Torsional rigidity is defined as how much an object of specified material resists twisting force, also known as torque. It is based on both the material of the object, as well as its shape.

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## What Is Torsional Rigidity?

Torsion, also known as twist or torque, occurs when a force is applied to rotate an object. Torsional rigidity is that object’s resistance to deformation brought on by torque forces. In a somewhat counter intuitive way, the rigidity can be measured by the amount of torque needed to deform the object. Torsional rigidity is said to be the amount of torque necessary to twist an object by one radian per unit length (of the object).

## How is Torsional Rigidity Measured?

Torsional rigidity is also thought of as the product of an object’s shear modulus and its polar moment of inertia. There are multiple ways to measure it, as it can also be derived from the force applied and resulting angle of twist. Let’s go through a few equations used to define torsional rigidity.

where:

- G = modulus of rigidity, or shear modulus (N/m
^{2}) - J = polar moment of inertia (m4)
*T*= applied torque (N-m)- L = length of object or beam (m)
- θ = angle of twist (radians)
- Torsional rigidity has units of (N-m
^{2})

As we can see, torsional rigidity can be calculated multiple ways, depending on which variables are known. Alternatively, the equation can be rearranged to find values for angle of twist (or angle per length), applied torque, or shear modulus:

Here we can clearly see how easy it us to rearrange if something else is needed to be solved for. It’s also quite clear what the relationship is between torsional rigidity, and some of the variables that affect it.

The higher the modulus of rigidity or the polar moment of inertia, the higher the torsional rigidity. These qualities have a direct relationship. Naturally, the inverse of that holds true as well. If either of these values are lowered, an object will have a lower torsional rigidity, and be more susceptible to physically twisting.

The other equation to calculate torsional rigidity is based around the physical measurements due to applied torque. As the angle of twist due to torque increases, that means less torsional rigidity of the structure. Alternatively, if length and angle of twist remain constant and if applied torque increases, then torsional rigidity will increase as well. This is because angular deflection has not changed, which means more resistance to deflection (though in a real-world example this would likely mean that the material has changed, if all else remained constant).

Finally, the equation that solves for theta, the angle of twist or angle of deflection, can be thought of as the equation that solves for the torsional flexibility of a beam under a particular torque. Torsional flexibility is the inverse to torsional flexibility and therefore their values are inversely proportional.

## Torsional Rigidity vs Lateral Rigidity

Torsional rigidity is not the same thing as lateral rigidity, though both measure the deflection of a beam (or other structure) from its original position under different forces. While torsional rigidity measures deflection from twisting force or torque, lateral rigidity is the resistance against a pushing or bending force along the lateral axis. Note that it must be along the lateral axis, not longitudinal, because that is a compressing force.

Understanding how both of these forces will affect a beam of certain rigidity is important to the design process. However, the rigidity measures are not the same, and should not be treated as the same when choosing material or design or solving problems of beam deflection.

## Why Is Torsional Rigidity Important?

Torsional rigidity is an important consideration in many design engineering spaces. Having more torsional rigidity obviously means more resistance to torque forces, holding up stronger against unexpected forces. One area where this matters a lot is when designing the chassis of a car.

Sometimes car manufacturers will claim that they have increased a car’s torsional rigidity by 25%, or maybe even doubled it. Generally, this is done for drivability purposes.

The importance of torsional rigidity here, in the case of normal driving on bumpy roads and the like, is that the chassis remains more stable, and therefore acts as a more stable base for the suspension of the car. This in turn makes the car more predictable for the driver, easier for them to react to unexpected situations, and ultimately safer to control. Also, from a comfort standpoint this is a huge benefit: a more rigid chassis will have less rattling and vibrating, making for a smoother and quieter ride.

## How Do You Increase Torsional Rigidity?

There are several ways one could increase torsional rigidity in a design. Round shafts will always have a higher torsional rigidity, due to having a higher polar moment of inertia.

Another way to increase torsional rigidity is to increase the diameter of a shaft. A shaft that is twice the diameter of another will have just 1/16th the angle of twist if both shafts receive the same amount of torque. Again, this is due to larger shafts having a higher polar moment of inertia.

Changing the material will change the shear modulus, and this can have a huge impact on torsional rigidity. However, there is a lot more than can go into material considerations. Often changing to a material for one benefit (like torsional rigidity) will lead to a drawback of another variety. Different materials will deform under stress in different ways, so there are safety concerns to consider when making sure deformation does not occur in some catastrophic or dangerous way. Engineers today will use CAD programs to study simulations of stresses on designs of varying material and shape.

If you know you want to decrease the angle of twist of a constant force, one way to do so is to decrease the length of a shaft. Finally, a way to increase torsional stiffness of a hollow shaft is to add in diagonal “ribs” crossing each other like trusses of a bridge. This is a structural addition that increases the shear modulus without changing material.

## Drawbacks of High Torsional Rigidity?

Typically, increasing torsional rigidity, whether through increased shaft thickness, or stronger material, will result in a heavier weight. One of the challenges for engineers when designing car chassis, is how to obtain a high torsional rigidity without sacrificing its light weight. With increasingly stringent regulations for fuel consumption and efficiency, it is not only important but necessary for car manufacturers to find materials and chassis designs that shave weight without compromising the torsional rigidity, and therefore safety, of the car.

Other, much less frequent downsides to having a high torsional rigidity in specific design is if a material structure is meant to return to its original shape rather than deform after torsional forces are applied. In the case of say, twisting a towel to wring out water, it wouldn’t be so great if that towel snapped if you twisted it hard enough, right? This is a rather silly example, but there are more practical reasons you’d occasionally want to have a low torsional rigidity in a design.