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Saturday, 22 April 2017

BENDING STRESS ANALYSIS FOR SYMMETRICAL AND UNSYMMETRICAL CROSS-SECTIONS

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 expression for bending stress in pure bending during our last session. 

Now we are going ahead to start new topic i.e. bending stress analysis for symmetrical and unsymmetrical cross-sections with the help of this post. 

First of all we must have to understand here the meaning of symmetrical sections and unsymmetrical sections. 

In case of symmetrical sections, neutral axis will pass through the geometrical center of the section.
Cross-sections such as circular cross-section hollow circular cross-section, square cross-section, hollow square cross-section, rectangular cross-section, hollow rectangular cross-section and I cross-section are the best examples of symmetrical sections. 

In case of unsymmetrical sections, neutral axis will not pass through the geometrical center of the section. Cross-sections such as T cross-section and L cross-section are the best examples of unsymmetrical sections.

Bending stresses in symmetrical sections

Neutral axis of symmetrical sections such as for circular section will lie at a distance d/2 from the outermost layer of the section. Where d will be diameter of the circular cross-section.

As we have discussed the formula for bending stress in pure bending of beams, we have concluded that stress acting on layer of the beam will be directionally proportional to the distance y of the layer from the neutral axis.

Let us see the formula for bending stress in pure bending of beams
If we consider the value of stress at neutral axis, we can easily say that it will be zero because value of y will be zero here.
Stress acting on layer of the beam will be directionally proportional to the distance y of the layer from the neutral axis; hence maximum stress will occur at the outermost layer of the section.

If we consider the case of simply supported beam, we must note it here that due to bending action, top portion of the beam will be in compression whereas bottom portion of the beam will be in tension.

If we will draw the diagram for stress distribution, we will have following figure showing the stress distribution for symmetrical sections.

Bending stresses in unsymmetrical sections

In case of unsymmetrical sections, neutral axis will not pass through the geometrical centre of the section and therefore value of y, which is the distance of the layer from the neutral axis, for outermost layers i.e. for topmost layer and bottom layer of the section will not be same.

In order to calculate the bending stress for unsymmetrical sections, we must have to find the value of centre of gravity of the given unsymmetrical section. 

As we know that neutral axis will pass through the center of gravity of the section and hence after determining the center of gravity of the section, we can have value of y for topmost layer and bottom layer of the section. 

In order to calculate the bending stress for unsymmetrical sections, we will use the bigger value of y.
If we will draw the diagram for stress distribution for unsymmetrical section such as for T section, we will have following figure showing the stress distribution for T sections.
We will discuss another topic i.e. bending equation or flexural formula for beam subjected to a simple 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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