BIZOMAT MAKEOS · Strength
Rectangular Bar in Torsion
What shear stress and twist does torque cause in this rectangular or square bar?
Maximum shear stress, angle of twist and torsional constant of a solid rectangular bar under torque, using Roark’s approximations.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Torque | N·m | Torque transmitted. |
| Long side | mm | Longer side of the section. |
| Short side | mm | Shorter side (equal to b for a square). |
| Length | mm | Length over which twist is wanted. |
| Shear modulus | GPa | Shear modulus (steel about 79 GPa, aluminium about 26 GPa). |
| Allowable shear stress | MPa | Allowable shear stress from your design code. |
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: Torque
- What it is
- The twisting moment carried by the bar about its long axis, for example the torque passed from a drive through a square or flat shaft.
- Units and format
- Enter a number in N·m: newton-metres. The form accepts 0.000001 N·m or more. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 200 N·m as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Shear stress and angle of twist both rise in direct proportion to torque.
- Get it right
- Enter newton-metres; the tool converts to N·mm internally. Use the peak torque, including starting or shock torque.
What this means: Long side
- What it is
- The longer side of the solid rectangular cross-section, measured across the bar at right angles to its axis.
- 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.
- Get it right
- If you swap the two sides by mistake, the tool takes the larger one as the long side anyway.
What this means: Short side
- What it is
- The shorter side of the rectangular cross-section; for a square bar it equals the long side.
- 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.
- Get it right
- The short side has the strongest influence: stress varies roughly with its square and stiffness with its cube.
What this means: Length
- What it is
- The length of bar between the points where the torque is applied and resisted, over which the angle of twist is wanted.
- 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 500 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Twist rises in direct proportion to length; shear stress does not depend on it.
What this means: Shear modulus
- What it is
- The shear modulus (modulus of rigidity) of the bar material, which sets how much the section twists under shear.
- Units and format
- Enter a number in GPa: gigapascals (1 GPa = 1000 MPa). The form accepts 0.000001 GPa to 500 GPa. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 79.3 GPa as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- A higher modulus reduces the angle of twist; it does not change the stress.
- Get it right
- Enter gigapascals, not megapascals. Take it from the material data sheet.
What this means: Allowable shear stress
- What it is
- The highest shear stress your design code, standard or company rule permits for this bar and duty; the calculated peak stress 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 100 MPa as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Use a shear allowable, not a tensile one, and include any fatigue reduction if the torque cycles.
Method and formulas
- τmax ≈ T(3b + 1.8c) ÷ (b² c²).
- K = b c³ [1/3 − 0.21 (c/b)(1 − c⁴ ÷ 12b⁴)].
- Twist θ = T L ÷ (K G).
- Example: 200 N·m on a 40 × 20 mm bar gives τmax = 48.75 MPa and 0.98649° of twist over 500 mm.
Limitations
- Approximate formulas (within a few per cent); stress concentrations at re-entrant corners are not included.
- The bar ends are assumed able to warp.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Check square drive shafts and flat bars in torsion
- Compare a flat bar with a round shaft
- Estimate twist of a torsion arm
Who it is for
- Design engineers
- Structural and mechanical engineers
- Engineering students
What you get
- Maximum shear stress (mid-point of the long side)
- Torsional constant K
- Angle of twist
Interpreting the result
- A rectangular section is much less stiff in torsion than a round or closed tube of the same area.
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 295–303. Topic reference; no source tables or recommended settings are reproduced.
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