Rotational Stiffness Calculator

Determine the resistance of a body to rotation using the rotational stiffness calculator.

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.