Torsional Stiffness Calculator

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.

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.

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.

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).

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.

To obtain the torsional stiffness in newton-meters per radian (N·m/rad), pound-feet per radian (lbf·ft/rad), and pound-inches per radian (lbf·in/rad), use the following units for G, L, and J.

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

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

For imperial or US units, we can use pound-feet per radian (lbf·ft/rad) or pound-inches per radian (lbf·in/rad).

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.

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

k = d⁴E/(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
• Nₐ — Equivalent number of active turns (dimensionless).

We obtain Nₐ with the number of body coils (N_b) and the lengths (l₁ and l₂, in m) of the ends where torque is applied:

Nₐ = N_b + (l₁ + l₂)/(3πD)