# Flexural rigidity

Flexural rigidity is defined as the force couple required to bend a non-rigid structure to a unit curvature or it can be defined as the resistance offered by a structure while undergoing bending.

## Flexural rigidity of a bar

In a beam or rod, flexural rigidity (defined as EI) varies along the length as a function of x shown in the following equation:

$\ EI {dy \over dx}\ = \int_{0}^{x} M(x) dx + C_1$

where $E$ is the Young's modulus (in Pa), $I$ is the second moment of area (in m4), $y$ is the transverse displacement of the beam at x, and $M(x)$ is the bending moment at x.

Flexural rigidity has SI units of Pa·m4 (which also equals N·m²).

## Flexural rigidity of a plate (e.g. the lithosphere)

Main article: Plate theory

The thin lithospheric plates which cover the surface of the Earth are also subject to flexure, when a load or force is applied to them. On a geological timescale, the lithosphere behaves elastically (in first approach) and can therefore bend under loading by mountain chains, volcanoes and so on.

The flexure of the plate depends on:

1. The plate thickness (usually referred to as mechanical thickness of the lithosphere).
2. The elastic properties of the plate
3. The applied load or force

As flexural rigidity of the plate is determined by the Young's modulus, Poisson's ratio and cube of the plate's elastic thickness, it is a governing factor in both (1) and (2).

Flexural Rigidity[1] $D = \dfrac{Eh_e^3}{12(1-\nu^2)}$

$E$ = Young's Modulus

$h_e$ = elastic thickness (~10–15 km)

$\nu$ = Poisson's Ratio

Flexural rigidity of a plate has units of Pa·m4, i.e. one dimension of length less from the one for the rod, as it refers to the moment per unit length per unit of curvature, and not the total moment. I is termed as moment of inertia.J is denoted as 2nd moment of inertia/polar moment of inertia.