, the vertical deflection and the horizontal deflection of the A end are as per the following formula when the P load acts as .

Formula 33

Formula 34

When the W load acts, the formula is as follows.

Formula 35

Formula 36

Figure 17

For the shape in Figure 17, when the load (P) acts, the formula is as follows.

Formula 37

Formula 38

When W acts, the formula is as follows.

Formula 39

Formula 40

It is represented here as .

  ・1-9-5 Calculation Formulas for Flat Springs with Arc and Straight Line Part (No.5)

Figure 18

For a spring with the shape shown in Figure 18 that combines an arc with a small curvature radius and a straight line, the deflection that excludes the radius of the arc is expressed by the following formula.

Formula 41

The maximum stress occurs at the BC part when , can be calculated with the following formula.

Formula 42

If , the maximum stress occurs at the fixed end, with the formula as follows.

Formula 43

  ・1-9-6 Calculation Formulas for Flat Springs with Complicated Combination of Arc and Straight Line Part

The shape of the flat spring is often a complex combination of an arced part and a straight part. The formulas introduced so far can be used for the shape of the thin leaf springs. The shapes and calculation formulas shown below can be used as practical concepts.

Figure 19

The shape in Figure 19 is considered to be a combination of the two shapes in Figure 13, and the deflection can be calculated by doubling that of Formula 28.

Figure 20

In the shape of Figure 20, both ends are similar to Figure 10, and the stress formula can be shown with Formula 19. The deflection on one side with respect to the axis of symmetry is an added part from Formula 18, and the deflection on one side is given by the formula below.

Formula 44

The deflection formula is:

Formula 45

２．Flat Spring Calculation Formulas by Characteristics and Applications

・2-1 Calculation Formulas for Flat Springs with Nonlinear Characteristics

Fifure 21

A flat spring with non-linear characteristics, as shown in Figure 21, is achieved by adopting an adhesive part structure where the fixed adhesive position changes in sequence due to deflection.

The formula for a flat spring with non-linear characteristics is as follows.

Formula 46

・2-2 Calculation Formulas for Flat Springs Which are Subject to Axial and Lateral Loads

Figure 22

Flat springs are found in applications such as load measuring equipment as the one shown in Figure 22. One end is fixed and the other end can move laterally but cannot rotate. In this case, assuming that the axial load (P) is smaller than the buckling load, the deflection and stress due to the lateral load (Q) are expressed by the following formulas.

Formula 47

Formula 48

If P is larger than the buckling load, the formula above is multiplied by the coefficients and determined by . Here, is Euler’s buckling load and . Here, the coefficients and are as per the following formulas.

Formula 49

Formula 50

・2-3 Calculation Formulas for Flat Springs with Large Deflection

・2-3-1 Flat Springs with a Rectangular Cross-section

Figure 23

When the deflection is large, changes to . Figure 24 shows the calculation result with this effect added.

Figure 24

The horizontal axis of the figure shows , and the vertical axis shows , . represents the bending stiffness of the plate, and when is large, . As is clear from Figure 24, when the value of is small, that is, when the load P is small, and are close to 1, and when , and . Therefore, in the case of this degree of deformation, it is deemed unnecessary to handle it as a large deflection in terms of the spring’s practical application.

  ・2-3-2 Flat Springs with Trapezoidal Cantilever

Figure 25

Figure 26

Figures 25 and 26 show the approximate results when the deflection of the flat spring with a trapezoidal cantilever is large. The horizontal axis is and the vertical axis shows the rate of decrease in deflection or stress with as a parameter. This just needs to be applied to the formula .