Rotational Stiffness Calculator

Rotational Stiffness Calculator finds stiffness in N·m/rad from moment and angle, a round shaft (G J ÷ L) or a beam end (4 E I ÷ L), with twist, shear stress and natural frequency.

Rotational Stiffness Calculator: k = M ÷ θ

The Rotational Stiffness Calculator finds how strongly a joint, shaft or beam end resists being turned. It divides the applied moment by the angle the part rotates through, with the angle taken in radians.

Rotational stiffness is the moment needed for each radian of rotation. A stiff part turns very little under a large moment. A soft one turns a lot under a small moment. A door closer, a torsion bar and the end of a bolted beam all have one.

The stiffness formula has the same shape as the one for an ordinary spring, where k is force divided by stretch. Here the force becomes a moment and the stretch becomes an angle.

$$k = \frac{M}{\theta}$$

In this formula k is the rotational stiffness in N·m/rad, M is the applied moment or torque in N·m, and θ is the angle of rotation in radians. The Rotational Stiffness Calculator opens with 16 N·m turning a part by 0.7 rad, which gives 22.86 N·m/rad.

The angle can also be typed in degrees, milliradians or arcminutes. Those are converted to radians before the division, so 16 N·m over 40.11° gives the same 22.86 N·m/rad.

Rotational Stiffness Units

The SI unit is the newton-meter per radian, N·m/rad. Structural work often uses kN·m/rad, and US practice uses lbf·ft/rad or lbf·in/rad. Catalogs for couplings and torsion springs often quote stiffness per degree instead.

The Result Unit list in the Rotational Stiffness Calculator offers seven units, and the first card restates the answer in others. These are the factors for one N·m/rad.

The default 22.86 N·m/rad is 0.3989 N·m/deg, 16.86 lbf·ft/rad and 3.531 lbf·in/deg. A per-degree figure is always about 57 times smaller than the per-radian one, so check which of the two a data sheet uses before comparing parts.

Moment or Angle From a Known Stiffness

The Calculate list turns the formula around. With the stiffness known, the Rotational Stiffness Calculator finds the moment that holds a given angle, or the angle a given moment produces.

With the default 50 N·m/rad, holding 0.7 rad takes 35 N·m. Going the other way, 16 N·m turns that part by 0.32 rad, or 18.33°.

The “Moment at Other Angles” panel lays out the straight-line rule for the stiffness in use. These are its rows for the default 22.86 N·m/rad.

The energy held in the twist is ½ × M × θ, which is 5.6 J at the default values. The second card also gives the compliance, the inverse of stiffness, as 43.75 mrad per N·m. The straight-line rule holds only while the part stays elastic, and the Rotational Stiffness Calculator adds a reminder at large angles.

Shaft Stiffness: Solid Shafts and Tubes

For a round shaft the stiffness can be worked out from the material and the size, with no test. The torsional stiffness of a shaft formula uses the shear modulus G, the polar moment of area J and the length L. The Boston University mechanics notes cover the torsion theory behind it.

$$k = \frac{G J}{L}$$

The polar moment for a round section comes from the diameters. The same expression is the torsional stiffness of a hollow shaft formula, and a solid shaft is the case d = 0.

$$J = \frac{\pi (D^4 – d^4)}{32}$$

Choose “Round shaft” and the Rotational Stiffness Calculator works as a shaft stiffness calculator. The default is a solid 25 mm steel shaft, 1.5 m long, with G = 79.3 GPa. Its polar moment is 38,350 mm⁴ and its stiffness is 2,027 N·m/rad.

Under the default 200 N·m torque that shaft twists 5.652°, or 3.768° per meter. The fourth card gives the surface shear stress as 65.19 MPa, the figure to compare with the strength of the material.

The Inner Diameter field makes it a tube torsional stiffness calculator. Boring the same shaft out to 20 mm leaves 36% of the metal but keeps 59% of the stiffness, 1,197 N·m/rad. Material near the center adds little, which is why drive shafts are hollow.

