DIN 6796 Mixed Series-Parallel Combinations: $n \times i$ Parallel and Series Collaborative Design

1. Necessity of Mixed Series-Parallel Combinations

The single-disc load capacity and elastic travel of DIN 6796 disc spring washers are limited. In certain bolted connections:

• The preload $F_{Mmax}$ may exceed the permissible load of a single washer → parallel arrangement is needed to increase force capacity.
• The expected axial settlement $\Delta f$ (embedding + thermal expansion + creep) may exceed the available elastic travel of a single washer → series arrangement is needed to increase travel.

When a single arrangement cannot satisfy both requirements simultaneously, a mixed series-parallel combination must be used: first increase load capacity via parallel arrangement, then extend compensation travel via series arrangement, ultimately forming an $n \times i$ washer stack.

2. Arrangement and Naming of Mixed Series-Parallel Combinations

• Parallel group: $n$ washers stacked in the same direction, forming one parallel unit. This unit has total force $= n \times F_{single}$ and travel $= s_{single}$.
• Series: $i$ of the above parallel units are arranged alternately in opposite directions (adjacent parallel groups with cone surfaces facing each other), connected in series to form the assembly.

Total number of washers $N = n \times i$, abbreviated as $n \times i$ mixed series-parallel.

Arrangement example ($n=2, i=3$, total 6 washers):

Bolt head
╱▔▔╲ ← 1st parallel group (2 washers same direction)
╱ ╲
╱ ╲
╲ ╱ ← 2nd parallel group (opposite direction)
╲ ╱
╲▁▁╱
╱▔▔╲ ← 3rd parallel group (same direction)
╱ ╲
╱ ╲
Nut

3. Mechanical Characteristics of Mixed Series-Parallel Combinations

Let the force of a single washer at compression $s$ be $F_{single}(s)$, free cone height $h_0$, stiffness $k_{single}$.

3.1 Total Travel-Total Load Relationship

The entire mixed stack bears total preload $F_{total}$ and total compression travel $s_{total}$. According to the superposition rules for series and parallel:

Interpretation: total travel $s_{total}$ is distributed among $i$ series groups, each compressing by $s_{total}/i$; each group consists of $n$ parallel washers, carrying $n$ times the single-washer force.

3.2 Total Flat Force and Total Flat Travel

• Total flat force $F_{flat,total} = n \cdot F_{flat,single}$ (contributed only by parallel washers, independent of series)
• Total flat travel $s_{flat,total} = i \cdot h_0$ (contributed only by series groups)

Must ensure: maximum preload $\le$ total flat force / $S_{flat}$, and total working travel $\le 0.75 \times s_{flat,total}$.

3.3 Total Stiffness and Flexibility

The tangent stiffness of the mixed stack at the working point is:

Corresponding flexibility:

This flexibility is used in the VDI 2230 flexibility chain to calculate preload loss and load coefficient $\Phi^*$.

4. Stress State and Fatigue

In a mixed series-parallel combination, the compression of each single washer is:

Therefore, the stress state of each washer depends only on $s_{single}$, independent of the number of washers $n$ in the group. This means:

• Even if the total load is large (due to large $n$), as long as the number of series groups $i$ is sufficient, the single-washer compression $s_{single}$ can be small, keeping the OM point stress at a low level.
• The fatigue stress amplitude is also reduced due to small $s_{single}$; mixed series-parallel combinations inherently exhibit excellent fatigue performance.

Check: Calculate OM stress and uM stress based on compression $s_{single}$ and compare with permissible values.

