A fundamental property in any structure is its ability to resist deflection or deformation. We call this rigidity (or stiffness) \(k\).

## Rigidity Types

Below are the types of rigidity present in any structure which would depend on the type of internal force:

### Flexural

This type measures a structure's resistance to flexural (or bending) loads. It is the product of Young's Modulus \(E\) and the moment of inertia \(I\) of a cross-section.

\(k_f=EI\)

### Axial

This type measures a structure's resistance to axial loads. It is the product of Young's Modulus \(E\) and the area \(A\) of a cross-section.

\(k_a=EA\)

### Torsional

This type measures a structure's resistance to torsion. It is the product of Shear Modulus \(G\) and the polar moment of inertia \(J\) of a cross-section.

\(k_t=JG\)

## Implications

From these different types, we can observe that rigidity is a product of two variables: (1) the inertial properties of the cross-section and (2) the mechanical property of a material.

It implies several things when it comes to the structure's deflection or deformation:

- Rigidity will depend on the cross-section. For example, if we have a tapered beam, the rigidity on its bigger end is more significant than its other end. It means that the smaller end is more prominent to deform or deflect.
- If we want our structure to deform/deflect less, for example, there are two things we can do. First, we may increase its mechanical properties \(E\) or \(G\). It implies that we must find a better material to carry the loads. Second, we may have a larger cross-section. What this does is increase its inertial properties. A bigger \(I\) or \(J\) means the structure will deform less.

## Summary

Let's summarize:

A fundamental property in any structure is its ability to resist deflection or deformation. We call this rigidity (or stiffness) \(k\).

There are several types of rigidity: flexural, axial, and torsional. Each rigidity respectively measures resistance with bending, axial, and torsion loads.

Rigidity is a product of two variables: (1) the inertial properties of the cross-section and (2) the mechanical property of a material.