A hollow shaft can weigh less than a solid shaft with the same outside diameter, but the bore also reduces bending and torsional rigidity. The table compares ideal circular cross sections by inner-to-outer diameter ratio. It holds outside diameter, material, and length constant and reports normalized section properties rather than an allowable load.

Read the ratios as a geometric comparison

Let k equal the inside diameter divided by the outside diameter. The hollow section area relative to the solid section is 1−k². With the same density and length, this is also the mass ratio. The mass reduction column is the complement of that area ratio. At a bore ratio of 0.5, the hollow member retains 75% of the cross-sectional area and therefore weighs 25% less than the same-length solid member at the same outside diameter.

For a circular shaft, the bending second moment and torsion polar moment each scale with 1−k⁴ when compared at the same outer diameter. These properties describe resistance to elastic deformation. At k=0.5, each ratio is 0.9375, so this idealized hollow section retains 93.75% of the solid section’s bending and torsional rigidity while reducing its mass by 25%. Material placed near the outside radius contributes strongly to these section properties.

Strength and stiffness answer different questions

Under equal allowable maximum surface stress and equal outer radius, bending and torsional moment capacity both fall in proportion to 1−k⁴. The hollow section reaches the same outer-fiber stress at a lower moment because its section modulus and polar moment are smaller. It will also deflect or twist more under the same load. This is why a statement such as “hollow shafts are stronger” needs a defined constraint. Equal mass, equal outside diameter, equal stiffness, equal stress, and equal manufacturing cost produce different comparisons.

The calculation does not include a keyway, spline, cross-hole, shoulder, weld, press fit, corrosion allowance, residual stress, or stress concentration. It also does not assess fatigue, impact, vibration, buckling, critical speed, bearing support, or local wall collapse. A thin wall can be sensitive to dents, clamping loads, and manufacturing variation even when the section formulas look favorable.

Use the table to screen concepts

Start with packaging limits and the required torque, bending load, and deflection. Select candidate outer and inner diameters, then calculate actual polar moment, second moment, torsional stress, bending stress, and angular or lateral deflection. Check combined loading and the load path through hubs and connections. If the shaft is hollow, assess tube availability, internal features, drainage or sealing, inspection access, and the manufacturing process.

For the same outside diameter, increasing the bore ratio reduces mass quadratically and reduces bending and torsional stiffness by the fourth-power relation. That relation is useful for a quick comparison but does not establish a safe design. Use the material’s certified properties, suitable fatigue data, appropriate safety factors, and applicable design code. Have the completed shaft checked by a qualified engineer before release.

The normalized values also help compare packaging options: if a solid design is too heavy, increasing the bore can remove material near the axis while retaining much of the outer-fiber contribution to bending and torsion. If deflection or twist is already near its limit, however, the lower section rigidity may govern before stress does. Evaluate both peak stress and serviceability for the actual span, support conditions, torque spectrum, and load direction. These ratios alone do not identify an optimum design.

For a concrete ratio check, let Do be 50 mm and Di be 25 mm. Then k=0.5, the area ratio is 1−0.5²=0.75, and the mass reduction is 25%. The bending and polar moment ratios are both 1−0.5⁴=0.9375. The dimensional result still depends on the actual diameters: compute I=π(Do⁴−Di⁴)/64 and J=π(Do⁴−Di⁴)/32 in consistent units before calculating stress, twist, or deflection. Ratios simplify comparison but do not replace dimensional calculations.

Source: independently calculated from circular-section area, bending second moment, and polar moment equations. The assumptions and equations are stated with the dataset so that each ratio can be reproduced from a selected diameter ratio.