5. Design Procedure

Input requirements:

• Maximum preload $F_{Mmax}$
• Total axial settlement to be compensated $\Delta f$ (including embedding, thermal, creep)

Steps:

1. Select base washer specification (according to bolt diameter), obtain single-washer $h_0, t, F_{flat,single}, F_{zul,single}$ (permissible load).
2. Determine number of parallel washers $n$:
   $$n_{min} = \left\lceil \frac{F_{Mmax}}{F_{zul,single}} \right\rceil$$

Take integer $n$, and $n \le 3$ (DIN recommendation; if $n>3$, increase washer specification or reduce preload). 3. Determine number of series groups $i$:

Take integer $i$. 4. Calculate total number of washers $N = n \times i$, total thickness $H = N \times t$, check if bolt thread length is sufficient. 5. Check strength and stress: - Verify $F_{Mmax} \le n \cdot F_{flat,single} / S_{flat}$ (flat safety). - Working travel $s_{work} =$ total compression corresponding to $F_{Mmax}$ (derived from mixed load formula), verify $s_{work} \le 0.75\,i\,h_0$. - Single-washer stress: calculate based on $s_{single} = s_{work}/i$, ensure permissible compressive/tensile stress is satisfied. 6. Update system flexibility: $\delta_W = i/(n \cdot k_{single})$, used for VDI 2230 calculations. 7. Check contact surface pressure: Total force $F_{Mmax}$ acts within the washer outer diameter range, verify surface pressure on the clamped parts according to VDI 2230 R10.

6. Calculation Example

Requirements:

• M12 bolt, maximum preload $F_{Mmax} = 35\,000\ \text{N}$
• Settlement to be compensated $\Delta f = 2.5\ \text{mm}$

Base washer (M12 using DIN 6796):

• $h_0 = 0.75\ \text{mm}$, $t = 1.5\ \text{mm}$
• Single-washer flat force $F_{flat} = 18\,000\ \text{N}$, permissible load $F_{zul} = 18\,000/1.3 \approx 13\,850\ \text{N}$

Number of parallel washers:

3 parallel washers total permissible load $= 3 \times 13\,850 = 41\,550\ \text{N} > 35\,000\ \text{N}$, satisfied.

Number of series groups:

Total number of washers $N = 3 \times 5 = 15$ washers, total thickness $15 \times 1.5 = 22.5\ \text{mm}$ (requires sufficiently long bolt).

Working condition:

• Total preload 35 000 N is carried by 3 parallel washers; each parallel group bears approximately 11 667 N, single-washer force approximately 11 667 N (since each group has 3 parallel washers, each washer force = 35 000 / 3 = 11 667 N). Single-washer force 11 667 N is less than single-washer permissible load 13 850 N, safe.
• From the single-washer force-deflection curve, 11 667 N corresponds to a single-washer compression of approximately $s_{single} \approx 0.5\ \text{mm}$ (example).
• Total compression $s_{total} = i \times s_{single} = 5 \times 0.5 = 2.5\ \text{mm}$.
• Total available elastic travel $0.75 \times i \times h_0 = 0.75 \times 5 \times 0.75 = 2.81\ \text{mm} > 2.5\ \text{mm}$, leaving margin after compensating settlement.
• Single-washer stress: OM stress calculated at $s=0.5\ \text{mm}$, approximately 1800 MPa; permissible compressive stress for spring steel is approximately 2200 MPa, satisfied.

Conclusion: Using a 3×5 mixed series-parallel combination successfully satisfies both high preload and large settlement compensation requirements.

7. Important Notes

• Parallel washer count limit: $n \le 3$ to ensure uniform load distribution; if exceeding 3, increase washer specification or switch to disc springs (DIN 2093).
• Series group count: No strict upper limit in principle, but consider total height, stability, and guidance. When $i$ is large, use guide sleeves or mandrels to prevent lateral instability.
• Total height check: Mixed stack total thickness $H = n \times i \times t$ may increase significantly; ensure sufficient bolt thread length, custom longer bolts if necessary.
• Installation direction: Strictly ensure all washers in the same parallel group are oriented identically, and adjacent parallel groups are opposite. Incorrect installation will completely alter characteristics.
• No arbitrary mixing: Washer specifications must be consistent; do not mix different sizes or thicknesses.
• Single-use recommendation: Mixed stacks with many washers have complex deformation histories after disassembly; reuse is not recommended.

Summary:
The $n \times i$ mixed series-parallel combination fully exploits the potential of DIN 6796 washers: increasing force capacity through parallel arrangement, extending travel through series arrangement, and adapting to demanding bolted connections with high preload and large settlement. Design requires precise determination of $n$ and $i$, and rigorous checking of strength, travel, and space to ensure the combination characteristics meet requirements.