Utilities U1 & U2 — Guide
Assembly & Groove Verification
You have a ring and a groove from a drawing or a machined part.
These two tools answer the questions every MRO and re-design engineer needs:
can the ring be assembled, and can the groove carry the load?
Open the Calculator →
The Two-Step Verification
A complete retaining ring assembly involves two distinct checks. Design tools
(Modules A, B, E) combine these automatically. When verifying an existing design,
reviewing a drawing, or diagnosing a field failure, you check them independently
with the actual dimensions.
Step 1 · Tab U1
Assembly Check
Given ring dimensions (d₁, d₃, b, s): what bending stress occurs during assembly? Does the ring clear all features in the path? What is the over-expansion fracture limit?
→
Step 2 · Tab U2
Groove Check
Given groove dimensions (d₁, d₂, n): what axial load can the groove carry? Is the collar length adequate? Does the wall thickness meet w ≥ 3?
Design vs verification: to design a ring for a given load use
Module A (grooved rings) or
Module E (snap rings) — those iterate ring width b.
U1/U2 work in the opposite direction: dimensions in, performance out.
Assembly Bending Stress
A retaining ring is stressed most highly during assembly, not in service. This bending
stress must stay within the elastic limit, otherwise the ring permanently deforms,
sits in the groove with reduced tension, and carries less load than designed.
Assembly bending stress — tapered ring, shaft (Eq. ⑫)
σ_b = (d₁ − d₃) × E × b / [(d₁ + 0.75b)(d₃ + 0.75b)]
Bore ring: swap sign, use (d₁ − 0.7b) and (d₃ − 0.7b) in denominator
Assembly bending stress — snap ring / circlip (Eq. ⑭)
σ_b = 1.15 × (d₁ − d₃) × E × b / [(d₁ + b)(d₃ + b)]
Factor 1.15 for pliers (non-circular deformation). Omit for mandrel/taper.
+b in denominator, not +0.75b — uniform section, no eccentricity.
Diameter change during assembly (Eq. ⑬)
ΔD = σ_b × D₃² / (E × b − σ_b × D₃) [shaft ring]
D₃ = d₃ + 0.75b (tapered) or d₃ + b (snap ring)
Bore ring denominator: E × b + σ_b × D₃
Permissible Assembly Stress Limits
| Nominal diameter d₁ | Permissible σ_b | Notes |
|---|
| ≤ 20 mm | 2500 N/mm² | Small rings — high hardness |
| ≤ 40 mm | 2000 N/mm² | Standard range |
| ≤ 100 mm | ~1500 N/mm² | Medium rings |
| ≤ 200 mm | ~900 N/mm² | Large rings |
| > 200 mm | ~500 N/mm² | Very large rings |
Assembly is the most critical moment in a ring's life.
Over-expansion causes permanent diameter increase, reducing groove tension and load capacity.
The effect is invisible — the ring looks correctly seated.
Use pliers with a stop screw or a correctly dimensioned mandrel for every installation.
Installation Clearance — The Path Check
Before a shaft ring reaches its groove, it travels along the shaft in its expanded state.
Any feature with a smaller bore than the ring's expanded outer diameter will block it.
This is frequently missed in design review and discovered only during physical assembly.
Outer diameter at assembly
Tapered ring (shaft):
d_assy = d₁ + 1.5b (z/b = 0.25)
Snap ring (shaft):
d_assy = d₁ + 2b (z = 0)
Feature bore must be > d_assy for the ring to pass.
Snap rings require 33% more radial clearance than equivalent tapered rings.
Over-expansion Risk
The absolute maximum expansion diameter is d₁ + 2b — where a correctly designed
mandrel makes circular contact and prevents further expansion.
Above ~2500 N/mm² fracture risk is significant. Above the elastic limit,
permanent deformation occurs and load capacity in service is reduced.
Groove Load Capacity
The groove transmits axial load into the shaft or housing material by shear across
the groove annular area A_N. For soft materials — aluminium, brass, cast iron —
the groove often governs before the ring does.
Groove load capacity — Eq. ③
F_N = (σ_s × A_N) / (q × S)
A_N = (π/4)(d₁² − d₂²) shaft · A_N = (π/4)(d₂² − d₁²) bore
σ_s = yield point of shaft or housing material [N/mm²]
q = load factor (from n/t ratio) · S = safety factor
The Collar Length Ratio n/t and Load Factor q
At n/t = 3, q = 1.20 — used in all standard ring catalogue data.
Below n/t = 3, q rises steeply, reducing effective groove capacity.
Above n/t = 5, q approaches 1.0.
Short collars matter most near shaft ends, adjacent to keyways, or next
to other stress concentrations. The groove checker calculates the actual q for the
real n/t, which can be significantly lower than catalogue values assume.
Wall Thickness Ratio w
Wall thickness ratio
Bore housing:
w = (d₀ − d₁) / (d₂ − d₁) ≥ 3
Hollow shaft:
w = (d₁ − d₀) / (d₁ − d₂) ≥ 3
d₀ = housing outer diameter (bore ring) or hollow shaft bore ID (shaft ring)
w < 3: Eq.③ overestimates groove capacity — reduce groove depth or increase wall.
