## What is Flexural Modulus?

In mechanics, the flexural modulus or bending modulus is an intensive property that is computed as the ratio of stress to strain in flexural deformation, or the tendency for a material to resist bending. It is determined from the slope of a stress-strain curve produced by a flexural test (such as the ASTM D790), and uses units of force per area. The flexural modulus defined using the 3-point bend test assumes a linear stress-strain response.

Flexural modulus is important in understanding the rigidity or stiffness of a material. Certain material applications may need strength and rigidity for structural support, while other applications may require flexibility in order to prevent damage during bending. As such, there are a few factors that modulate the flexural modulus of a material, and understanding these factors aids in material development and choice.

The base flexural modulus of the base material is determined by the fundamental properties of the material. For example, in plastics, the type of polymer, molecular weight, thickness, and shape all play a role in flexibility.

One way to modulate the flexibility is to add a fine mineral filler, such as talc powder to the plastic. Typically, adding these fine mineral fillers will increase the flexural modulus and stiffen the material. The ability of a mineral filler to modulate the flexural modulus is also dependent on the aspect ratio and particle size of the filler.

Higher aspect ratio particles increase the flexural modulus of material more so than lower aspect ratio particles. Decreasing the particle size of a mineral additive may increase flexural modulus if the particle size aspect ratio increases during the particle size reduction.

It is important to note that flexural modulus is a useful measure for materials that do not break or rupture upon the applied stress. Brittle materials that break upon a certain level of force are better evaluated by flexural strength.

## How Is the Flexural Modulus of a Material Determined?

Flexural modulus is tested from a 3-point analysis on a rectangular beam of the material, with width w and height h. A parameter L specifies the length between two support points placed on the bottom of the beam.

A force, F, is applied on a point on the opposite side and in-between the two supports, which creates a displacement in the material called the deflection, d. With these parameters, the flexural modulus, E_{bend}, is calculated as units of force per area as follows:

E_{bend} = (L^{3}F) / (4wh^{3}d)

Flexural modulus is an important calculation for engineers and architects as it relates to the amount of weight material can handle when used as structural support.

## The Relationship Between Flexural Modulus and Tensile Modulus

As bending occurs in the test sample, its top surface experiences compression forces while the opposite side undergoes tensile deformation, as such, flexural modulus measurements are best suited for isotropic materials, i.e., materials with uniform properties in all directions.

In ideal elastic conditions, the flexural and tensile modulus of material should be similar since they are both representations of mechanical strain. That is, they both express a material’s ability to resist deformation under loads, although the loads they are resisting are different.

From elastic beam theory, for a simply supported beam subjected to a concentrated load:

Deflection, d = L^{3}F/48EI

Transposing for E, we get

E = L^{3}F/48Id

For a rectangular section,

I = 1/12 wh^{3}

Substituting, I, into the equation for E, we get

E = L^{3}F/4wh^{3}d

Therefore, E = Ef

In reality, however, these two properties may differ if measurements occur under non-ideal, non-elastic conditions.