Shaft Torsional Shear Stress Calculator

Enter the torque magnitude, the shaft outer radius, and the cross-section's polar moment of inertia, then choose the units used for each quantity. The calculator applies the elastic torsion relation tau equals torque times radius divided by polar moment, normalizes the values to SI units, and reports the result in both pascals and megapascals. It is suitable for solid or hollow circular shafts when the correct polar moment is already known.

What the torsional shear stress result means

A torque twists a shaft and creates shear stress across its cross-section. For a circular shaft under ordinary elastic torsion, that stress increases linearly with distance from the centerline. It is zero at the center and reaches its largest magnitude at the outside surface, which is why this calculator uses the outer radius. The returned maximum value is the nominal torsional shear stress at that surface. Designers commonly compare it with an allowable shear stress after applying the safety factors, material criteria, and design rules appropriate to their project. The calculation applies to both solid and hollow circular shafts because the supplied polar moment of inertia describes the actual cross-section. However, it does not automatically account for keyways, splines, shoulders, holes, cracks, or other local features that concentrate stress. It also does not combine torsion with bending, axial load, contact stress, thermal effects, fatigue cycles, or residual stress. Treat the answer as a well-defined mechanics result, not as a complete declaration that a shaft is safe.

How to enter torque, radius, and polar moment

Enter torque as a positive magnitude and select its unit. The available choices cover newton-metres, newton-millimetres, pound-force feet, and pound-force inches. Enter the outer radius using the selected length unit. The polar moment must use that same length unit raised to the fourth power: if the radius is in millimetres, supply the polar moment in millimetres to the fourth power; if the radius is in inches, use inches to the fourth power. This shared length-unit rule prevents an easy but serious dimensional mistake. The calculator converts torque to newton-metres, radius to metres, and the polar moment to metres to the fourth power before evaluating the formula. All three magnitudes must be finite and greater than zero. Use the applied torque at the section being checked, not necessarily the motor nameplate torque, because transmitted torque can change along a shaft where gears, pulleys, couplings, or other components add or remove load. Obtain the polar moment from the actual circular geometry or a trusted section-property calculation.

Formula, assumptions, and interpreting the output

The governing relation is tau_max = T times r divided by J, where T is the internal torque at the section, r is the outer radius, and J is the polar second moment of area. In coherent SI units, newton-metres multiplied by metres and divided by metres to the fourth power produces newtons per square metre, or pascals. The calculator also divides that result by one million to display megapascals, a convenient unit for shaft design. The relation assumes a circular cross-section, elastic material response, small deformation, and Saint-Venant torsion away from disturbed end regions. It assumes the supplied J is correct for the section: for example, a hollow shaft requires the hollow-section polar moment rather than a solid-shaft value based only on its outer diameter. The stress magnitude does not show twist angle; calculating twist also requires shaft length and shear modulus. Before making a design decision, compare the nominal result with the governing material criterion and code, consider stress concentration and fatigue where applicable, and verify that all input units describe the same physical section.