Torsional stiffness and torsional rigidity are not the same thing. Rigidity is G × J, a property of the cross-section alone, in N·m². Stiffness is that rigidity divided by the length, in N·m/rad. The default shaft has a rigidity of 3,041 N·m², and halving its length doubles its stiffness.

The fourth power settles most questions about shaft size. Doubling the diameter makes a shaft 16 times stiffer in torsion. The “Shaft diameter for a k” option works backward from a target stiffness to the outer diameter that delivers it.

For 2,000 N·m/rad over 1.5 m of steel it gives a solid shaft of 24.92 mm. With an Inner Diameter of 20 mm the answer rises to 27.17 mm, a tube with about 55% of the solid shaft’s metal. A 200 N·m torque twists either one by 5.73°, and the solid shaft sees 65.86 MPa at its surface.

A square bar or a rectangular tube also follows k = G J ÷ L, but J is then the torsion constant of the section, which is smaller than its polar moment. For a solid square of side a it is about 0.1406 a⁴. A 20 mm square bar has a torsion constant of 22,496 mm⁴, against a polar moment of 26,667 mm⁴.

Choose “Any section” and type that J yourself. The default 25,000 mm⁴ over 1.5 m of steel gives 1,322 N·m/rad, and the fourth card shows that a 22.46 mm round bar has the same J. For round and hollow sections, the Polar Moment of Inertia Calculator gives J directly.

Torsion Spring Rate From Wire Size

A wire torsion spring, the kind in a clothespin or on a garage door, resists being wound up. Its rate depends on Young’s modulus E, the wire diameter d, the mean coil diameter D and the number of active coils Nₐ.

$$k = \frac{E d^4}{64 D N_a}$$

Young’s modulus appears here, not the shear modulus, because winding the spring bends the wire. Choose “Torsion spring” in the Rotational Stiffness Calculator and the default 2 mm steel wire on a 20 mm coil with 5 active coils gives 0.5 N·m/rad.

That is 3.142 N·m for one full turn of wind-up. A moment of 0.2 N·m winds the spring by 22.92°. The spring index, D ÷ d, is 10, and the active wire is 314.2 mm long.

The wire stress on the fourth card is 32 M ÷ (π d³), which is 254.6 MPa here. It is labeled uncorrected because the curve of the coil raises the stress on the inside of the wire. At a spring index of 10 the standard curvature factor adds about 8%, so the peak is near 275 MPa.

Stiffness in Series and Parallel

A shaft joined end to end with a coupling makes two rotational springs in series. Both carry the same moment and their twists add, so the combined stiffness is lower than either part alone.

$$\frac{1}{k} = \frac{1}{k_1} + \frac{1}{k_2}$$

Two parts that turn through the same angle side by side are in parallel. Each takes a share of the moment, and the stiffnesses simply add, so k = k₁ + k₂.

The “Two stiffnesses combined” option covers both arrangements. The defaults of 50 and 80 N·m/rad give 30.77 N·m/rad in series and 130 N·m/rad in parallel. In series under 16 N·m, the softer part takes 18.33° of the 29.79° total. In parallel, the stiffer part carries 9.846 N·m of the 16 N·m.

A stepped shaft is a series case. Work out each step with the “Round shaft” option, then combine the two in the Rotational Stiffness Calculator. A 25 mm steel step 1.5 m long has 2,027 N·m/rad, and a 30 mm step 1 m long has 6,306 N·m/rad. Together they give 1,534 N·m/rad, less than the thinner step alone.

Rotational Stiffness of a Beam

In structures, the rotational stiffness of a beam is the moment needed to turn one end by one radian. It depends on Young’s modulus E, the second moment of area I, the length L, and how the far end is held. Prof. Jeffrey Erochko’s structural analysis notes derive both cases.

$$k = \frac{4 E I}{L}$$

That is the value when the far end is fixed. When the far end is pinned the factor drops to 3, so k = 3 E I ÷ L. The “Far End of the Beam” setting in the Rotational Stiffness Calculator switches between them.

