Symbol	Name	Unit
material	material	—
nominal_dia	nominal_dia	mm

Spring Stiffness Z/M/L Comparison: NFE 25-511 Single-Side Tooth Conical Spring Washers

1. Model Definition and Geometric Characteristics

NFE 25-511 divides single-side tooth conical spring washers into three series based on radial width:

Model	Name	Radial Width Characteristic	Relationship between Outer Diameter $$D_e$$ and Inner Diameter $$D_i$$
Z	Série Z (Étroite / Narrow)	Smallest width	$D_e \approx D_i + (2\sim3)t$
M	Série M (Moyenne / Medium)	Medium width	Between Z and L
L	Série L (Large / Wide)	Largest width	$$D_e$$ significantly larger than $$D_i$$ , large bearing area

Under the same nominal diameter (matching bolt size), the three models have essentially the same inner diameter $$D_i$$ , while the outer diameter $$D_e$$ increases sequentially, thus the width $$(D_e - D_i)/2$$ increases. The cone height $$h$$ and thickness $$t$$ vary slightly with the series, but the main difference lies in the outer diameter.

2. Theoretical Basis for Stiffness Differences

From the Almen‑Laszlo formula, the stiffness of a smooth disc spring is proportional to:

k \propto \frac{t^3}{K_1 \cdot D_e^2}

where the shape factor $$K_1$$ depends on the outer-to-inner diameter ratio $$c = D_e/D_i$$ :

K_1 = \frac{1}{\pi} \cdot \frac{[(c-1)/c]^2}{(c+1)/(c-1) - 2/\ln c}

When the inner diameter $$D_i$$ is constant and the outer diameter $$D_e$$ increases:

$$c$$ increases → $$K_1$$ increases (the denominator term decreases faster)
$$D_e^2$$ also increases

The superposition of these two effects causes a significant decrease in stiffness. The wider the washer, the "softer" the conical washer.

After incorporating the tooth correction factor $\beta_{strié}$ , the actual stiffness is:

k_{dent} = \beta_{strié} \cdot k_{Almen}

Typically, wide washers have more teeth, and $\beta_{strié}$ is also slightly smaller (softer), further widening the stiffness gap with the narrow type.

3. Stiffness Comparison Table (Same Bolt Size, e.g., M10)

The following are typical numerical comparisons (based on NFE 25-511 standard dimensions and material 50CrV4, $E = 206\,000$ MPa):

Parameter	Z Type (Narrow)	M Type (Medium)	L Type (Wide)
Inner Diameter $$D_i$$ (mm)	10.5	10.5	10.5
Outer Diameter $$D_e$$ (mm)	16	20	25
Thickness $$t$$ (mm)	1.2	1.5	1.8
Cone Height $$h$$ (mm)	0.8	1.0	1.2
Outer-to-Inner Diameter Ratio $$c = D_e/D_i$$	1.52	1.90	2.38
Shape Factor $$K_1$$	≈ 0.55	≈ 0.65	≈ 0.78
Tooth Correction $\beta_{strié}$	0.85	0.78	0.72
Flat Load $F_{flat}$ (N)	≈ 5 200	≈ 7 800	≈ 10 500
Working Point Stiffness $$k$$ (N/mm)*	≈ 8 500	≈ 10 800	≈ 13 200

*Working point taken at $s \approx 0.5h$ , i.e., half-compressed state.

Trend Summary:

Z → M → L: Stiffness gradually increases (narrow type is softest, wide type is stiffest)
Due to the larger thickness $$t$$ of wide washers, although the increased outer diameter reduces stiffness, the cubic influence of thickness dominates, ultimately resulting in wide type stiffness being greater than narrow type
Flat load (load capacity) is largest for L type and smallest for Z type

4. Force‑Deflection Characteristic Curve Comparison

Under the same bolt size, the approximate relationship of the three curves is:

F ↑
│ ┌── L (Wide, high load, steeper)
│ ┌─┼── M (Medium)
│ ┌─┼─┼── Z (Narrow, low load, shallower)
│ ┌─┼─┼─┼
│ ┌─┼─┼─┼─┼
│ ┌┼─┼─┼─┼─┼
│┌┼┼─┼─┼─┼─┼
└──────────────→ s (Deflection)
0 h_Z h_M h_L

Z Type: Smaller slope in the low-force range, lowest flat load; suitable for light loads and compact spaces.
M Type: Medium stiffness and load capacity, most widely used.
L Type: Steepest force‑deflection curve, strong elastic recovery force; suitable for compensating larger preload losses or high-vibration conditions.

5. Selection Guide

Application Requirement	Recommended Model	Reason
Space-constrained, light-load connections	Z	Smallest outer diameter, no interference with surrounding structures
General industrial connections	M	Balance point of stiffness, load capacity, and size
High preload / large relaxation compensation	L	High flat load, provides ample elastic reserve
Soft material clamped parts (aluminum, plastic)	L	Wide area reduces surface pressure, prevents crushing
Severe vibration, high anti-loosening requirement	M or L	Higher elastic recovery force maintains more stable clamping force
Precision instruments, small bolts	Z	Compact size, force matches small bolt specifications

6. Stiffness Relationship with DIN 25201 Wedge Washers

DIN 25201 wedge washers are not elastic elements; their stiffness is extremely high (equivalent to rigid washers), providing almost no elastic compensation.
NFE 25-511 single-side tooth conical spring washers are elastic elements that can compensate for relaxation through compressive deformation.
If a connection requires both anti-loosening (wedge locking) and relaxation compensation (elastic recovery), the two can be used in series: the wedge washer provides anti-loosening, and the conical spring washer provides elastic force compensation. In this case, the total system stiffness is determined by the elastic washer.

Selection Process:
Determine bolt size → Select Z/M/L series based on required elastic recovery force → Calculate working deflection and force → Check washer stress → Combine with wedge washer if necessary.

Spring Stiffness Z/M/L Comparison

Formula Expression

Parameters

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Detailed Calculation Guide