Saturday, 22 April 2017

FORMULA FOR BENDING STRESS IN A BEAM

FORMULA FOR BENDING STRESS IN A BEAM

We were discussing basic concept of bending stress in our previous session. We have also discussed assumptions made in the theory of simple bending and formula for bending stress or flexural formula for beams during our last session.

Now we are going ahead to start new topic i.e. expression for bending stress in pure bending of beam in the strength of material with the help of this post.

Let us go ahead step by step for easy understanding, however if there is any issue we can discuss it in comment box which is provided below this post.

Let us consider one structural member such as beam with rectangular cross section, we can select any type of cross section for beam but we have considered here that following beam has rectangular cross section. 

Bending stress

 
Let us assume that following beam PQ is horizontal and supported at its two extreme ends i.e. at end P and at end Q, therefore we can say that we have considered here the condition of simply supported beam.
Once load W will be applied over the simply supported horizontal beam PQ as displayed above, beam PQ will be bending in the form of a curve and we have tried to show the condition of bending of beam PQ due to load W in the above figure.

Now let us consider one small portion of the beam PQ, which is subjected to a simple bending, as displayed here in following figure. Let us consider two sections AB and CD as shown in following figure.
Now we have following information from the above figure.

AB and CD: Two vertical sections in a portion of the considered beam
N.A: Neutral axis which is displayed in above figure
EF: Layer at neutral axis
dx = Length of the beam between sections AB and CD

Let us consider one layer GH at a distance y below the neutral layer EF. We can see here that length of the neutral layer and length of the layer GH will be equal and it will be dx.

Original length of the neutral layer EF = Original length of the layer GH = dx

Now we will analyze here the condition of assumed portion of the beam and section of the beam after bending action and we have displayed here in following figure. 
As we can see here that portion of the beam will be bent in the form of a curve due to bending action and hence we will have following information from above figure.

Section AB and CD will be now section A'B' and C'D'

Similarly, layer GH will be now G'H' and we can see here that length of layer GH will be increased now and it will be now G'H'

Neutral layer EF will be now E'F', but as we have discussed during studying of the various  assumptions made in theory of simple bending, length of the neutral layer EF will not be changed.

Length of neutral layer EF = E'F' = dx

A'B' and C'D' are meeting with each other at center O as displayed in above figure
Radius of neutral layer E'F' is R as displayed in above figure
Angle made by A'B' and C'D' at center O is θ as displayed in above figure
Distance of the layer G'H' from neutral layer E'F' is y as displayed in above figure

Length of the neutral layer E'F' = R x θ

 
Original length of the layer GH = Length of the neutral layer EF = Length of the neutral layer E'F' = R x θ

Length of the layer G'H' = (R + y) x θ

As we have discussed above that length of the layer GH will be increased due to bending action of the beam and therefore we can write here the following equation to secure the value of change in length of the layer GH due to bending action of the beam.

Change in length of the layer GH = Length of the layer G'H'- original length of the layer GH
Change in length of the layer GH = (R + y) x θ - R x θ
Change in length of the layer GH = y x θ

Strain in the length of the layer GH = Change in length of the layer GH/ Original length of the layer GH
Strain in the length of the layer GH = y x θ/ R x θ
Strain in the length of the layer GH = y/R

As we can see here that strain will be directionally proportional to the distance y i.e. distance of the layer from neutral layer or neutral axis and therefore as we will go towards bottom side layer of the beam or towards top side layer of the beam, there will be more strain in the layer of the beam.

At neutral axis, value of y will be zero and hence there will be no strain in the layer of the beam at neutral axis.

Let us recall the concept of Hook’s Law
According to Hook’s Law, within elastic limit, stress applied over an elastic material will be directionally proportional to the strain produced due to external loading and mathematically we can write above law as mentioned here.

Stress = E. Strain
Strain = Stress /E
Strain = σ/E
Where E is the Young’s Modulus of elasticity of the material

 
Let us consider the above equation and putting the value of strain secure above, we will have following equation as mentioned here.
σ/E = y/R
σ= (y/R) x E
Therefore, bending stress on the layer will be given by following formula as displayed here
We can conclude from above equation that stress acting on layer of the beam will be directionally proportional to the distance y of the layer from the neutral axis.


We will discuss another topic i.e. derivation of flexure formula or bending equation for pure bending in the category of strength of material in our next post.

Reference:

Strength of material, By R. K. Bansal
Image Courtesy: Google

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