Shaft Combined Bending and Torsion Calculator

Combined-stress equations

For a solid or concentric hollow circular shaft, the centroidal area second moment I and polar second moment J are:

At outer radius c = D/2, the nominal stress magnitudes are:

The calculator combines the tensile-side outer-surface state σx = σ_b, σy = 0, and τxy = τ_t:

Worked solid-shaft example

Use the default inputs: M = 750 N·m, T = 500 N·m, D = 40 mm, and d = 0.

1. The section properties are I = 125,663.706 mm⁴ and J = 251,327.412 mm⁴.
2. The tensile-side outer-surface stresses are σ_b = 119.366207 MPa and τ_t = 39.788736 MPa.
3. The principal stresses are σ₁ = 131.413267 MPa and σ₂ = −12.047060 MPa.
4. The maximum shear stress is 71.730164 MPa, and the principal plane is 16.845034° from the bending normal plane for the reported positive shear convention.
5. The nominal equivalent stresses are σ_vM = 137.832224 MPa and σ_Tresca = 143.460327 MPa.

No material strength is entered, so the example does not state whether the shaft is acceptable.

Interpret von Mises and Tresca without inventing an allowable

Von Mises and Tresca reduce a multiaxial nominal stress state to scalar values commonly used with ductile-material yielding criteria. They are not material properties and do not become safety factors by themselves. Compare only the criterion required by the applicable design method with verified material data, temperature and condition adjustments, load factors, concentration treatment, and the required safety basis.

The Tresca equivalent stress reported here is σ₁ − σ₂, equal to twice the maximum shear for this plane-stress state. It is normally at least as large as the corresponding von Mises result, but that does not authorize choosing either criterion without a project-specific basis.

Pure bending and pure torsion limits

Set T = 0 for pure bending. The von Mises and Tresca equivalent stresses then both equal |σ_b|. Set M = 0 for pure torsion. Von Mises becomes √3 τ_t, while Tresca becomes 2τ_t. At least one load must remain positive.

If bending occurs about two perpendicular axes, first determine the resultant bending moment magnitude at the evaluated section from the complete shaft free-body diagram. This tool does not derive bearing reactions, gear forces, belt forces, moment diagrams, or the critical section.

Nominal stress is not peak local stress

The uniform circular-section equations do not represent peak stress at a keyway, shoulder, snap-ring groove, thread, spline, cross-hole, press fit, weld, or fillet. Apply the theoretical and fatigue concentration treatment required by the chosen shaft-design method. Do not insert an unverified factor merely to make the displayed value appear conservative.

Engineering scope and limitations

This calculator covers a straight, prismatic, homogeneous, isotropic, solid or concentric hollow circular shaft under a resultant bending-moment magnitude and a torque magnitude at one section. It uses linear-elastic nominal stress superposition. It excludes:

• axial force, transverse shear stress, bearing contact, local load introduction, and noncircular sections;
• keyways, splines, cross-holes, shoulders, grooves, threads, fillets, welds, fits, notches, and stress-concentration factors;
• material selection, yield or ultimate strength, allowable stress, safety factors, yielding, plasticity, fracture, and final acceptance;
• mean and alternating stress, fatigue, load spectra, reversal, shock, impact, surface finish, size effect, reliability, and residual stress;
• deflection, slope, bearing reactions, gear or belt forces, critical speed, lateral whirl, torsional vibration, resonance, damping, and transient response;
• tolerances, corrosion allowance, heat treatment, inspection, stock availability, manufacturing feasibility, cost, certification, and engineering approval.

Use a complete shaft free-body diagram, the governing design standard or company method, verified project data, and qualified engineering review before releasing a drawing or selecting a safety-critical shaft.