Shaft Torque and Stress Calculator
Calculate the maximum torsional shear stress in a solid circular shaft given applied torque and shaft diameter. Used in mechanical design to verify shafts against yield and fatigue limits before manufacture.
About this calculator
For a solid circular shaft subjected to pure torsion, the shear stress is maximum at the outer surface and is given by: τ = Tc / J, where T is the applied torque, c is the outer radius, and J = πd⁴/32 is the polar second moment of area. Substituting and simplifying gives the working formula: τ = 16T / (πd³), where T is in N·mm and d is the shaft diameter in mm, yielding τ in N/mm² (MPa). In this calculator torque is entered in N·m and converted: τ = (16 × T × 1000) / (π × d³). The result must be compared against the allowable shear stress for the shaft material, typically taken as 0.577 × yield strength (von Mises criterion) divided by a safety factor.
How to use
A steel shaft with diameter d = 40 mm transmits a torque T = 200 N·m. Using the formula: τ = (16 × 200 × 1000) / (π × 40³). Numerator: 16 × 200,000 = 3,200,000. Denominator: π × 64,000 = 201,062. τ = 3,200,000 / 201,062 ≈ 15.9 MPa. For a steel shaft with yield strength 250 MPa, the allowable shear stress is about 144 MPa, giving a generous safety factor. Enter T = 200 and d = 40 into the calculator to confirm τ ≈ 15.9 MPa.
Frequently asked questions
What is torsional shear stress and where does it occur in a shaft?
Torsional shear stress is the internal shear force per unit area induced in a shaft when a twisting moment (torque) is applied. It acts tangentially on cross-sectional planes perpendicular to the shaft axis. For a solid circular shaft the shear stress is zero at the centre and reaches its maximum value at the outer surface, which is why surface cracks and fatigue failures typically initiate there. The distribution is linear from centre to surface, a result that follows directly from the assumptions of plane sections remaining plane under torsion.
How does shaft diameter affect torsional shear stress?
Shear stress is inversely proportional to the cube of the diameter (τ ∝ 1/d³). This means doubling the diameter reduces shear stress by a factor of eight, making diameter the most powerful design lever for reducing stress. Conversely, a 10% reduction in diameter increases shear stress by roughly 33%. This cubic relationship explains why even small errors in measuring or manufacturing shaft diameter can significantly affect the actual stress level relative to design predictions.
How do I calculate the angle of twist for a shaft under torsion?
The angle of twist φ is given by φ = TL / (GJ), where T is torque (N·mm), L is the shaft length (mm), G is the shear modulus of the material (MPa), and J = πd⁴/32 is the polar second moment of area (mm⁴). The result is in radians; multiply by 180/π to convert to degrees. For steel G ≈ 80,000 MPa. Excessive twist can misalign coupled components or cause torsional vibration resonance, so checking angle of twist is equally important as checking stress, especially for long shafts or precision machinery.