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Monday, 24 April 2017

MOMENT OF RESISTANCE OF BEAM SECTION

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. moment of resistance of a 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.

Moment of resistance

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 as displayed in following figure.
As we have discussed that when a beam will be subjected with a pure bending, as displayed in above figure, layers above the neutral axis will be subjected with compressive stresses and layers below the neutral axis will be subjected with tensile stresses.

Therefore, there will be force acting on the layers of the beams due to these stresses and hence there will be moment of these forces about the neutral axis too.

Total moment of these forces about the neutral axis for a section will be termed as moment of resistance of that section.

As we have already assumed that we are working here with a beam having rectangular cross-section and let us consider the cross-section of the beam as displayed here in following figure.
Let us assume one strip of thickness dy and area dA at a distance y from the neutral axis as displayed in above figure.

Let us determine the force acting on the layer due to bending stress and we will have following equation.
dF = σ x dA

Let us determine the moment of this layer about the neutral axis, dM as mentioned here
dM = dF x y
dM = σ x dA x y
Recall here the concept of bending stress, which is mentioned below, and use the value of bending stress (σ) in above equation and we will have equation for bending moment of the layer about the neutral axis.
σ = (E/R) x y
dM = (E/R) x y x dA x y
dM = (E/R) x y2 dA

Total moment of the forces on the section of the beam around the neutral axis, also termed as moment of resistance, could be secured by integrating the above equation and we will have
dM = (E/R) x y2 dA
Where M will be termed as moment of resistance

We will discuss another topic i.e. Section modulus 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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