Thursday, 20 April 2017

WHAT ARE THE ASSUMPTIONS MADE IN THE THEORY OF SIMPLE BENDING?

WHAT ARE THE ASSUMPTIONS MADE IN THE THEORY OF SIMPLE BENDING?

We were discussing basic concept of bending stress in our previous session. We have also discussed strain energy and expression for strain energy due to various types of loading such as strain energy stored due to gradual applied load during our last to last session.

Now we are going ahead to start new topic i.e. Assumptions made in the theory of simple bending in strength of material with the help of this post.

Assumptions made in the theory of simple bending

Before understanding the theory of simple bending, we must have to be aware about the various assumptions made, as mentioned here, in the theory of simple bending.

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

First assumption

Material of the beam will be homogenous and isentropic.

Now you might be thinking that what is the meaning of the terms homogeneous and isentropic used here in first assumption.

Homogeneous term is used here to indicate that material of the beam will be same throughout or we can say more specifically that material composition of the beam will be same throughout the beam i.e. material of the beam will not be changing throughout.

Isentropic term used here to indicate that elastic properties of the material will be same in all the directions i.e. modulus of elasticity of the material will be same in X-direction, in Y-direction and in Z-direction.

Second assumption

Young’s modulus of elasticity of the material of the beam will be same in tension and compression.

As we have discussed in our previous post that due to bending action, top portion of the beam will be in compression whereas bottom portion of the beam will be in tension. Value of Young’s modulus of elasticity of material of the beam will be same for tension and compression.

 

Third assumption

Beam will be straight before loading and will remain straight once load will be removed.

Let us assume that we have following horizontal beam AB as displayed here. Beam AB is straight before loading, now once load W will be applied over the simply supported horizontal beam AB as displayed here, beam AB will be bending in the form of a curve.
If we have removed the load W, beam AB must be straight i.e. there must be elastic deformation.

Fourth assumption

The sections of the beam which were plane before bending, must remain plain after bending too.

Fifth assumption

Beam material must be stressed within its elastic limit and therefore beam material must follow the principle of Hooke’s law.

Bending stress developed in the beam, once beam will be loaded, must be within elastic limit or we can say that there must be elastic deformation in the beam.

 

Sixth assumption

The radius of curvature, during bending of the beam, will be large as compared with the dimensions of the cross-section of the beam and beam will have symmetrical cross-section.

Seventh assumption

Beam will be subjected with the pure bending action.

As we have already discussed that when horizontal simply supported beam AB will be loaded with a load, there might be shearing action and beam might be sheared. 

We will not have to consider here the shearing action, but also we will consider here only bending action i.e. we have assumed that beam AB will be loaded with a load and there will be only bending action.

Eighth assumption

Load will be applied in the plane of bending and each layer of the beam will be free to expand or contract, independently of the layer, above or below it.

We will discuss theory of 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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