Torsional Stiffness

Torsional stiffness is the measure of how much radial torque force an object can withstand, or has the ability to experience without deforming. This is a central principal to both civil and mechanical engineering, and plays a key component when designing and testing structural parts or tools. Anything that is subjected to a rotational force will react in a way based on its torsional stiffness.

What is Torsional Stiffness?

In more concrete terms, torsional stiffness is defined as the amount of radially applied torque required to “twist” an object by one radian unit (radian in this case being the unit measure based on the radius of a circle).

Practically speaking, torsional stiffness is a material property that defines how rigid an object is when torque is applied. Is it easy to twist the object, does the object return to its original state once the force stops being applied? If so, then this object would be said to have low torsional stiffness.

Why is Torsional Stiffness Important?

An objects ability to resist torsion, or twisting, it highly important in many design instances. Let’s look at the example of a frame or chassis of a car or other automobile. In the case of a car frame, the metal supporting the structure of the car is subjected to a number of forces as a car zips along roads. A strong frame resists falling apart and the shearing off of connected beams from all this energy.

In the extreme and unfortunate case of an accident or rollover, high torsional stiffness in the chassis of the car will hold the frame together as the cars rolls and is subjected to shearing and twisting forces.

Another everyday example of torsional stiffness are the bridges our cars drive across. Any amount of bridge extension hanging off the beam supporting the underside of the bridge will be inducing a “torsional moment” to the beam or pillar below. This torsion will have a strong twisting force acting on the support pillars from just the weight of the bridges and the cars traveling on it, so it is important to use stiff materials and appropriate designs to prevent collapse!

Often times the requirements need to be even more stiff for the case of a seismic event, preparing for the worst possible scenario.

How is Torsional Stiffness Measured?

Remember that torsional stiffness is a resistance to a shear stress supplied by torque, measured in torque per radian. But torsional stiffness is also a material property, meaning it is highly dependent on the material makeup of the object being twisted. It is also highly dependent on the shape of the object that is being twisted, such as the amount of curvature or whether or not the object is hollow.

Here are a few common examples of how to calculate the torsional stiffness of typical objects both in textbooks and the real world.

Torsional Stiffness Equation

torsional stiffness equation

where:

  • τ = torsional stiffness (N-m/radian)
  • T = torque applied (N-m)
  • θ = angular twist (radians)
  • G = modulus of rigidity (Pa)
  • J = polar moment of inertia (m4)
  • L = length of shaft (m)

So there are two ways to calculate torsional stiffness, to be used depending on which variables are known. Additionally, if you need to calculate stiffness over radius and know the angle of twist in radians for example, the equation can be rearranged to bring variables such as θ over to the other side of what you are solving for.

The modulus of rigidity, also known as the shear modulus, is essentially the elasticity coefficient for a shear force, and is defined as the ration of shear stress per unit length. This is a material property that has been likely already experimentally determined and can be looked up when dealing with a certain type of material.

J, the polar moment of inertia, is based on a material’s shape. In many classic textbook example problems, torque is applied to a shaft or rod. The polar moment of inertia and its equation depends on whether or not this shaft is hollow.

As we will see in a second, a hollow shaft will have a much different torsional stiffness than a solid one. Lastly, these terms in the equation are over an object’s length L because torsional stiffness can also be defined as torsional rigidity per unit length.

Torsional Stiffness of a Solid Shaft

For a solid circular shaft the polar moment of inertia J is:

polar moment of inertia for a solid circular shaft

where d is the diameter of the shaft.

When substituted back into the main equation for J we have

torsional stiffness of solid shaft derivation

Finally, we can simplify and solve for torsional stiffness, leaving us with this final equation for a solid circular shaft:

torsional stiffness of a solid shaft equation

Torsional Stiffness of a Hollow Circular Shaft or Tube

The polar moment of inertia of a hollow circular shaft is similar to that of a solid shaft, but has to take into account both the inner and outer diameter of the solid object. It is as follows:

polar moment of inertia of hollow circular tube

If we substitute this definition of J back into the original equation we get:

torsional stiffness of hollow circular tube derivation

Finally we can simplify the equation and are left with the following equation to find the torsional stiffness of a hollow circular shaft:

torsional stiffness of hollow circular tube equation

While at first glance thought these equations seem quite similar, the fact is that because the diameter terms are raised to a power of four, subtracting the inner distance from the outer will have a profound effect on final torsional stiffness! This will leave the sufficient stiffness much lower than if the shaft under question was solid.

What is a Good Torsional Stiffness?

What a “good torsional stiffness is is highly subjective depending on the situation. For something like a personal car chassis or a bridge, you’d obviously want a relatively high torsional stiffness to prevent against sheers and breaks under normal and even extreme forces that might occur.

However, when engineers design a race car, they are searching for the optimal package of a low weight for a fast car, while still keeping things safe enough. Because higher torsional stiffness often means heavier, those designing race care chassis are not looking for the stiffest material, but one that is light and just stiff enough.

Keeping with the example of cars for torsional stiffness, there is a rough range that most car makers keep within when building their cars’ chassis. Generally speaking, car manufacturers have an overall total chassis torsional stiffness somewhere above 15-20,000 N*m/degree.

This is good for the consumer, with higher stiffness leading to less rattling of the car on the road, which is why many sports and luxury cars have even higher torsional stiffness of chassis. For convertibles, this figure is closer to 10,000 N*m/degree radians, as the open top of the car is less structurally sound.

How Do You Increase Torsional Stiffness?

There are a number of ways to increase the torsional stiffness of a design. Just looking at the equations above, increasing the diameter of a given shaft or object should increase the stiffness of any material. By increasing diameter (or better yet using a solid instead of hollow shaft), the polar moment of inertia has increased, thereby increasing stiffness.

In the example of an I-beam (commonly used in structures and bridges), torsional stiffness can be gained by adding a brace to the “open sections” (the overhangs that make it an I-beam). Finally, by choosing a material with a high intrinsic modulus of rigidity, high torsional stiffness can be baked into a design.

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