# Torsional Stiffness Calculator

Created by Luis Hoyos
Reviewed by Steven Wooding
Based on research by
Budynas RG, Nisbett JK Shigley's mechanical engineering design, 11th edition McGraw-Hill Education (2020)
Last updated: Dec 21, 2022

With this tool, you'll be able to calculate the torsional stiffness of beams subjected to torsional loads. You can use the torsional stiffness formula that uses experimental values (k = T/ϕ) or the one that uses the geometrical and material properties (k = GJ/L).

The torsional stiffness equation that uses geometrical properties only applies to straight beams. On the other hand, with the k = T/ϕ equation, we can even obtain the spring rate (stiffness) of torsion springs, as the units and definitions are the same.

## What is torsional stiffness?

Torsional stiffness is the resistance to angular deformation in a member subjected to a torsional load. It is the analog to the spring constant we saw in our Hooke's law calculator, but in this case, for torque-subjected systems.

## The formulas for torsional stiffness calculation

From the previous definition, the torsional stiffness equation is established as the ratio of the torque applied to the angular deformation:

k = T/ϕ

The previous equation is the general definition of stiffness applied to torsional loads. We can use it for any system that experiences these types of loads, such as beams and torsion springs. For beams, ϕ is known as the angle of twist, while for torsional springs, ϕ is known as the angular deflection of the coil body.

Inserting the angle of twist formula for beams (ϕ = TL/JG) in the equation above, we can obtain a more useful torsional stiffness formula for beams:

k = GJ/L,

where:

• k — Beam torsional stiffness, in newton-meters per radian (N m/rad);
• G — Shear modulus, in pascals (Pa);
• L — Beam length, in meters (m); and
• J — Polar moment of inertia, in m⁴.

🙋 Polar moment and shear modulus:

• To know what is the polar moment of inertia and how to calculate it, look at the FAQ section below (or use our polar moment of inertia calculator). For non-circular beams, the torsional constant is used instead of the polar moment; you can also obtain it with our torsional constant calculator (link under calculator).
• The shear modulus is a material property that indicates its resistance to shear stress (the stress caused by torsion). To learn more about it, visit our shear modulus calculator.

As you may have noted, with the first formula, we use experimental or measured data (T and ϕ) to obtain k. In contrast, the second equation (k = GJ/L) simply requires information about the geometry and material of the beam (usually easier to get).

## Torsional stiffness units

With our calculator, you can input data in your desired units and get the correct result. In contrast, dealing with the torsional stiffness equation above requires caring about the units used in the input variables.

G

Pa

lbf/ft²

psi

L

m

ft

in

J

m⁴

ft⁴

in⁴

## FAQ

### How do I calculate the torsional stiffness?

There are two ways to calculate the torsional stiffness (k) of a beam:

1. Using the torque applied (T), the angle of twist (ϕ), and the formula k = T/ϕ; or

2. Using the shear modulus (G), the polar moment of inertia (J), the beam length (L), and the formula: k = GJ/L.

For example, for a beam that experiences a torque of 80 000 N·m that causes a 0.02037 rad twist angle, k = (80 kN·m)/(0.02037 rad) = 3 927 344 N m/rad

### What are the torsional stiffness units?

We can express the torsional stiffness in units of newton-meters per radian (N·m/rad) in SI units.

### What is the polar moment of inertia?

The polar moment of inertia is a geometrical property that indicates the torsional stiffness of circular beams. To calculate it, use the following formula:

J = (π/32)(D⁴ - d⁴),

where:

• D — Outer diameter, in meters (m);
• d — Inner diameter, in m; and
• J — Polar moment of inertia, in m⁴.

If the beam is solid or non-hollow, just let d = 0 and use the same formula.

### How do I calculate the stiffness of a torsional spring?

To calculate the stiffness of a torsional spring (the spring rate), use the following equation:

k = dE/(64DNₐ),

where:

• k — Torsional spring stiffness, in N·m/rad;
• d — Wire diameter, in m;
• E — Young's modulus, in Pa;
• D — Mean coil diameter, in m; and
• N — Equivalent number of active turns (dimensionless).

We obtain N with the number of body coils (Nb) and the lengths (l and l, in m) of the ends where torque is applied:

Nₐ = Nb + (l₁ + l₂)/(3πD)

Luis Hoyos
You can calculate the torsional stiffness in two ways:
Using the k = T/ϕ formula
Internal torque (T)
lbf·in
Angle of twist (ϕ)
or the k = GJ/L formula (for beams)
Material (optional)
Custom ▾
Shear modulus (G)
psi
Polar moment of inertia (J)
in⁴
Beam length (L)
ft
Torsional stiffness (k)

If you don't know the value of J, our polar moment of inertia calculator can be useful for circular beams, and the torsional constant calculator if the beam is noncircular.

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