Flexural Rigidity: Definition, Formula, Derivation, Calculation, Application

what is flexural rigidity ?

Flexural rigidity, also known as bending stiffness, is a mechanical property used to describe the resistance of a material or structure to bending. When a force is applied perpendicular to the longitudinal axis of a beam or plate-like structure, it causes it to bend. Flexural rigidity measures the material’s ability to resist this bending deformation.

The Flexural Rigidity is The product of

1. The Product of elasticity and Mass moment of Inertia

2. Modulus of rigidity and area moment of Inertia

3. Modulus of rigidity and mass Moment of inertia

4. Modulus of Elasticity and Area Moment of Inertia

Answer is 4. Modulus of Elasticity and Area Moment of Inertia

Mathematically, flexural rigidity (D) is given by the formula:

D = E * I

Where:
E is the elastic modulus of the material (a measure of its stiffness).
I is the moment of inertia of the cross-sectional area of the structure about the bending axis.

Must Read : Mechanical properties of materials

flexural rigidity formula

The formula for calculating flexural rigidity (D) of a beam or plate-like structure is:

D = E * I

Where:
E is the elastic modulus (Young’s modulus) of the material, which represents the material’s stiffness.
I is the moment of inertia of the cross-sectional area of the structure about the bending axis.

As  we know from the bending equation, 

M/I = σ/y = E/R

Where,

R = Radius of curvature of the beam(mm)

M = Bending moment (N-m)

E = Young’s modulus (N/m^2)

I = Area moment of inertia (m^4)

σ = Bending Stress (N/m^2)

Y = Distance of Beam Subjected to bending from neutral axis(m)

So,

M/I = E/R

MR = EI

EI = Flexural Rigidity

So,

Mathematically it can be defined as ” Flexural Rigidity is the

This formula relates the material properties (E) and the geometric properties (I) of the structure to its flexural rigidity. The higher the value of D, the stiffer the structure will be, indicating its ability to resist bending deformation when subjected to external loads.

flexural rigidity units

The units of flexural rigidity (D) depend on the units used for the elastic modulus (E) and the moment of inertia (I) in the formula.

As a result, the units of flexural rigidity (D) are:

It is essential to ensure consistency in units when using the flexural rigidity formula to obtain accurate and meaningful results.

flexural rigidity of a beam

To calculate the flexural rigidity (D) of a beam, you need two key pieces of information:

Once you have the values of E and I, you can calculate the flexural rigidity (D) using the formula:

D = E * I

For example, if the elastic modulus of the material is given as 200 GPa (200,000 MPa) and the moment of inertia of the beam’s cross-section is 0.003 m^4 (3,000 mm^4), the flexural rigidity would be:

D = 200,000 MPa * 0.003 m^4 = 600,000 N·m^2 (or 600,000 N·mm^2, since 1 m^4 = 1,000,000 mm^4)

The value of flexural rigidity provides an indication of the beam’s ability to resist bending under external loads. A higher flexural rigidity means the beam is stiffer and will deflect less when subjected to bending forces.

Derivation of Flexural Rigidity of Beam

Derivation of Flexural Rigidity of a Beam:

Let’s consider a beam that deflects from its original position (x, y) to a new position (x’, y’) under the influence of an applied external force or self-load. The beam has a length R, and a small angular deflection θ occurs due to the bending.

In the figure provided:

Before bending, the elongation of (x’, y’) is equal to the elongation of (x, y), so:

x – y = x’ – y’

The elongation of (x’, y’) after bending can be expressed as:

Elongation of (x’, y’) = Elongation of (x, y) = x – y

Considering the deflection angle θ, we can relate the elongation to the bending radius R as follows:

Elongation = R × θ

So, (x – y) = R × θ

Similarly, for the deflected position (x’, y’):

Elongation of (x’, y’) = (R + x)θ

Now, equating the elongation of (x’, y’) before and after bending:

(R + x)θ = R × θ

Simplifying, we get:

x × θ = (R + x)θ – R × θ

Now, strain is defined as the change in dimension divided by the original dimension:

Strain = (x × θ) / (R × θ) = x / R

Using the relationship between stress and strain:

Stress = Modulus of Elasticity × Strain

We can express the stress as:

Stress = E × (x / R)

Now, let’s consider a small area da on the cross-section of the beam from (x, y) to (x’, y’). An external force df is applied on this small area. The force is given by:

df = E × (x / R) × da

The moment of this force about the deflection point (x, y) is given by:

M = df × x = E × (x^2 / R) × da

The total bending moment (bending moment) can be obtained by summing up the moments of all such small forces over the entire cross-section:

M = Σ {E × (x^2 / R) × da}

M = E/R × Σ(x^2 × da)

M = E/R × a × K^2

Where:
a = Total cross-sectional area
K = Radius of gyration

Since a × K^2 is equal to the moment of inertia of the beam (I), we have:

M = E × I / R

The term EI is defined as the Flexural Rigidity of the beam.

If R = 1 (unit radius of curvature), then:

M = E × I

Hence, Flexural Rigidity can be defined as “The bending moment required to produce a unit radius of curvature in the beam.”

Factor Affecting Flexural Rigidity

Flexural rigidity is influenced by several factors that play significant roles in determining the beam’s resistance to bending. Let’s discuss each of these factors in detail:

Overall, these factors interact in determining the flexural rigidity of a beam. Engineers consider these factors during the design phase to select appropriate materials, beam dimensions, and support configurations to meet the desired performance criteria and ensure the structural integrity of the beam under various loading conditions.

Significances of Flexural rigidity:

Flexural rigidity is a fundamental mechanical property that describes the stiffness or resistance to bending of a material or structure. It plays a crucial role in various engineering applications and has significant importance in different fields. Here are some key points highlighting the significance of flexural rigidity:

In summary, It is a significant mechanical property that impacts the structural integrity, stability, and performance of various materials and structures. Engineers and designers consider it a fundamental parameter in their work to ensure safe and efficient designs.

applications of flexural rigidity

Flexural rigidity finds applications in a wide range of fields and engineering disciplines. Here are some specific applications of flexural rigidity:

In summary, the application of flexural rigidity extends across various industries and disciplines, playing a key role in designing and analyzing structures, materials, and products to meet specific performance and safety requirements.

advantages of flexural rigidity

Flexural rigidity offers several advantages in different engineering applications due to its influence on the behavior of materials and structures. Some of the key advantages include:

In summary, the advantages of flexural rigidity lie in its ability to enhance structural integrity, stability, load-bearing capacity, and overall performance of materials and structures. It plays a vital role in achieving optimal design and efficient functionality in various engineering applications.

disadvantages of flexural rigidity

While flexural rigidity offers numerous advantages, there are also some potential disadvantages associated with this property, depending on the specific context and application:

It is important to note that the disadvantages associated with flexural rigidity are context-dependent, and there is no one-size-fits-all approach to engineering design. Engineers and designers must carefully consider the specific requirements of each application and strike a balance between the benefits and limitations of flexural rigidity when selecting materials and designing structures.

Reference : https://en.wikipedia.org/wiki/Flexural_rigidity

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