Hollow Shaft Diameter Calculator to DIN 743-1

Why does the outer diameter grow more than for a solid shaft?

A bore removes material where it contributes least to bending and torsional capacity -- at the shaft centre -- but it still lowers the section modulus, because that grows with the fourth power of diameter. The factor (1-beta^4) captures exactly this loss: at beta=0.6 the section modulus falls to 87% of the full section, at beta=0.8 to just 59%. The outer diameter must grow accordingly for the stress to return to the allowable level.

Formula and variables

do = [sqrt((32*Mb/pi)^2 + 3*(16*T/pi)^2) / (Re/S * (1-beta^4))]^(1/3)

• beta = di/do (freely chosen)
• di = beta*do
• sigma_b = 32*Mb / (pi*do^3*(1-beta^4))
• tau_t = 16*T / (pi*do^3*(1-beta^4))
• sigma_v = sqrt(sigma_b^2 + 3*tau_t^2) (von Mises)

Choose the inputs correctly

Bending moment Mb, torque T, yield strength Re, required safety S, and the freely chosen bore ratio β=di/do.

How to use the calculator

Enter the load, material yield strength, required safety and the desired bore ratio beta. The calculator returns the required outer and corresponding inner diameter, plus the area comparison to an equal-strength solid shaft.

Worked example

Mb=500 N·m, T=300 N·m, Re=355 MPa, S=1.5, beta=0.6: required outer diameter do=30.3 mm, inner diameter di=18.2 mm -- with 30% less cross-section area than an equal-strength solid shaft (do=28.9 mm).

When is a hollow shaft still worthwhile?

The cross-section area ratio shows whether the larger bore pays off: at beta=0.6, area -- and approximately mass -- falls to about 70% of an equal-strength solid shaft despite the larger outer diameter. At small beta values (below about 0.3) the effect is small and a bore usually only makes sense for functional reasons (a line, cabling), not weight saving. At very large beta values (above about 0.8) the wall becomes thin enough that buckling or local wall failure needs checking separately -- this calculator covers only the stress calculation, not that.

Moments in newton-metres, stresses in MPa, diameters in millimetres, beta and the area ratio are dimensionless.

Typical applications

Pre-designing shafts with a lubricant or cable pass-through, weight optimization of long, moderately loaded shafts, an initial estimate before a complete DIN 743 verification with notch effects and fatigue.

Assumptions, limits and common mistakes

DIN 743-1:2012-12, nominal stress approach for annular sections.

Common mistake: Don't mistake beta for a standard or catalogue value -- it is a free design decision, typically set by the required bore diameter (a line, a shaft-in-shaft arrangement), not a table. For very thin-walled hollow shafts (beta near 1), also check local buckling, which this calculator does not cover.