Flexural Rigidity: Definition, Formula, Derivation, Calculation

Hello, friends today we are going to discuss flexural rigidity in which you will learn what is flexural rigidity, formula, and derivation with their unit, various affecting factor, calculation, and many more.

Flexural rigidity plays the most important role while designing any type of structure or beam as flexural rigidity indicates the deflection of the beam which we will know further in detail.

So without wasting time let's get started.

What is Flexural Rigidity?

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

Flexural rigidity is resistance to twisting. If any structure like a rod, the beam has higher flexural rigidity which means it's not going to twist easily.

Formula of Flexural Rigidity

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 product of modulus of elasticity and moment of inertia of a beam or structure".

Unit of Flexural Rigidity

As we know in the SI system,

Unit of modulus of elasticity E = N/m^2

Unit of the moment of inertia = m^4

So,

(N/m^2) × m^4

After solving the final result,

N- m^2

So the unit of Flexural Rigidity will be N- m^2.

Derivation of Flexural Rigidity of Beam

Let us consider a beam deflect from x - y to x'- y' at a distance of x during applied external force or self-load.

Show in the figure,

flexural rigidity


From the above figure clearly see that,

x - y = R × θ

Similarly,

x'- y' = (R + x)θ

Before bending

x - y  = x'- y'

So,

Elongation of x'- y' after bending = Elongation of x'- y' before bending (x'- y' = x - y )

So,

(R + x)θ = R × θ

So,

Elongation,

x × θ =  (R + x)θ - R × θ

As we know,

Strain = Change in Dimension/Original Dimension 

So,

Strain = (x × θ)/(R × θ) = x/R

As we know,

Stress = modulus of elasticity × strain

So,

Stress = E × (x/R)

Now consider a cross-section with a small area of da from x - y to x'- y'.

Now, external force df is applied on a small area da.

Since we know,

Force = Stress × Area

So,

df =  E × (x/R) × da

Now we know,

Moment of force = Force × Distance

So,

M = df × x = E × (x^2/R) × da

So,

Total moment of force(bending moment)

M = Σ {E × (x^2/R) × da} 

     = E/R × Σ(x^2 × da) 

     = E/R × a × K^2

Where,

a = Total Area

K = radius of gyration

As we know,

a × K^2 = I (moment of inertia of the beam)

So,

M = E × I/R 

EI = Flexural rigidity

If R = 1

Then,

M = E × I

Hence, Flexural rigidity can also be defined as " The bending moment required to produce a unit radius of curvature in the beam is known as flexural rigidity".

Factor Affecting Flexural Rigidity

The following factors affect flexural rigidity:
  1. Span of Beam
  2. Area moment of inertia
  3. Modulus of Elasticity
  4. Types of Load
  5. Location of load

Span of Beam

If the length of the beam increases then the flexural rigidity of the beam will decrease.

Area moment of inertia

If the area moment of inertia increases then the flexural rigidity of the beam will be increased.

Modulus of Elasticity

If the modulus of elasticity increases then the flexural rigidity of the beam will be increased.

Types of Load

Flexural rigidity also depended on the types of load such as udl load, point load, etc.

Location of load

If the location of the load is away from the center of gravity, the flexural rigidity decreases.

Calculation of Flexural Rigidity

Question

Find the flexural rigidity of the beam if the beam is made of steel with the modulus of elasticity of 30 MPa and the moment of inertia of 3 x 10-3 m^4?

Solution 

Given Data,

E = 30 MPa = 30 × 10^6 N/m^2

I = 3 x 10-3 m4

As we know,

Flexural Rigidity = E × I

So,

Flexural Rigidity = 30 × 10^6 × 3 x 10-3

= 30 × 10^3 × 3
= 90 × 10^3

So here you have to know all aspects related to the flexural rigidity if you have any doubt then you are free to ask me by mail or on the contact us page.

Thank You.

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