A solid circular shaft under bending and torque develops different surface stresses: bending creates longitudinal normal stress, while torsion creates shear stress. The equations below summarize section properties, stress, twist, power-to-torque conversion and one common combined-stress measure. They are derived from elementary elastic mechanics of materials and apply only when the shaft geometry and loading match the assumptions stated for each relationship.

Solid circular shaft formulas

Use consistent units throughout. In an SI calculation, for example, use metres, newtons, pascals, newton-metres and watts consistently. The von Mises expression combines bending normal stress and torsional shear stress at the same outer surface; it does not include a safety factor or material design allowable.

How bending and torsion differ

For a solid round section of diameter d, the centroidal bending second moment is I = pi d^4/64, and the polar moment is J = pi d^4/32. The elastic section modulus is S = pi d^3/32. With a bending moment M, the maximum bending stress at the outer fiber is sigma = M c/I, where c=d/2. Under torque T, the maximum shear stress at the outside radius is tau = T c/J. These separate relationships are why a shaft subject to both loads must be evaluated for combined stress rather than checked with only a bending or torsion equation.

For uniform torque over a prismatic shaft length L, elastic twist is theta = T L/(G J), where G is shear modulus. The diameter enters the polar moment to the fourth power, so a small change in diameter can materially affect twist. This equation assumes constant section and material and does not account for a keyway, shoulder, spline, local notch, variable torque, or flexible coupling.

Converting power to torque

For steady power transmission, torque is power divided by angular speed: T=P/omega. When power is expressed in kilowatts and speed in revolutions per minute, T in newton-metres is approximately 9550 times power divided by speed. The conversion is useful for a first pass from motor output to shaft torque. It does not describe startup torque, impact, braking, torque ripple or a gearbox’s transient load. Use the worst credible load case and the system’s actual service factors when designing a component.

Combining surface stresses

At a point on the outside surface of a circular shaft under bending and torsion, a common ductile-material screening measure is von Mises equivalent stress: the square root of sigma-b squared plus three times tau squared. This is a yield criterion under a plane-stress assumption. The result can be compared with an appropriate material yield strength only after the correct material condition, design factor, load combination and governing design method are established.

The equation is not a complete shaft-design method. It does not add stress concentration factors for a keyseat or shoulder, assess fatigue from rotating bending, determine deflection or critical speed, check bearing reactions, or address buckling and misalignment. For cyclic service, a shaft may fail in fatigue at a nominal stress well below yield. Use detailed geometry, load history, material data and an applicable design standard for a real machine.

Common mistakes and limitations

Do not substitute diameter where a formula requires radius, or mix N·m with millimetres and MPa without converting units. Keep bending moment and torque distinct, and evaluate stresses at the same section and load case. A solid-shaft formula cannot be applied unchanged to a hollow tube; its section properties must be recalculated using the inside and outside diameters. Non-circular bars also do not follow the same elementary Saint-Venant stress distribution.

These are reference equations for an idealized solid circular member, not a recommendation for shaft size. The MIT and University of Illinois mechanics materials describe the underlying torsion and beam relations and their assumptions. A qualified engineer should verify shaft strength, fatigue life, stiffness, bearings, fits, manufacturing details and code requirements before a design is released.

Related engineering references

Browse more mechanical design topics in the Mechanitec Engineering category.

Sources

• MIT OpenCourseWare, Solid Mechanics, readings on shafts in torsion and beams in bending.
• University of Illinois Urbana-Champaign, TAM 251 formula sheet, circular-shaft torsion relations.
• Missouri S&T, Torsional Stresses, circular-shaft stress equations and applicability.