KEVOS® MAKEOS · Strength
Thin Ring Loaded Across a Diameter (Bending Stress, Diameter Changes, Safe Load)
When two opposite forces squeeze or pull my thin ring across a diameter, what is the bending stress, how much do the diameters change, and what load can it take at my allowable stress?
Stresses and deflections of a thin ring loaded by two equal and opposite forces across a diameter, by the handbook formulas: the maximum bending stress S = P D d ÷ (4π I) at the load points for a round, rectangular or other section, the increase of the diameter along the load and the decrease across it, the load that brings the stress to your allowable value, and checks of the stress and diameter change against your limits.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Ring section | choice | Shape of the bar the ring is made from; for other sections enter the second moment of area and radial depth. |
| Mean diameter of the ring | mm | Diameter of the ring measured to the centre of the bar section. |
| Bar diameter or radial depth of the section | mm | Diameter of a round bar, or the depth of the section measured radially (in the plane of the ring). |
| Axial width (rectangular bar) | mm | Width of a rectangular section along the ring's axis. |
| Second moment of area (other section) | mm⁴ | Second moment of area of the section about the axis that bends in the plane of the ring (used for other sections only). |
| Modulus of elasticity | MPa | Elastic modulus of the ring material, from the material data. |
| Load on the ring (each of the two opposite forces) | N | Each of the two equal and opposite forces acting across the diameter. |
| Allowable bending stress | MPa | Largest bending stress you allow in the ring, from your material data and design code. |
| Largest change of diameter accepted | mm | Largest increase or decrease of a diameter your fit or clearance can accept. |
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: Ring section
- What it is
- Shape of the bar the ring is made from.
- Units and format
- Choose one of 3 options from the list: Round bar; Rectangular bar; Other section (enter I and depth).
- Example
- The form opens with “Round bar” selected as a worked example. Choose the option that matches your case.
- Effect on the result
- Sets the second moment of area that resists bending.
- Get it right
- Choose other section for tubes or shapes, and enter their second moment of area and depth.
What this means: Mean diameter of the ring
- What it is
- Diameter of the ring measured to the centre of the bar.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 1000000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 200 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- The stress grows with the diameter and the diameter change with its cube.
- Get it right
- Average the inside and outside diameters.
What this means: Bar diameter or radial depth of the section
- What it is
- Bar diameter for a round section, or the section depth measured radially in the plane of the ring.
- 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 20 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- A deeper section lowers the stress sharply.
- Get it right
- For a flat band bent the easy way this is the thickness, not the width.
What this means: Axial width (rectangular bar)
- What it is
- Width of a rectangular section along the axis of the ring.
- 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 10 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- A wider bar lowers the stress in proportion.
- Get it right
- Used for rectangular bars only.
What this means: Second moment of area (other section)
- What it is
- Second moment of area of an other-shaped section about the axis that bends in the ring's plane.
- Units and format
- Enter a number in mm⁴: millimetres to the fourth power (second moment of area). The form accepts 0.000001 mm⁴ or more. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 7854 mm⁴ as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- A larger value lowers the stress and the diameter change.
- Get it right
- Used for other sections only; take it from section tables or CAD.
What this means: Modulus of elasticity
- What it is
- Elastic modulus of the ring material.
- Units and format
- Enter a number in MPa: megapascals (1 MPa = 1 N/mm²). The form accepts 100 MPa to 1000000 MPa. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 200000 MPa as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Only changes the diameter changes, not the stress.
- Get it right
- Take it from your material data; about 200 000 MPa for steel.
What this means: Load on the ring (each of the two opposite forc…
- What it is
- Each of the two equal and opposite forces acting across a diameter of the ring.
- 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 3000 N as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- The stress and diameter changes are in direct proportion to the load.
- Get it right
- Use the largest pinch or pull the ring sees in service.
What this means: Allowable bending stress
- What it is
- Largest bending stress you allow in the ring.
- 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 150 MPa as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Only moves the stress check and sets the load at the allowable stress.
- Get it right
- Base it on yield strength with your design code's factor, or on fatigue for repeated loads.
What this means: Largest change of diameter accepted
- What it is
- Largest increase or decrease of a diameter your fit or clearance can accept.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 10000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0.5 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Only moves the diameter-change check.
- Get it right
- Take it from the clearance around the ring or the fit it must keep.
Method and formulas
- Second moment of area: π d⁴ ÷ 64 for a round bar, b d³ ÷ 12 for a rectangular bar.
- Maximum bending stress S = P D d ÷ (4π I); for a round bar this is S = 1.621 P D ÷ d³, or P = 0.617 S d³ ÷ D.
- Increase of the diameter along the load = (π ÷ 4 − 2 ÷ π) P R³ ÷ (E I) with R = D ÷ 2, which the handbook gives as 0.0186 P D³ ÷ (E I).
- Decrease of the diameter across the load = (2 ÷ π − 1 ÷ 2) P R³ ÷ (E I), about 92 % of the increase as the handbook states.
- Load at the allowable stress = 4π I S_allow ÷ (D d).
- Example: 3000 N across a 200 mm ring of 20 mm bar gives 121.59 MPa and opens the loaded diameter by 0.28415 mm; the ring takes 3,701.1 N at 150 MPa.
Limitations
- Thin rings only: the section depth must be small compared with the diameter, as the handbook states; thick rings need curved-beam theory.
- Direct and shear stresses at the load points, and local contact stress, are not included.
- Linear elastic material; the ring must not yield or buckle.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Check a ring, link, collar or band loaded across a diameter
- Estimate how much a ring goes oval under a pinch load
- Find the load a ring can take at an allowable stress
Who it is for
- Machine designers
- Mechanical engineers
- Tooling designers
What you get
- Second moment of area of the section
- Maximum bending stress at the load points
- Increase of the diameter along the load and decrease across it
- Load at the allowable stress
- Section depth as a share of the diameter
Interpreting the result
- Stress grows with the ring diameter and falls with the cube of the bar size, so a slightly thicker bar helps a great deal.
- A ring flexes far more than a straight bar of the same length: the diameter change grows with the cube of the diameter.
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 219–220. Topic reference; no source tables or recommended settings are reproduced.
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Open the Thin Ring Loaded Across a Diameter (Bending Stress, Diameter Changes, Safe Load) calculator