DIN 2093 Mixed Stacking (Parallel + Series)

1. Definition and Arrangement

Mixed stacking combines parallel stacking (same direction) with series stacking (opposing arrangement). It simultaneously achieves:

• Parallel portion: $n$ disc springs stacked in the same direction, increasing load capacity (force multiplied by $n$)
• Series portion: $i$ groups of the above parallel stacks arranged in opposition, increasing elastic travel (travel multiplied by $i$)

Total number of disc springs $N = n \times i$, where $n$ is the number of springs per parallel group and $i$ is the number of series groups.

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

Load F
↓
╱╲ ← 1st parallel group (2 springs same direction, cone facing up)
╱╲

╲ ╱ ← 2nd parallel group (2 springs same direction, cone facing down, opposite to upper group) ╲ ╱ ╱ ╲ ↩ ← 3rd parallel group (2 springs same direction, cone facing up) ╱ ╲
─────────────────

Important rules:

• The $n$ springs within the same parallel group must be arranged in the same direction
• Adjacent parallel groups must be arranged in opposite directions (cones facing each other or away)
• The ends of the entire stack typically contact flat support surfaces

2. Force-Travel Relationship

The mechanical characteristics of mixed stacking are a superposition of parallel and series rules.

Let the axial force of a single disc spring at deflection $s$ be $F_{single}(s)$, then:

• Total travel (series effect): $s_{total} = i \cdot s_{single}$
• Total load (parallel effect): $F_{total}(s_{total}) = n \cdot F_{single}\!\left(\dfrac{s_{total}}{i}\right)$

Expanded into complete form:

where $F_{single}(s)$ is given by the Almen-Laszlo formula:

Flat Force and Flat Travel

• Total flat force: $F_{flat,total} = n \cdot F_{flat,single}$
• Total flat travel: $s_{flat,total} = i \cdot h_0$

3. Stiffness and Energy Storage

3.1 Tangent Stiffness

From the chain rule of the force-travel derivative, the tangent stiffness of the mixed stack at total travel $s_{total}$ is:

where $k_{single}(s)$ is the tangent stiffness of a single disc spring at deflection $s$ (see DIN 2093 stiffness section).

3.2 Elastic Energy Storage

Energy storage is the area under the force-deflection curve, with parallel contribution factor $n$ and series contribution factor $i$:

Total energy storage at full flattening is:

Total number of springs $N = n \times i$ is the key factor determining total energy storage—more springs mean greater total energy storage.

4. Stress State

In a mixed stack, the actual deflection of each individual disc spring is only $1/i$ of the total travel:

Therefore, the OM point stress, I point stress, etc., for each spring are calculated based on $s_{single}$, independent of the parallel number $n$. This means:

• The larger the series number $i$, the lower the individual spring stress and the longer the fatigue life
• The parallel number $n$ only affects total load, not the individual spring stress level
• This is the most significant advantage of mixed stacking: achieving high load and large travel while keeping individual spring stress controllable

5. Hysteresis Effects Due to Friction

Mixed stacks have numerous contact surfaces (between parallel springs, between series groups), making friction effects more significant than in pure parallel or pure series stacks.

Sources of friction:

• Contact surfaces between parallel springs ($n-1$ internal contact surfaces per parallel group)
• Edge contacts between series groups ($i-1$ inter-group contact surfaces)
• Contact with support surfaces at both ends

Hysteresis effects:

• During loading: Friction opposes the external force, requiring a higher actual load than the theoretical value, typically +5 % to +15 %
• During unloading: Friction reverses direction, resulting in a lower actual released load than the theoretical value; the area enclosed by the hysteresis curve represents energy dissipation

More springs mean more significant friction. The total number of contact surfaces in a mixed stack is approximately $(n \times i - 1)$, far more than in a single stacking method.