Worked Example
A 30 mm shaft with ring dimensions d₁=30, d₃=27.9, b=4.0, s=1.2 mm (spring steel,
tapered). Groove at d₂=28.4 mm, collar n=4.5 mm. Shaft: medium carbon steel (σ_s=320).
A 32 mm bearing bore lies between shaft end and groove. S=1.5.
U1 — Assembly Check
σ_b = (30−27.9) × 210,000 × 4.0 / [(30+3.0)(27.9+3.0)]
= 1,764,000 / 1,019.7 = 1,730 N/mm²
d_assy = 30 + 1.5×4.0 = 36.0 mm
Feature clearance: 32 mm < 36 mm ✗ ring cannot pass bearing
σ_b1,730 N/mm² ✓ (limit 2,000)
Clearance (32 mm bore)FAIL — ring outer ⌀ 36 mm > 32 mm
Assembly stress passes but the ring cannot travel past the bearing in its expanded state.
Options: fit the ring before the bearing, or use a narrower ring so d_assy < 32 mm.
U2 — Groove Check
t = (30−28.4)/2 = 0.8 mm
A_N = (π/4)(30²−28.4²) = 73.3 mm²
n/t = 4.5/0.8 = 5.625 → q = 1.025
F_N = (320 × 73.3) / (1.025 × 1.5) = 15,254 N
n/t5.625 — q = 1.025 ✓
F_N Groove capacity15,254 N ✓
Groove is adequate. The assembly clearance issue from U1 is the binding constraint.
Symbol Reference
| Symbol | Unit | Tool | Description |
|---|
| d₁ | mm | U1, U2 | Nominal shaft or bore diameter |
| d₂ | mm | U2 | Groove diameter |
| d₃ | mm | U1 | Ring unstressed diameter |
| d₀ | mm | U2 | Housing OD or hollow shaft bore ID — for wall check |
| b | mm | U1 | Ring maximum radial width |
| s | mm | U1 | Ring thickness (axial) |
| n | mm | U2 | Collar length — distance from groove to nearest free surface |
| t | mm | U2 | Groove depth = (d₁−d₂)/2 |
| A_N | mm² | U2 | Groove annular area |
| q | — | U2 | Load distribution factor (from n/t) |
| w | — | U2 | Wall thickness ratio — must be ≥ 3 |
| σ_b | N/mm² | U1 | Assembly bending stress |
| σ_s | N/mm² | U2 | Yield point of shaft or housing material |
| F_N | N | U2 | Groove load capacity |
| d_assy | mm | U1 | Ring outer diameter during assembly (d₁+1.5b tapered; d₁+2b snap) |
| ΔD | mm | U1 | Diameter change during assembly |
| K | N·mm | U1 | Ring spring constant (shown for reference) |
Frequently Asked Questions
The ring passes the stress check but fails the clearance check — what are my options?
Four options: (1) sequence the assembly — fit the ring before the blocking feature; (2) use a narrower ring (smaller b) so d_assy is smaller, accepting reduced load capacity; (3) use a
radially-assembled ring which does not need to slide over the shaft at all; (4) redesign the blocking feature with a larger bore. The calculator shows the exact margin on each constraint.
What does over-expansion actually do to a ring?
Expanding a ring beyond its elastic limit causes it to spring back to a larger d₃ than intended — it has been permanently stretched. The ring seats in the groove but with less radial tension, giving a lower loosening speed and higher risk of working loose under vibration. The effect is invisible. The only reliable check is to measure the ring's free diameter before and after installation; if it has grown, over-expansion occurred.
Why does groove width not appear in the groove capacity calculation?
Groove width has no effect on axial load capacity. The load is carried by the annular area A_N perpendicular to the shaft axis, which depends only on d₁ and d₂. A wider groove gives the same A_N and the same capacity. The only function of groove width is to accommodate ring thickness with clearance, and to allow the ring to dish without bottoming on the groove walls.
My collar length ratio n/t is less than 1 — is the formula valid?
At n/t < 0.7 the formula is not reliable — the groove may fail by tearing rather than shearing, and the calculated q value is highly uncertain. Redesign to increase collar length if at all possible. If the groove cannot be moved, consider a
radially-assembled ring which does not require a collar behind the groove in the same way.
Does the wall thickness check apply to solid shafts?
No — w only applies to hollow shafts and thin-walled housings. For a solid shaft or a thick housing, leave d₀ blank. Enter d₀ only when the shaft has a central bore reducing the material below the groove, or when checking a bore ring in a thin-walled housing.
U2 shows a high F_N but the ring pulled out — why?
U2 only checks the groove. The ring itself may have a lower load capacity F_R — check this in
Module A or
Module E. Also consider: was the ring over-expanded during assembly (permanent d₃ increase reduces F_R)? Is the load alternating (requires 30% F_R reduction)? Is the contact point chamfered (increases lever arm h, reduces F_R)?
AbarTech