Flexural Formula - Bending theory -Derivation

The flexural formula is given by the relation:
M/I = E/R = 𝛔/y 


Assumptions in Flexural Formula/Simple Bending Theory

The assumptions in simple bending theory are:
  1. The material of the beam is homogeneous and isotropic
  2. The transverse section of the beam remains plane before and after bending.
  3. The value of young's modulus is the same in tension and compression
  4.  The beam is initially straight and all the longitudinal filaments bend into circular arcs with a common center of curvature
  5. The radius of curvature is large compared with the dimensions of the cross-section.
  6. Every layer of the beam is free to expand or contract independent of the layer below it.

Derivation of Flexural Formula



Simple bending theory can be explained by following beam element consideration, as shown in figure-2 below.


As shown in the figure above, consider a beam small beam section ABCD with a length of dx. The N-N forms the neutral axis of the beam element. Section AB and CD are perpendicular to the neutral axis N-N.  The beam element under the action of bending gets deformed as shown in figure-2(b).

The layers of the beam before bending do not remain the same after bending. The layer AC and BD have deformed to A'C' and B'D' respectively. The layer gets shorten at top and expands at the bottom layer. The beam layer does not undergo any change at the neutral layer. This means,
 NN = dx = N'N'
The layers above the neutral axis are shortened, hence subjected to a compressive force. The layer below the neutral axis is elongated, thus subjected to tensile stress. Hence, the compressive stress is maximum at the top layer and the tensile stress is maximum at the bottom layer. 

As we move from the bottom layer to the neutral layer, the length of layers decreases. Hence, the increase or decrease of length of the layer is dependent on its distance from the neutral axis. This theory of bending is called the theory of simple bending.

Derivation of Bending Equation

As shown in figure-2(a), consider a layer EF from a distance 'y' from the neutral axis. After bending, EF gets deformed to E'F' as shown in 2(b). 

Given:
  1. The radius of the Neutral layer = R
  2. The angle subtended by A'B' and C'D' at O = θ
Then,

  • Strain Variation along Depth of Beam

Original length of layer = EF = dx;
Original length of Neutral layer = NN = N'N' = dx
From Figure,

N'N' = R x  θ = dx
E'F' = (R + y ) θ

Strain in the layer EF = Increase in the length of layer EF / (Original Length)
                                   = (E'F' -EF)/EF 
                                   = ( (R +y) θ - (Rx θ))/dx
                                   = (R θ + y θ - R θ)/ dx
                                   = y θ/dx
                                   = y θ/R θ
                                   =y/R
Hence,
the strain in the layer EF is directly proportional to the distance of the layer from the neutral axis. This relation shows the vairation of the strain along the depth of the beam. The variation of strain is linear.

  • Stress Variation along Depth of Beam

  Young's modulus E = stress in layer EF / Strain in layer EF =  𝛔 /(y/R)

=> E = 𝜎R/y

=> E/R = 𝛔/y
  • Moment of Resistance
The layer above the neutral axis experience compressive force Fc and the layers below the neutral axis experiences tensile force Ft. For equilibrium:
  1. Fc= Ft
  2. The moments caused due to these internal forces must be equal and opposite the BM caused at section f the beam.
Moment of resistance is defined as the algebraic sum of moments about the neutral axis of the internal forces developed in the beam.

If the cross-sectional area of the element is dA, then 

Total Moment of resistance M = ∫ (𝛔 dA . y) = ∫(Ey. y. dA)/R
=> M = (E/R)∫y^2 . dA

Moment of Inertia I = ∫y^2. dA

Hence, we get
 M = EI/R

Hence, we get

M/I = E/R = 𝛔/y ;

The above equation is called as the Bending Equation/ Flexura Formula.

Also Read: Simple Bending Theory-Bending Stress-Bending Equation-Derivation


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