# Flexural Modulus

## Definition - What does Flexural Modulus mean?

The flexural modulus of a material is a mechanical property that measures a material’s stiffness or resistance to a bending action. It is typically measured when a force is applied perpendicular to the long edge of the sample.

The flexural modulus of a material may be calculated graphically by measuring the slope of the linear portion of a typical stress-strain curve. In other words, it is the change in stress divided by the corresponding change in strain.

Figure 1: Stress-strain graph showing the flexural modulus on the straight line segment of the curve (source)

Ideally, the flexural modulus of a material is equivalent to its Young’s modulus. In practical terms, the higher the flexural modulus of a material, the harder it is to bend. Conversely, the lower the flexural modulus is, the easier it is for the material to bend under an applied force.

Since the flexural modulus is expressed as the ratio of stress to strain, the standard unit of measurement for flexural modulus is the Pascal (Pa or N/m2). However, in materials, such as steel and concrete, this property is typically expressed as Megapascals or gigapascals (MPa or GPa). The equivalent US customary unit for flexural modulus is pounds per square inch (psi).

The flexural modulus is also known as bend modulus or bending modulus of elasticity.

## Trenchlesspedia explains Flexural Modulus

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

The flexural modulus of a specimen is determined by subjecting it to a 3-point flexural test. During this test, a specimen of a fixed length, L, rests on two supports while a concentrated force, F, is applied to its center. The second moment of area, I, of the specimen section is also calculated. The 3-point Flexural test is typically carried out in accordance with ASTM D790.

Figure 2: Illustration of the 3-point flexural test (source)

The concentrated force is applied to the specimen, and the resulting deflection is noted. Using the previously mentioned parameters, the flexural modulus, Ef is expressed as the formula below:

Ef = L3F/4wh3d

Where:

w = width of the test section

h = depth or thickness of the test section

Alternatively, flexural modulus may also be expressed as:

Ef = L3m/4wh3

Where:

m = the gradient (or slope) of the linear-elastic portion of the load-deflection curve.

### 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 a 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 = L3F/48EI

transposing for E, we get

E = L3F/48Id

For a rectangular section,

I = 1/12 wh3

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

E = L3F/4wh3d

Therefore, E = Ef

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