Module C — Design Guide
L-ring & W-ring
End Play Compensation
L-rings and W-rings are spring retaining rings that take up axial play in
bearing and spindle assemblies. Unlike flat retaining rings which are rigid,
these rings deflect axially under load — compensating the accumulated
tolerances of all components in the stack and maintaining a controlled
preload regardless of manufacturing variation.
Open the Calculator →
What End Play Compensation Means
In a bearing assembly, the total axial play is the sum of the dimensional tolerances
of every component in the stack — bearing width, housing depth, shaft shoulder height,
spacer length, ring groove position. Even with tight tolerances on each individual
component, the accumulated variation (ΣΔ) can produce a measurable axial gap
or preload variation.
A flat retaining ring provides a rigid stop — it transmits whatever force is applied
but cannot compensate play. An L-ring or W-ring sits in the same groove but deforms
elastically, generating a spring force that takes up the stack variation and maintains
a defined preload range regardless of how the individual tolerances combine.
The key question: does the maximum possible end play (ΣΔ_max) fit
within the ring’s useful compression range? If yes — the ring can always compensate
the play. If no — there will be conditions where the ring bottoms out or loses contact,
and the preload guarantee fails.
L-ring vs W-ring — Which to Use?
L-ring (conical)
- Conical shape — like a Belleville washer
- Linear spring: F_L = C × f
- Constant C given in catalogue for each size
- Flattens completely at f = L − u
- Good for single bearing, one component
- Lever arm h = a (groove width dimension)
- Shaft rings: h = a; Bore rings: h = a − t
W-ring (convex / bowed)
- Convex shape — two-point contact
- Progressive spring: non-linear F vs deflection
- F₁ at W₁ and F₂ at W₂ given in catalogue
- Can locate two components simultaneously
- Two-point contact — axially symmetric pressing
- Volume-produced — lower cost for large quantities
- Harder to calculate exactly — interpolate between F₁/F₂
The Precondition — Will the Ring Work at All?
Before calculating forces, you must check whether the ring can physically compensate
the tolerance stack. This is the single most important check:
L-ring precondition
ΣΔ ≤ L − u
ΣΔ = total tolerance stack of the assembly [mm]
L = L-ring free height [mm] — from catalogue
u = minimum compression clearance [mm] — assembly cannot fully flatten the ring
W-ring precondition
ΣΔ ≤ W₂ − W₁
W₁ = ring height at minimum height (maximum force F₁) [mm]
W₂ = ring height at maximum height (minimum force F₂) [mm]
The ring must operate between W₁ and W₂ for the entire tolerance range
If the precondition fails: the ring cannot guarantee compensation.
At worst-case dimensions the ring either bottoms out (applying excessive force) or
loses contact (zero preload). Options: (1) select a ring with larger L or wider
W₂ − W₁; (2) tighten tolerances to reduce ΣΔ; (3) use a different preload method.
L-ring Force Calculation
L-ring behaviour is linear. The catalogue gives spring constant C in N/mm.
The actual compression f in service varies between f_min (loosest stack) and
f_max (tightest stack):
L-ring compression and force
f_max = L − u (tightest stack — maximum compression)
f_min = L − ΣΔ − u (loosest stack — minimum compression)
F_max = C × f_max = C × (L − u)
F_min = C × f_min = C × (L − ΣΔ − u)
F_max occurs when all dimensions are at maximum (tight stack — ring most compressed)
F_min occurs when all dimensions are at minimum (loose stack — ring least compressed)
C values in catalogue are for spring steel; multiply by E’/210,000 for other materials
W-ring Force Calculation
W-rings have a non-linear (progressive) spring characteristic. The catalogue gives
two force/height points: F₁ at height W₁ and F₂ at height W₂. In service the ring
operates between these two points, so the preload range is simply:
- Maximum preload F_max = F₁ (tightest stack — ring compressed to W₁)
- Minimum preload F_min = F₂ (loosest stack — ring at W₂)
For intermediate conditions, force can be interpolated linearly between the two
catalogue points as a reasonable approximation. For precise values the full
spring curve from the catalogue data charts should be used.
Groove Dimension Range
The groove width a must be specified as a range such that the ring always operates
within its useful compression zone regardless of which tolerance extreme applies.
For an L-ring retaining a single bearing (Fig.59 arrangement):
L-ring groove width range (single bearing arrangement)
a_min = Σb_max + u + s_ring (parts at max, ring barely compressed)
a_max = Σb_min + L + s_ring (parts at min, ring at free height)
Σb_max = sum of all component nominal dimensions at their maximum [mm]
Σb_min = sum of all component nominal dimensions at their minimum [mm]
s_ring = ring’s own axial width as a spacer element [mm]
The groove width tolerance Δa = a_max − a_min should be achievable in machining
Worked Example
A ball bearing assembly requires end play compensation using an L-ring.
