Torsional Stiffness Calculator

Background.

Twist a steel bar and it behaves like a spring, winding up an amount of rotation in proportion to the torque on it. Torsional stiffness, k_t, is the constant of that spring: the torque needed to turn one end of a member one radian relative to the other, in newton-metres per radian. It decides how far a drive shaft winds up under load, how accurately a machine spindle holds its angle, whether a car body feels taut or loose, and at what frequency a rotating system will vibrate.

For a uniform bar three things set it. The shear modulus G measures how strongly the material resists being sheared; steel is around 79 GPa, roughly three times typical aluminium alloys. The torsion constant J captures the size and shape of the cross-section. Length works against both, since k_t = GJ/L: a shaft twice as long twists twice as far under the same torque.

Geometry is usually the strongest lever. For a solid round shaft J = πd⁴/32, so stiffness climbs with the fourth power of diameter, and a shaft 25% thicker is about 2.44 times as stiff. Tubes exploit this by placing material far from the axis, where it resists twisting most effectively.

The classic mistake is using the wrong J. For solid circles and round tubes the torsion constant equals the polar moment of inertia; for every other shape it is smaller, sometimes drastically so. Open sections such as channels, angles or a tube slit along its length keep only a small fraction of the stiffness of the closed tube they resemble.

The model behind the result is Saint-Venant torsion: a straight member of constant cross-section, linear-elastic material, free to warp, and twisted through small angles.

What is torsional stiffness calculator?

Torsional stiffness is the ratio of applied torque to the resulting angle of twist, k_t = T/φ, for a member loaded in pure torsion. Its SI unit is the newton-metre per radian. For a straight bar of constant cross-section made of a linear-elastic material, Saint-Venant theory gives k_t = GJ/L, where G is the shear modulus, J the torsion constant of the section and L the length between the two sections whose relative rotation is of interest. A high value means a member that barely rotates under load.

How to use this calculator.

1. 01Enter the Shear modulus in pascals: 79,000,000,000 for typical steel. If you only know Young's modulus E and Poisson's ratio ν for an isotropic material, use G = E ÷ (2(1 + ν)).
2. 02Enter the Torsion constant J in m⁴. For a solid round shaft J = πd⁴/32 and for a round tube J = π(D⁴ − d⁴)/32, with diameters in metres; a diameter typed in millimetres makes J a trillion times too large. For other shapes take J from a section table, not the polar moment.
3. 03Enter the Member length in metres, measured between the two sections whose relative twist you care about.
4. 04Read the stiffness in N·m/rad. Divide a torque by it for the twist in radians, or multiply it by π/180 for newton-metres per degree.

The formula.

k_t = G × J ÷ L. G, the shear modulus in pascals, is a material property: shear stress divided by shear strain in the elastic range. J, the torsion constant in m⁴, depends only on the shape and size of the cross-section. L is the length in metres over which twist accumulates. Pascals times metres to the fourth, divided by metres, leaves newton-metres, and because the angle is in radians the stiffness is quoted per radian. The equation is the angle-of-twist formula for a prismatic bar, φ = TL/(GJ), rearranged as T/φ. The result is proportional to G and J and inversely proportional to L, and since J grows with the fourth power of a shaft's diameter, size is usually the dominant term. It holds only while the material stays elastic and the section is free to warp; tapered members, restrained warping in open sections and yielding all fall outside it.

A worked example.

Frequently asked questions.

References& sources.

1. [1]Gere and Goodno, Mechanics of Materials, 9th ed. (PRINT), chapter 3, angle of twist for prismatic bars.
2. [2]MIT OpenCourseWare, Unified Engineering, torsion of shafts.
3. [3]BIPM, The International System of Units (SI Brochure), 9th ed., version 3.01, coherent derived units and quantity equations.