CTR K

Torsion of Sections Calculator — Shear Stress, Angle of Twist & Fatigue (Roark / Bredt-Batho)

Governing standard: Roark / Bredt-Batho· Saint-Venant / Timoshenko / Roark torsion for solid sections · Bredt-Batho shear-flow for closed thin-walled sections · Goodman/Soderberg torsional fatigue (Shigley §6)

The MechanixCalc torsion calculator computes the torsional shear stress, angle of twist, polar moment of inertia and yield/ultimate safety factors for eight standard cross-section types — solid and hollow circular shafts, solid and hollow squares, solid rectangles, thin-walled circular tubes and thin-walled rectangular boxes. Solid sections follow Saint-Venant torsion theory with Timoshenko/Roark coefficients; closed thin-walled sections use Bredt-Batho shear-flow theory. The result includes a Mohr's circle for the combined stress state, a torsional fatigue check against Modified Goodman and Soderberg diagrams, and a combined bending-plus-torsion analysis using Von Mises, Tresca and the ASME equivalent-moment method.

It is built for structural and mechanical engineers who need to verify a shaft, structural hollow section (SHS/RHS) or frame member under a known torque — and who need to hand a reviewer a complete, standards-cited calculation with a worked safety factor rather than a spreadsheet. The tool covers everything from a quick polar-moment check to a full torsional fatigue life assessment.

What this calculator does

  • Torsional shear stress and angle of twist for 8 cross-section types (Saint-Venant / Timoshenko / Roark)
  • Polar moment of inertia J and torsional section modulus Wt per section type
  • Yield and ultimate safety factors via Von Mises and Tresca criteria
  • Bredt-Batho shear-flow theory for closed thin-walled sections (box, tube, open channel) with efficiency comparison
  • Mohr's circle for the full combined stress state under bending plus torsion
  • Torsional fatigue life check using Modified Goodman and Soderberg diagrams (Shigley §6)
  • Branded PDF engineering report with the full method and section sketch

Method & formulas

Torsional shear stress and angle of twist (Saint-Venant / Roark)

For circular cross-sections (solid and hollow) the torsional shear stress is exact: τ = T·c / J, where J = π·D⁴/32 for a solid shaft and J = π·(Do⁴ − Di⁴)/32 for a hollow shaft. The angle of twist is φ = T·L/(G·J). For non-circular solid sections (square, rectangle) the maximum shear stress does not occur at the furthest point from the centroid — Saint-Venant's theory gives τ_max = T / Wt, where the torsional section modulus Wt and the torsion constant J are tabulated functions of the section aspect ratio (Timoshenko/Roark α and β coefficients, with J = β·a·b³ and Wt = α·a·b² for long side a, short side b).

Circular section — shear stress and twist
τ = T·c / J; φ = T·L / (G·J)

where τ = maximum shear stress (MPa); T = applied torque (N·mm); c = outer radius (mm); J = polar moment of inertia (mm⁴); φ = angle of twist (rad); L = shaft length (mm); G = shear modulus (MPa)

Non-circular solid section (Roark / Timoshenko)
τ_max = T / Wt; Wt = α·a·b²; J = β·a·b³

where α, β = Roark/Timoshenko torsion coefficients (function of a/b ratio); a = long side (mm); b = short side (mm); Wt = torsional section modulus (mm³)

Bredt-Batho theory for closed thin-walled sections

For closed thin-walled sections (hollow square tube, circular tube, rectangular box) the Bredt-Batho theory gives the shear flow q = T / (2·A_m) and the shear stress τ = q / t, where A_m is the area enclosed by the section mid-line and t is the wall thickness. The torsion constant is J = 4·A_m²·t / s, where s is the perimeter of the mid-line. Open sections (slotted tube, open channel) carry torque very inefficiently by Saint-Venant warping — the calculator compares τ and φ between closed and open forms for the same envelope dimensions to show the stiffness penalty of opening the wall.

Bredt-Batho closed thin-wall
q = T / (2·A_m); τ = q / t; J = 4·A_m²·t / s

where q = shear flow (N/mm); A_m = area enclosed by mid-line (mm²); t = wall thickness (mm); s = mid-line perimeter (mm); J = torsion constant (mm⁴)

Torsional fatigue (Modified Goodman / Soderberg)

When the torque has both a steady (mean) and cyclic (alternating) component the fatigue safety factor is found from the Modified Goodman line in shear-stress space. The torsional endurance limit Sse = 0.577 × 0.5 × Sut (von Mises conversion) and the shear yield is Ssy = 0.577 × Sy. A Sines-criterion check is also included: because pure torsion produces zero hydrostatic mean stress, the mean torque does not shift the Sines endurance limit (Shigley §6-14) — only the alternating component matters for Sines.

