Calculator
Formula Used
Chord + Sagitta
For a circular arc defined by chord c and sagitta s:
R = c² / (8s) + s/2
θ = 2 asin( c / (2R) )
L = R θ
Arc Length + Angle
If arc length L and central angle θ are known:
R = L / θ
Curvature:
κ = 1 / R
Curvature Method
If curvature κ is measured or modeled:
R = 1 / κ
Optional mechanics (small strain):
ε ≈ t / (2R)
σ ≈ E ε
Minimum Radius Rule
A common design rule relates minimum bend radius to diameter:
Rmin = k · D
Pick k from manufacturer guidance for your material.
How to Use This Calculator
- Select a method that matches your known measurements.
- Choose units for inputs and desired output display.
- Fill only the fields required for your chosen method.
- Optionally add thickness and modulus for strain and stress.
- Press Compute to view results above the form.
- Use the download buttons to export the last result.
Example Data Table
| Scenario | Method | Inputs | Expected Output |
|---|---|---|---|
| Tube bend check | Chord + Sagitta | c = 120 mm, s = 15 mm | R ≈ 127.5 mm |
| Arc-based design | Arc + Angle | L = 200 mm, θ = 90° | R ≈ 127.3 mm |
| Measured curvature | Curvature | κ = 0.8 1/m | R = 1.25 m |
| Cable routing rule | Minimum Radius | D = 6 mm, k = 10 | Rmin = 60 mm |
Examples are illustrative; always follow manufacturer limits.
Professional Notes on Bend Radius
1) Why bend radius matters
Bend radius controls how sharply a material or cable can curve without damage. When radius is too small, the outer fibers stretch, the inner fibers compress, and defects like cracking, kinking, ovalization, and conductor fatigue become more likely. A larger radius typically improves reliability.
2) Geometry-based radius from field measurements
In site work, you often know the chord c and sagitta s from a quick measurement. The circle-segment relation R = c²/(8s) + s/2 lets you estimate radius directly. If the same segment is reused, tracking θ = 2 asin(c/2R) helps compare bend severity across layouts.
3) Arc length and angle for design drawings
CAD drawings frequently report a bend as arc length L and central angle θ. The relation R = L/θ (with θ in radians) is numerically stable and easy to audit. For example, L = 200 mm and θ = 90° implies R ≈ 127.3 mm.
4) Curvature connects to modeling and sensing
Many simulations and sensors use curvature κ in 1/m. Because κ = 1/R, you can compare bends across different sizes without unit confusion. A curvature of 0.8 1/m corresponds to R = 1.25 m, a gentle bend for many applications.
5) Minimum radius rules and safety factors
Many cables and tubes are guided by a minimum rule Rmin = k·D. Typical planning values for k may range from about 6 (flexible) to 20 (more sensitive), but the authoritative value is the manufacturer limit for your exact construction. Use a higher k when cyclic bending is expected.
6) Strain estimate from thickness
For small strains, a simple outer-fiber estimate is ε ≈ t/(2R), where t is thickness. If t = 2 mm and R = 100 mm, then ε ≈ 0.01 (1%). This quick check helps compare candidate radii before deeper analysis.
7) Stress estimate using elastic modulus
When elastic modulus E is known, stress can be approximated as σ ≈ Eε. For E = 70 GPa and ε = 0.01, the estimate is σ ≈ 700 MPa, highlighting why small radii can become risky for stiff materials. Treat this as a screening estimate, not a full design code check.
8) Practical workflow for documentation
Use this calculator to compute radius by the method that matches your data, then export CSV or PDF for traceability. Record the measurement method, units, and any chosen k factor. For production, keep a short checklist: verify minimum radius, confirm routing constraints, and document the final geometry for repeatable installation.
FAQs
1) What is bend radius?
Bend radius is the radius of the circle that best matches a curved path. Smaller radius means a tighter bend and typically higher strain on the outer surface.
2) Which method should I use?
Use chord+sagitta for quick field measurements, arc+angle for drawings, curvature for modeling/sensing outputs, and k·D when you only have diameter and a minimum-rule factor.
3) Why must angle be in radians for R = L/θ?
Because the derivation assumes θ is dimensionless in radians. If you enter degrees, convert using θ(rad) = θ(deg)·π/180 to avoid scaling errors.
4) What does curvature mean physically?
Curvature κ is how quickly the path turns per unit length. Higher κ means a tighter bend. Since κ = 1/R, it provides a convenient, unit-consistent way to compare bends.
5) How do I choose the k factor in Rmin = k·D?
Start with manufacturer guidance for your exact product. If unavailable, use conservative planning values and increase k for repeated flexing, cold conditions, or critical reliability requirements.
6) Are the strain and stress results exact?
No. They are screening estimates based on simple bending assumptions. Real behavior depends on geometry, neutral axis shift, plasticity, layered construction, and installation details.
7) Why do my results differ from a handbook chart?
Handbooks may use different assumptions, define radius at different locations, or apply empirical limits. Ensure consistent units, measurement definitions, and whether the chart reports minimum recommended or absolute limit.