Engineering design recommendations:

• Lubrication: Apply high-pressure grease or molybdenum disulfide coating to disc spring contact surfaces and edges to reduce friction and stabilize hysteresis characteristics
• Experimental calibration: For precise preload control, the force-travel curve must be determined through actual loading/unloading tests
• Conservative estimation: If testing is not possible, multiply the theoretical force value by 1.1 to 1.2 as the actual force to be applied during loading

6. Guiding Clearance and Tilt

When the total height of a mixed stack is large, if there is significant clearance between the disc spring inner/outer diameters and the guide element (mandrel or sleeve), tilt may occur during compression, leading to:

• Highly uneven force distribution among springs
• Excessive edge stress on some disc springs, causing premature fatigue
• Deviation of the travel-force characteristic from the theoretical value

Guiding methods:

• Mandrel guiding: A cylindrical shaft passing through the disc spring inner diameter; recommended clearance between disc spring inner diameter and mandrel is 0.1 to 0.5 mm (depending on specification)
• Sleeve guiding: A cylinder surrounding the disc spring outer diameter; recommended clearance between disc spring outer diameter and sleeve is 0.2 to 0.8 mm

Too small clearance increases friction; too large clearance loses the guiding function. DIN 2093 provides standard recommended clearance values.

7. Stability Criteria (Column Buckling)

The total free height $L_0$ of a mixed stack is the sum of the axial lengths of all disc springs in the free state:

To prevent lateral buckling (Euler instability) of the stack during compression, DIN 2093 specifies:

When the ratio exceeds these limits:

• Unguided → A guiding device must be added
• Guided → The number of springs must be reduced or the disc spring outer diameter increased

Note: Even under guided conditions, an excessively long spring column may still experience local instability (certain springs "jumping out") during compression. Therefore, $L_0/D_e > 6$ is not recommended, even with guiding; the stack should be arranged in sections.

8. Design Procedure and Selection Parameters

Given requirements: Maximum load $F_{max}$, required elastic travel $\Delta s_{req}$

Steps:

1. Select single spring specification: Based on space and outer diameter, preliminarily select disc spring $D_e, D_i, t, h_0$
2. Determine parallel number $n$:
   $$n_{min} = \left\lceil \frac{F_{max}}{F_{zul,single}} \right\rceil, \quad F_{zul,single} = \frac{F_{flat,single}}{S_F},\; S_F \ge 1.3$$

and $n \le 4$ 3. Determine series number $i$:

1. Check total height $L_0$ and verify stability ($L_0/D_e \le 3$ or 6)
2. Calculate actual operating point: From $F_{max}$ and the stack formula, back-calculate total travel $s_{total}$ and confirm individual spring stress safety
3. Consider friction: Loading force ≈ theoretical force × 1.1~1.2

9. Calculation Example

Requirements: Maximum load $F_{max} = 18\,000\ \text{N}$, elastic travel $\Delta s_{req} = 2.0\ \text{mm}$

Single spring parameters (DIN 2093 Series A, 20×10.2×1.5):

• $F_{flat,single} = 11\,000\ \text{N}$, $F_{zul,single} \approx 8\,500\ \text{N}$ (safety factor 1.3)
• $h_0 = 1.0\ \text{mm}$, $t = 1.5\ \text{mm}$, $D_e = 20\ \text{mm}$

Parallel number:

3 springs in parallel: total allowable load = 3 × 8 500 = 25 500 N > 18 000 N, satisfied.

Series number:

Stack specification: 3 × 3, total springs $N = 9$.

Total free height:

Stability: $L_0 / D_e = 15.0 / 20 = 0.75 \le 3$, unguided use is acceptable.

Operating point: At a total load of 18 000 N, each parallel group experiences 6 000 N (individual spring force = 18 000 / 3 = 6 000 N). From the single spring force-deflection curve, 6 000 N corresponds to a deflection of approximately $s_{single} \approx 0.48\ \text{mm}$. Total travel $s_{total} = 3 \times 0.48 = 1.44\ \text{mm}$.

Remaining elastic travel: Effective maximum travel = 3 × 0.75 = 2.25 mm, remaining 0.81 mm, can compensate for additional settling.

Friction effect: Loading force needs to be increased by approximately 1.1 times → actual applied force ≈ 19 800 N.

10. Key Points Summary

• Mixed stacking = parallel (n) × series (i), simultaneously amplifying load and travel
• Key formulas: $F_{total} = n \cdot F_{single}(s_{total}/i)$, $k_{total} = (n/i) \cdot k_{single}$
• Friction: Loading force is 5 % to 15 % higher; more springs mean more significant effects; precision applications require experimental calibration
• Guiding: Clearance must be strictly controlled to avoid both tilt and excessive friction
• Stability: $L_0/D_e \le 3$ (unguided) or $\le 6$ (guided); if exceeded, must be sectioned
• Stress advantage: The larger the series number $i$, the lower the individual spring stress and the longer the fatigue life