Modified Goodman in shear space
τ_a / Sse + τ_m / Sus = 1 / SF_Goodman

where τ_a = alternating shear stress (MPa); τ_m = mean shear stress (MPa); Sse = torsional endurance limit = 0.577 × 0.5 × Sut (MPa); Sus = torsional ultimate = 0.577 × Sut (MPa); SF_Goodman = torsional fatigue safety factor

Worked example

A solid circular steel shaft, D = 40 mm, carries a steady torque T = 200 N·m over a length L = 500 mm. The shear modulus of steel is G = 80,000 MPa. Find the maximum torsional shear stress and the total angle of twist.

Given

  • Shaft diameter D40 mm
  • Applied torque T200 N·m = 200,000 N·mm
  • Shaft length L500 mm
  • Shear modulus G80,000 MPa (steel)

Result

  • Polar moment J251,327 mm⁴ (≈ 251.3 × 10³ mm⁴)
  • Maximum shear stress τ≈ 15.9 MPa
  • Angle of twist φ≈ 0.285° (0.00497 rad)
  1. Compute the polar moment of inertia: J = π·D⁴/32 = π × 40⁴ / 32 = π × 2,560,000 / 32 = 251,327 mm⁴.
  2. The outer radius (distance from centre to surface): c = D/2 = 20 mm.
  3. Maximum torsional shear stress: τ = T·c / J = 200,000 × 20 / 251,327 = 4,000,000 / 251,327 ≈ 15.9 MPa.
  4. Angle of twist: φ = T·L / (G·J) = 200,000 × 500 / (80,000 × 251,327) = 1.00 × 10⁸ / 2.011 × 10¹⁰ ≈ 0.004974 rad.
  5. Convert to degrees: φ = 0.004974 × (180/π) ≈ 0.285°.

This is an illustrative example for a solid circular shaft only — the calculator handles all 8 section types, applies the correct Saint-Venant or Bredt-Batho formula for each, and computes the safety factor against your material's shear yield strength.

Frequently asked questions

Which standard does this torsion calculator use?

Solid circular and hollow circular sections use the exact elasticity result (τ = T·c/J; φ = T·L/(G·J)) as given in Roark's Formulas for Stress and Strain and Timoshenko's Strength of Materials. Non-circular solid sections (square, rectangle) use the Saint-Venant torsion coefficients tabulated by Roark/Timoshenko. Closed thin-walled sections (hollow square, rectangular box, circular tube) use the Bredt-Batho shear-flow theory. Torsional fatigue follows the Modified Goodman and Soderberg criteria from Shigley's Mechanical Engineering Design §6. The governing method and coefficients are shown in the generated PDF report.

What is the difference between the torsion constant J and the polar moment of inertia?

For circular sections (solid or hollow) the torsion constant J is identical to the polar second moment of area (J = π·D⁴/32). For non-circular sections, however, the polar moment does not govern torsion — the Saint-Venant torsion constant (also called J or C) is a different, smaller quantity that must be obtained from Roark's tables or computed from the Timoshenko α/β coefficients. This calculator uses the correct constant for each section type, not the naive polar moment.

Why are open thin-walled sections so much less stiff in torsion than closed ones?

A closed cross-section (hollow box or tube) carries torque by a continuous shear flow q = T/(2·A_m) around the perimeter — the shear stress is low and the torsional stiffness is high. If the wall is cut (open section), the shear flow is interrupted and the section can only resist torque by Saint-Venant warping shear, which is proportional to t³. For the same wall thickness and envelope dimensions the shear stress in an open section is typically 10–100× higher and the twist is 100–1000× greater than in the closed form. The calculator's Bredt-Batho comparison panel shows the ratio explicitly.

Can I check a shaft under combined bending and torsion?

Yes — the combined-loading tab takes a bending moment M, axial force N and torque T for a solid or hollow circular shaft, and computes the Von Mises equivalent stress, the Tresca equivalent stress and the maximum principal stresses. It also draws the Mohr's circle for the combined state and recommends a minimum shaft diameter (ASME equivalent-moment method). A separate sub-panel sizes a solid circular shaft for a given bending moment, torque and stress-concentration factors.

Is the torsion calculator free?

You can use it during a free 30-minute preview with no sign-up, and a free 14-day account trial unlocks every calculator with no credit card required. The branded PDF engineering report and saved calculations are part of a paid plan.

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