Calculate the behavior of a Belleville spring stack made of several, possibly different groups, arranged in series (common force) or parallel (common displacement), at a given target displacement or target force. Each group is evaluated internally through the validated single-group core.
This is a numerical coupling of the existing single-group core, not a dedicated EN 16983/EN 16984 method for progressive stacks: friction and hysteresis are modeled per group only, not between groups.
Set the groups and target and calculate.
Calculate the force-displacement curve, active groups, transition points, strain energy and maximum stress of a progressive Belleville spring stack made of multiple groups, by numerically coupling the validated EN 16983/EN 16984 single-group core.
For each group, the same geometry data as the existing Belleville-spring calculator (outside/inside diameter, nominal thickness, free height, parallel discs, series packs), the arrangement (series or parallel), and a target displacement or target force.
Each group is evaluated through the existing single-group core. In a series arrangement, all groups carry the same force and total displacement is the sum of the individual displacements; in a parallel arrangement, all groups share the same displacement and total force is the sum of the individual forces. A group that reaches its flat position (bottomed) no longer contributes further displacement (series) or force increase (parallel). The target quantity is found by bisection when not given directly.
Two groups of different thickness in series, at a 1.5 mm target displacement: the thinner, softer group reaches its flat position first and, from that transition point on, no longer contributes to further displacement -- the remaining group takes over the rest.
Sources and limits: Numerical coupling of the existing, validated EN 16983/EN 16984 single-group core; no dedicated standard method for progressive multi-group stacks, since EN 16983/16984 cover only uniform stacks.
Calculate the force-displacement curve, active groups, transition points, strain energy and maximum stress of a progressive Belleville spring stack made of multiple groups, by numerically coupling the validated EN 16983/EN 16984 single-group core.
A progressive Belleville spring stack consists of several groups of different geometry arranged so that its effective stiffness increases with displacement -- typically because softer groups reach their flat position earlier and drop out of further elastic deformation.
Series: s_total = sum s_i(F), F common; Parallel: F_total = sum F_i(s), s common
A group with s_i = s_i,max contributes no further displacement in series| Symbol / input | Meaning |
|---|---|
| s_i(F), F_i(s) | Group i's force-displacement relationship, taken unchanged from the validated single-group core. |
| s_i,max | Group i's flat position (maximum elastic displacement), beyond which it is considered bottomed. |
For each group, the same geometry data as the existing Belleville-spring calculator (outside/inside diameter, nominal thickness, free height, parallel discs, series packs), the arrangement (series or parallel), and a target displacement or target force.
Enter the geometry data for each group as in the existing Belleville-spring calculator. Choose the arrangement (series or parallel) and provide either a target displacement or a target force, not both.
Two groups of different thickness in series, at a 1.5 mm target displacement: the thinner, softer group reaches its flat position first and, from that transition point on, no longer contributes to further displacement -- the remaining group takes over the rest.
The count of active groups (not yet bottomed) relative to the total shows how many groups still contribute elastically at the target state. Transition points mark the force or displacement at which the stack's effective stiffness changes.
Lengths in mm, force in N; results in mm, N, MPa, Nmm.
Designing a Belleville-spring stack with a progressive (increasingly stiff) force-displacement curve by combining groups of different geometry so that softer groups reach their flat position first and the remaining groups take over the stiffness.
Numerical coupling of the existing, validated EN 16983/EN 16984 single-group core; no dedicated standard method for progressive multi-group stacks, since EN 16983/16984 cover only uniform stacks.
Common mistake: Don't confuse series and parallel -- in a series arrangement, group displacements add up at a common force; in a parallel arrangement, group forces add up at a common displacement. A target displacement exceeding the fully-bottomed travel of the whole stack is physically unreachable and is rejected.
Because EN 16983/EN 16984 explicitly cover only uniform stacks of identical discs. This calculator numerically couples multiple calls to the validated single-group core to combine different groups.
The group has reached its flat position (maximum displacement) and cannot deform elastically any further. In a series arrangement it stops contributing further displacement from that point on; in a parallel arrangement its force stays constant.
For each group, the force (series) or displacement (parallel) at which it reaches its flat position is determined; these points are sorted ascending and reported as transitions of the overall curve.
For each group, at its actual displacement in the target state, the validated five-location stress core (OM, I, II, III, IV) is evaluated; the largest absolute stress across all groups and locations is reported as governing.
It is the trapezoidal integral of the computed force-displacement curve over 41 sample points -- a numerical value, not a closed-form one. For a single group it agrees with the underlying core's analytic spring work to about 0.002%, and it stays within about 0.01% of a ten-times-finer integration even across a transition point, where the curve has a kink.