Torsional Stiffness Calculator

Use this tool to calculate the torsional stiffness by inputting the torque and the twist angle or the shear modulus, polar moment, and beam length.

Rotational stiffness9,549.2966 N·m/rad250 N·m ÷ 0.02618 rad

Per degree of twist166.6667 N·m/°

Angle of twist1.5°0.02618 radians

Energy stored in the twist3.2725 J½kθ² — the rotational analogue of a spring

Stiffness from the geometry33,217.1063 N·m/radGJ ÷ L for a solid round shaft

Polar second moment J2.513e-7 m⁴

Twist this torque would actually cause0.4312°against the 1.5° you entered

Maximum shear stress19.8944 MPaτr/J, at the outer surface where it is highest

At 10% more diameter48,633.1654 N·m/rad1.4641× stiffer — J goes as the fourth power of diameter, so small increases matter enormously

At twice the length16,608.5532 N·m/radhalf as stiff — length is only linear

The formula

k = τ ÷ θ; for a round shaft k = GJ ÷ L with J = πd⁴/32

Diameter to the fourth power

Rotational stiffness is torque per radian of twist, exactly as a spring rate is force per metre of extension. For a round shaft it is GJ/L: the shear modulus of the material, times the polar second moment of area, divided by the length.

J for a solid round shaft is πd⁴/32, and that fourth power dominates everything. A 10% thicker shaft is 46% stiffer; a 20% thicker one is more than twice as stiff. Length, by contrast, is only linear. This is why shaft design is overwhelmingly about diameter, and why a hollow tube is so efficient — removing the material near the axis, which contributes almost nothing to J, saves a great deal of weight for very little stiffness.

Note that stiffness and strength are different questions. The stiffness figures say how much the shaft twists; the maximum shear stress says whether it will fail. A shaft can be perfectly stiff enough and still yield, or flex unacceptably while remaining far below its strength limit.