Multiple Belleville washers may be stacked to modify the spring constant or amount of deflection. Stacking in the same direction will add the spring constant in parallel, creating a stiffer joint (with the same deflection). Stacking in an alternating direction is the same as adding springs in series, resulting in a lower spring constant and greater deflection. Mixing and matching directions allow a specific spring constant and deflection capacity to be designed.

If n = number of springs in a stack, then:

*Parallel Stack* (n in parallel, 1 in series) - Deflection is equal to that of one spring; Load is equal to that of n x 1 spring.

*Series Stack* (1 in parallel, n in series) - Deflection is equal to n x 1 spring; load is equal to that of one spring.

Consequently, depending upon the application, the designer can:

• Stack in "parallel" to increase load

• Stack in "series" to increase deflection

• Adjust the load and deflection of a washer stack by adding or removing individual washers and/or the sequence in which they are used, whether in series or parallel.

For intermediate loads, refer to the Load Deflection chart.

**Load**

In a parallel stack, load losses will occur due to friction between the springs. This loss due to friction can be calculated using hysteresis methods. Ideally, no more than 4 springs should be placed in parallel. If a greater load is required, then factor of safety must be increased in order to compensate for loss of load due to friction. Friction loss is not as much of an issue in series stacks

**Deflection**

In a series stack, the deflection is not exactly proportional to the number of springs. This is because of a bottoming out effect when the springs are compressed to flat. The contact surface area increases once the spring is deflected beyond 95%. This decreases the moment arm and the spring will offer a greater spring resistance. Hysteresis can be used to calculate predicted deflections in a series stack. The number of springs used in a series stack is not as much of an issue as in parallel stacks.

**Calculation**

If friction and bottoming-out effects are ignored, the spring rate of a stack of identical Belleville washers can be quickly approximated. Counting from one end of the stack, group by the number of adjacent washers in parallel. For example, in the stack of washers below, the grouping is 2-3-1-2, because there is a group of 2 washers in parallel, then a group of 3, then a single washer, then another group of 2.

The total spring coefficient for this group of Belleville washers is calculated as below:

Where:

n_{i} = the number of washers in the i^{th} group

g = the number of groups

k = the spring constant of one washer

So, a 2-3-1-2 stack (or, since addition is commutative, a 3-2-2-1 stack) gives a spring constant of 3/7 that of a single washer. These same 8 washers can be arranged in a 3-3-2 configuration (K = 6/7*k), a 4-4 configuration (K = 2*k), a 2-2-2-2 configuration (K = 1/2*k), and various other configurations. The number of unique ways to stack n washers increases dramatically with n, allowing fine-tuning of the spring constant. However, each configuration will have a different length, requiring the use of shims in most cases.