The default is a 1.5 m steel beam with E = 200 GPa and I = 833 cm⁴. Its flexural rigidity E × I is 1,666 kN·m². With the far end fixed the stiffness is 4,443 kN·m/rad, and with it pinned the stiffness is 3,332 kN·m/rad.

A fixed far end also feels the moment. Half of the applied moment carries over to it, and the fourth card lists that figure. A pinned far end carries none and rotates by half the angle in the opposite sense.

Bending stiffness units depend on which stiffness is meant. Flexural rigidity E × I is in N·m². Rotational stiffness is in N·m/rad. The stiffness of a beam against a point load is in N/m, with 3 E I ÷ L³ at the tip of a cantilever and 48 E I ÷ L³ at the middle of a simply supported span.

Natural Frequency and the Lever Arm

A mass attached to a rotational spring swings back and forth as a torsional oscillator. Its angular frequency is the square root of stiffness divided by the attached moment of inertia.

$$\omega_n = \sqrt{\frac{k}{I}}$$

With the default 22.86 N·m/rad and an Attached Inertia of 0.5 kg·m², the third card of the Rotational Stiffness Calculator shows 6.761 rad/s, which is 1.076 Hz with a period of 929.3 ms. The same shaft stiffness that looks ample for static twist can put this frequency close to a machine’s running speed.

The attached inertia comes from the Mass Moment of Inertia Calculator. For the frequency side alone, the Angular Frequency Calculator converts between rad/s, hertz and period, and the Damping Ratio Calculator covers how fast the swing dies away.

The Lever Arm field in the Rotational Stiffness Calculator turns the rotational spring into a straight-line one. At an arm r the linear stiffness is k ÷ r² and the force is M ÷ r. For the default values and a 0.25 m arm that is 0.3657 N/mm, a force of 64 N and 175 mm of travel along the arc.

Stiffness Types, Factors and Measurement

Stiffness is always a load divided by the movement it causes. The answers below cover the forms that come up next to rotational stiffness.

What are the different types of stiffness?

There are four common ones for a bar. Axial stiffness is E A ÷ L, for stretch along its length. Bending stiffness resists sideways deflection. Torsional stiffness is G J ÷ L, for twist about its own axis. Rotational stiffness is the moment per radian at a joint or support. The unit of the first two is N/m, and of the last two N·m/rad.

How is stiffness measured?

Apply a known load and measure the movement. For rotational stiffness, hang a weight on an arm to make a known moment, read the rotation with a dial gauge or an inclinometer, and divide. Several load steps plotted on a graph give a line whose slope is the stiffness. The Rotational Stiffness Calculator does the division for a single pair of readings.

What are the stiffness factor and the stiffness ratio?

In frame analysis the stiffness factor of a member is its rotational stiffness, 4 E I ÷ L or 3 E I ÷ L. The stiffness ratio at a joint is one member’s factor divided by the sum for all members meeting there. It sets the share of a moment that each member takes.

What is the stiffness method?

It is the standard way to analyze frames by computer. Each member gets a stiffness matrix that links its end forces to its end movements. For end rotations alone, a beam’s matrix has 4 E I ÷ L on the diagonal and 2 E I ÷ L off it, the same 4 and carry-over that appear above.

How do you calculate specific stiffness?

Specific stiffness is Young’s modulus divided by density. It compares materials by stiffness per unit of weight. Steel at 200 GPa and 7,850 kg/m³ comes to about 25.5 MN·m/kg.

Rotation in Degrees, Radians and Turns

Rotation is measured in degrees, radians or full turns. One full rotation is 360°, or 2π radians, and the formula for a rotation angle in radians is arc length divided by radius. The Rotational Stiffness Calculator needs radians, and converts the other units itself.

A 180° rotation is half a turn. In coordinate geometry the rule for it is that the point (x, y) moves to (−x, −y). A 90° counterclockwise rotation moves (x, y) to (−y, x).

For angles from arc length, speed or time, the Angular Displacement Calculator does the conversion. More tools for turning bodies are on the Rotational Motion Calculators hub.