BIZOMAT MAKEOS · Strength
Curved Beam and Hook Stress
What are the inner and outer fibre stresses in this curved beam or hook section?
Stresses at the inner and outer fibres of a curved rectangular section (crane hook, C-frame, ring) by the Winkler–Bach theory, with the straight-beam comparison.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Load (line of action through the centre of curvature) | N | Load pulling the section apart, acting along a line through the centre of curvature (as on a hook). |
| Inner radius | mm | Radius from the centre of curvature to the inside face. |
| Section depth (radial) | mm | Radial depth of the rectangular section. |
| Section width | mm | Width of the section. |
| Allowable stress | MPa | Allowable stress from your design code or lifting standard. |
What each input means
Open an input to see what it is, its units, the worked example and the common mistakes, with a small diagram.
What this means: Load (line of action through the centre of curv…
- What it is
- The load pulling on the curved section, assumed to act along a line through the centre of curvature, as on a crane hook carrying a sling.
- Units and format
- Enter a number in N: newtons. The form accepts 0.000001 N or more. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 20000 N as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- All the fibre stresses rise in proportion to the load.
- Get it right
- Use newtons: a mass in kilograms must be multiplied by g. The tool takes the bending moment as load times the centroid radius.
What this means: Inner radius
- What it is
- The radius from the centre of curvature to the inside face of the section, which is where the highest stress usually occurs.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 100000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 40 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- A smaller inner radius means sharper curvature and a larger rise in inner-fibre stress over the straight-beam value.
- Get it right
- Measure to the inside face, not to the centroid or the outside; the centroid radius is ri plus half the depth.
What this means: Section depth (radial)
- What it is
- The depth of the rectangular section measured radially, from the inside face to the outside face of the curve.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 100000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 60 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
What this means: Section width
- What it is
- The width of the rectangular section measured parallel to the axis of curvature, that is, across the thickness of the hook or frame.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 100000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 30 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Stresses fall in inverse proportion to the width.
What this means: Allowable stress
- What it is
- The highest fibre stress permitted by your design code or lifting standard; the larger of the inner and outer stress magnitudes is compared with it.
- Units and format
- Enter a number in MPa: megapascals (1 MPa = 1 N/mm²). The form accepts 0.000001 MPa to 100000 MPa. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 200 MPa as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Lifting hooks are rated and proof-tested to their standard; this check does not replace that rating.
Method and formulas
- ro = ri + h; centroid radius rc = ri + h/2; neutral radius rn = h ÷ ln(ro/ri); e = rc − rn.
- Moment M = P rc. Inner σ = P/A + M(rn − ri) ÷ (A e ri); outer σ = P/A − M(ro − rn) ÷ (A e ro).
- Straight beam: σ = P/A ± 6M ÷ (b h²).
- Example: 20000 N on the default hook section gives 120.76 MPa at the inner fibre against 88.889 MPa by straight-beam theory.
Limitations
- Rectangular section; hooks with trapezoidal or round sections need their own section integrals.
- Elastic theory; lifting hooks must follow their standard and proof testing.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Check hooks, clamp frames and C-frame presses
- See how curvature raises inner-fibre stress
- Size a curved bracket
Who it is for
- Design engineers
- Structural and mechanical engineers
- Engineering students
What you get
- Inner fibre stress
- Outer fibre stress
- Neutral axis shift
- Straight-beam stress and curvature factor
Interpreting the result
- The inner fibre governs: curvature raises its stress above the straight-beam value.
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 212–218, 257–260. Topic reference; no source tables or recommended settings are reproduced.
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