Flexural rigidity

Encyclopedia

**Flexural rigidity**is defined as the force

Force

In physics, a force is any influence that causes an object to undergo a change in speed, a change in direction, or a change in shape. In other words, a force is that which can cause an object with mass to change its velocity , i.e., to accelerate, or which can cause a flexible object to deform...

couple

Couple (mechanics)

In mechanics, a couple is a system of forces with a resultant moment but no resultant force. Another term for a couple is a pure moment. Its effect is to create rotation without translation, or more generally without any acceleration of the centre of mass.The resultant moment of a couple is called...

required to bend a non-rigid

Rigid

In mathematics, a rigid collection C of mathematical objects is one in which every c ∈ C is uniquely determined by less information about c than one would expect....

structure to a unit curvature

Curvature

In mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry. Intuitively, curvature is the amount by which a geometric object deviates from being flat, or straight in the case of a line, but this is defined in different ways depending on the context...

.

## Flexural rigidity of a BAR

In a beamBeam (structure)

A beam is a horizontal structural element that is capable of withstanding load primarily by resisting bending. The bending force induced into the material of the beam as a result of the external loads, own weight, span and external reactions to these loads is called a bending moment.- Overview...

or rod

Cylinder (geometry)

A cylinder is one of the most basic curvilinear geometric shapes, the surface formed by the points at a fixed distance from a given line segment, the axis of the cylinder. The solid enclosed by this surface and by two planes perpendicular to the axis is also called a cylinder...

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

where is the Young's modulus

Young's modulus

Young's modulus is a measure of the stiffness of an elastic material and is a quantity used to characterize materials. It is defined as the ratio of the uniaxial stress over the uniaxial strain in the range of stress in which Hooke's Law holds. In solid mechanics, the slope of the stress-strain...

(in Pa), is the 2nd moment of inertia (in m

^{4}), is the transverse displacement of the beam at

**x**, and is the bending moment

Moment (physics)

In physics, the term moment can refer to many different concepts:*Moment of force is the tendency of a force to twist or rotate an object; see the article torque for details. This is an important, basic concept in engineering and physics. A moment is valued mathematically as the product of the...

at

*x*.

Flexural rigidity has SI units of Pa

Pascal (unit)

The pascal is the SI derived unit of pressure, internal pressure, stress, Young's modulus and tensile strength, named after the French mathematician, physicist, inventor, writer, and philosopher Blaise Pascal. It is a measure of force per unit area, defined as one newton per square metre...

·m

^{4}(which also equals N·m²).

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

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:

- The plate thickness (usually referred to as elastic thickness of the lithosphere).
- The elastic properties of the plate
- The applied load or force

As flexural rigidity of the plate is determined by the Young's modulus, Poisson's ratio

Poisson's ratio

Poisson's ratio , named after Siméon Poisson, is the ratio, when a sample object is stretched, of the contraction or transverse strain , to the extension or axial strain ....

and cube of the plates elastic thickness, it is a governing factor in both (1) and (2).

Flexural Rigidity

= Young's Modulus

= elastic thickness (~10–15 km)

= Poisson's Ratio

Flexural rigidity of a plate has units of Pa

Pascal (unit)

The pascal is the SI derived unit of pressure, internal pressure, stress, Young's modulus and tensile strength, named after the French mathematician, physicist, inventor, writer, and philosopher Blaise Pascal. It is a measure of force per unit area, defined as one newton per square metre...

·m

^{3}, 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.