Calculate torsional shear stress, angle of twist, and polar moment of inertia for circular shafts and hollow tubes.
Choose a section — solid circular, hollow circular, or rectangular — then enter the applied torque, the member length, and the shear modulus G (about 79,300 MPa for steel and 26,000 MPa for 6061 aluminum in the material database).
For a solid shaft the polar moment is J = πd⁴/32 and the peak shear stress is τ = 16T/(πd³); a hollow shaft uses J = π(D⁴ − d⁴)/32 with stress evaluated at the outer radius. Because stress falls with the cube of diameter, a modest size increase buys a large stress reduction.
Rectangular bars have no simple polar moment: the tool interpolates the coefficients α and β from the classical aspect-ratio table (α = 0.208, β = 0.141 for a square, both approaching 0.333 for a thin strip) and applies τ = T/(α·b·t²) and θ = TL/(β·b·t³·G). The angle of twist is reported in both radians and degrees.
Shear stress varies linearly with radius, so material near the center carries almost nothing yet adds weight. A tube keeps material at the outside where it works hardest: boring out a core of half the outer diameter removes only 1/16 of J (which depends on D⁴ − d⁴) while saving a quarter of the weight.
θ = TL/(GJ) in radians is how far one end rotates relative to the other, also shown in degrees. It grows linearly with length and torque, and inversely with the stiffness GJ — useful for drive shafts and machine elements where alignment matters as much as strength.
Round-shaft formulas do not apply. The calculator interpolates the coefficients α and β from the aspect ratio b/t — 0.208 and 0.141 for a square, rising toward 0.333 as the bar flattens into a strip — then uses τ = T/(α·b·t²). Peak stress occurs at the midpoint of the long side, not at the corners.
Use G, not Young's Modulus. From the built-in material database: steel 79,300 MPa, stainless 304 77,000 MPa, 6061 aluminum 26,000 MPa, Titanium Gr5 44,000 MPa, and Douglas Fir about 810 MPa. Entering E by mistake would understate the twist of a steel shaft by roughly 2.5 times.