Bending Stress: Definition, Application, Formula, Derivation

In this article, you will learn what is bending stress? As well as you will know its application, units, assumption theory for bending, and the last also derive the bending equation.

So, let's get started to know step by step all things related to bending stress.

Pure Bending or Simple Bending

Simple bending or pure bending A beam or a part of it is said to be in a state of pure bending when it is bent under the action of a uniform or constant bending moment without any shear force.  

Alternatively, a portion of the beam is said to be in a state of simple bending or pure bending, when the shear force on that portion is zero.  In that case, there is no possibility of shear stress in the beam. 

What is Bending Stress?

When a beam is loaded with external loads all the sections will experience a bending moment. 

The bending moment at a section tends to bend or deflect the beam and the internal stresses resist its bending. 

The resistance, offered by the internal stresses to the bending, is called bending stress.

So,

Bending stresses are the internal resistance to external force which causes bending of a member.

It is denoted by σ.

Its unit will be N / mm².

Some practical applications of bending stresses are as follows:

1. Moment carrying capacity of a section.
2. Evaluation of excessive normal stress due to bending.
3. Design of beam for bending.
4. Evaluation of the load-carrying capacity of the beam.

Assumptions for Theory of Simple Bending 

1. The material of the beam is perfectly homogeneous and isotropic.
2. The beam material is stressed within its elastic limit and thus, obeys Hooke's law.
3. The transverse sections, which were plane before bending, remain plane after bending also. Each layer of the beam is free to expand or contract, independently, of the layer above or below it. 
4. The value of E ( Young's modulus of elasticity ) is the same in tension and compression. 
5. The beam is in equilibrium i.e., there is no resultant pull or push in the beam section.

Neutral Axis 

In the bending process, the inner surfaces of the beam contract, and the outer surfaces expand.  

Therefore, there must be a surface somewhere in the middle of the beam, whose length does not change, this surface is called the "neutral layer". There is no stress on this surface.

So,

The neutral axis is the axis through a beam where the stress is zero, that is there is neither compression nor tension.

Bending Equation

M/I = σ/y = E/R

Where, 

M = Bending Moment (N - mm)

I = Moment of Inertia mm⁴

σ =   Bending Stress N / mm² 

y = ( D / 2 ) Distance From Neutral Axis (mm) 

E = Modulus of Elasticity (N /mm²)

R = Radius of Curvature (mm)

Derivation of Bending Equation

Consider an elemental length AB of the beam. Let EF be the neutral layer and CD the bottom-most layer. If GH is a layer at a distance y from neutral layer EF. Show in Figure.

Let After bending A, B, C, D, E, F, G, and H takes positions A', B', C', D', E', F', G', and H ' respectively.

Let,

R = radius of curvature of the neutral 

θ = angle subtended by the beam length at O.      

σ = longitudinal stress

 EF = E'F '  since, EF is a neutral axis

As we know,

Angle = Arc/Radius 

So, in ∠OE'F'

θ = E'F '/R

E'F ' = Rθ 

Clearly shown in the figure,

EF = GH = E'F' = Rθ 

Again in ∠OG'H'

θ = G'H'/(R+y)

G'H' = (R+y)θ

Now, Strain in GH 

 = (Final length - Initial Length)/Initial Length

 = (R+y)θ - Rθ/Rθ

 = Rθ + yθ - Rθ/Rθ

 = y/R

As we know strain in GH is due to tensile force according to hook's law,

Strain = σ/E

So, Strain in layer GH = σ/E

y/R  = σ/E

σ/y = E/R

if E and R Constant

 then, σ ∝ y

Thus stress is proportional to the distance from the neutral axis.

Now,

Consider an elemental area Sa at a distance y from the neutral axis. 

So, from the above equation,

σ = E/R × y × δa

Now, force on this element,

= σ × δa

= E/R × y × δa

The moment of resistance of this elemental force about the neutral axis,

(E/R × y × δa) × y

E/R × y ² × δa

The total moment resisted by section M' is given by,

M' =  ∫(E/R × y ² × δa)

M' = E/R ∫( y ² × δa)

Where,

I = ∫( y ² × δa)

moment of inertia or second moment of area of the section.

So,

M' = (E/R) × I

From equilibrium conditions,

M' = M (applied moment)

So,

M = (E/R) × I

M/I = E/R

So,

M/I = σ/y = E/R

This is known as the bending equation.

Related Article:

Principal Stress

Modulus of Resilience

Section Modulus

Theories of Failure

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

Thank You.