Bearing width tolerance: ±0.10 mm (ΣΔ₁ = 0.20 mm). Housing depth tolerance:
±0.08 mm (ΣΔ₂ = 0.16 mm). Shaft shoulder: ±0.05 mm (ΣΔ₃ = 0.10 mm).
Total ΣΔ = 0.46 mm. Groove width tolerance Δa = 0.15 mm.
L-ring selected: L = 1.2 mm, u = 0.25 mm, C = 3.0 N/mm.
Step 1 — Precondition check
Available compensation = L − u = 1.2 − 0.25 = 0.95 mm
ΣΔ = 0.20 + 0.16 + 0.10 = 0.46 mm
0.46 ≤ 0.95 ✓ Precondition satisfied
Margin = 0.95 − 0.46 = 0.49 mm
Precondition0.46 ≤ 0.95 ✓
Step 2 — Preload force range
f_max = L − u = 1.2 − 0.25 = 0.95 mm
f_min = L − ΣΔ − u = 1.2 − 0.46 − 0.25 = 0.49 mm
F_max = 3.0 × 0.95 = 2.85 N (tight stack)
F_min = 3.0 × 0.49 = 1.47 N (loose stack)
F_max (tight stack)2.85 N
F_min (loose stack)1.47 N
Preload always ≥1.47 N ✓
Symbol Reference
| Symbol | Unit | Description |
|---|
| ΣΔ | mm | Total tolerance stack — sum of all component tolerances in the assembly |
| Δa | mm | Groove width tolerance |
| L | mm | L-ring free height (uncompressed state) — from catalogue |
| u | mm | Minimum compression clearance — ring cannot be fully flattened; from catalogue |
| C | N/mm | L-ring spring constant — from catalogue (proportional to E) |
| f_max | mm | Maximum compression = L − u (tightest stack) |
| f_min | mm | Minimum compression = L − ΣΔ − u (loosest stack) |
| F_max | N | Maximum preload force = C × f_max |
| F_min | N | Minimum preload force = C × f_min |
| W₁ | mm | W-ring height at lower height (higher force F₁) |
| W₂ | mm | W-ring height at upper height (lower force F₂) |
| F₁ | N | W-ring pressing force at W₁ — from catalogue |
| F₂ | N | W-ring pressing force at W₂ — from catalogue |
| a | mm | Groove width dimension |
Frequently Asked Questions
What is the difference between an L-ring and a K-ring?
K-rings are flat retaining rings with additional lugs distributed around the circumference, providing more evenly distributed abutment for components with large corner radii. L-rings are K-rings deformed into a conical shape to give them spring properties. An L-ring in the fully flattened state behaves identically to a K-ring — its axial load capacity in the flattened state is calculated the same way as a standard grooved axial ring (Module A). The spring properties exist only while the ring is between its free height L and its flattened state (L−u).
How do I measure ΣΔ for my assembly?
ΣΔ is the sum of the tolerance ranges (max − min) of every component in the axial stack between the groove and the fixed reference. For a bearing on a shaft retained by an L-ring: ΣΔ = tolerance of bearing width + tolerance of shaft shoulder height + any spacer tolerances. It is the worst-case axial play that could occur when all dimensions are simultaneously at their extreme values. Do not use RSS (root-sum-square) combination for safety-critical preload calculations — use the arithmetic sum.
What does the dimension u represent?
u is the “residual clearance” — a small compression the ring must always retain so it cannot be fully flattened during assembly. If the ring were pressed completely flat during assembly it would spring back into the groove rather than sitting correctly against the component. u ensures the ring is always under some axial compression, maintaining its spring function. Its value is typically 0.1–0.5 mm depending on ring size and is given in the manufacturer’s catalogue.
The precondition fails but only just — can I still use the ring?
If ΣΔ slightly exceeds L−u, the ring will occasionally bottom out when all dimensions happen to be at maximum simultaneously. Whether this is acceptable depends on the application. For a lightly loaded bearing with occasional axial play in one direction, it may be tolerable. For precision spindles or assemblies where the preload must always be maintained within a tight band, it is not. The safest fix is to choose a ring with a slightly larger L value or reduce one of the tighter tolerances in the stack.
Can L-rings and W-rings carry axial loads as well as compensate end play?
Yes — in the flattened state an L-ring has the same load capacity as a standard K-ring (same profile). The load capacity is calculated using Module A with the flat ring dimensions. The W-ring in its compressed state also carries axial load — its two-point contact provides a more uniform force distribution than an L-ring. The end play calculator here determines the spring behaviour; for load capacity of the ring in its flattened state, use
Module A.
AbarTech