A hollow shaft removes material near the centre, where it contributes less to bending and torsional section properties than material near the outside surface. The table compares a solid 25 mm shaft with two tubes that share the same outside diameter. It reports cross-sectional area, which is proportional to mass per unit length for the same material, and the geometric second moments used in basic bending and torsion equations. The rows isolate geometry so the tradeoff is easy to see; they do not claim that any example is a safe or interchangeable machine shaft.

Hollow Versus Solid Shaft Strength and Weight

Read the section columns

The inside-diameter ratio is the bore divided by the 25 mm outside diameter. At a ratio of 0.50, the bore is 12.5 mm; at 0.80, it is 20 mm. Area falls with the square of diameter, while the bending and polar moments depend on the fourth power. That fourth-power term explains why removing material near the axis can reduce mass more quickly than it reduces these idealized section properties. The table uses rounded calculated values, so recomputing with the displayed rounded diameters can differ slightly in the last decimals.

A worked comparison

For the 25 mm solid section, A=πDo²/4, I=πDo⁴/64, and J=πDo⁴/32. For the 20 mm bore case, multiply the solid area by 1−(20/25)²=0.36, giving about 176.7 mm². Its I and J are multiplied by 1−(20/25)⁴=0.5904, so their idealized values retain about 59% of the solid section values. If both shafts use the same material and length, the hollow section has about 36% of the material area, but it does not retain 100% of bending or torsional stiffness.

What the fourth-power result does not say

The table does not convert section moments into allowable load. A strength calculation also needs material yield and fatigue properties, shaft length, support conditions, applied bending and torque, and a design factor. Deflection may govern before stress, particularly for long shafts or precision alignment. A hollow section also introduces inner-surface condition and wall-thickness concerns, and real manufacturing can add eccentricity, weld seams, residual stress, or local thinning. Check those details before replacing a solid shaft with tubing.

Choose geometry for the actual load

Use bending I when assessing bending stress or deflection in the relevant plane; use polar J for elementary torsion of a circular shaft. Combined loading may require both, plus axial load and stress concentration effects at shoulders, keyways, cross holes, splines, and retaining-ring grooves. The same outside diameter is only one comparison basis. If packaging permits a larger outside diameter, a hollow shaft can gain section efficiency, but bearing fits, seals, critical speed, and connection details may then change. Compare complete designs at equal mass or equal envelope when that reflects the real constraint.

Common mistakes and references

Do not call the area ratio a strength ratio, and do not use J as though it were an allowable torque. They are geometric terms used inside separate equations. Keep the inside and outside diameters in the same unit before raising them to the fourth power. The equations and shaft-property meanings are summarized in the MIT OpenCourseWare circular-shaft material linked below. The dataset rows are independent calculations from those equations rather than a transcription of the cited handbook extraction. A qualified design check must still include the actual material, loads, supports, fatigue environment, and manufacturing features.

References

• MIT OpenCourseWare, circular shaft section properties
• Original calculations for circular shaft geometry

This reference is limited to the conditions named in the data and sources. Confirm the governing specification, complete geometry, material, operating environment, and service loads before using any value for design, procurement, inspection, or safety-critical decisions.