Monday, 19 June 2017

WHAT ARE THE ASSUMPTIONS MADE IN EULER’S THEORY?

WHAT ARE THE ASSUMPTIONS MADE IN EULER’S THEORY?


Today we will see here one very important topic in strength of material i.e. Assumptions made in the Euler’s column theory with the help of this post.

Assumptions made in the Euler’s column theory

Before understanding the Euler’s column theory, we must have to be aware about the various assumptions made, as mentioned here, in the Euler’s column theory.

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

 
The column will be initially perfectly straight and load will be applied axially.

Column will be initially perfectly straight i.e. column is not bend before loading but also column will be completely straight before loading.

Load which is applied over the column will pass through the axis of the column i.e. load will not be eccentric.

Second assumption

Material of the column will be homogenous and isentropic.

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

Homogeneous term is used here to indicate that material of the column will be same throughout or we can say more specifically that material composition of the column will be same throughout the column i.e. material of the column 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.

Third assumption

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

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

Fourth assumption

Direct stress developed in the column will be very small as compared with the bending stress.

 
When a column will be loaded axially, there will be produced compressive stress in the column as one end of the column will be fixed and load will be applied axially on the other end of the column. But this compressive stress developed in the column will be negligible as compared to bending stress.
In simple, we can say that Euler had neglected the direct compression of the column.

Fifth assumption

Length of the column will be very large as compared with other dimensions of the column i.e. length of the column will be very large as compared with lateral dimensions of the column.

Sixth assumption

Self weight of the column will be negligible i.e. Euler had neglected the own weight of the column.

Seventh assumption

 
Column will fail by buckling alone

Eighth assumption

Cross-section of the column will remain uniform throughout the length of the column.

We will discuss failure of column and difference between column and strut 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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