Sunday, 15 January 2017

ELONGATION OF UNIFORMLY TAPERING CIRCULAR ROD

ELONGATION OF UNIFORMLY TAPERING CIRCULAR ROD

We were discussing the concept of stress and strain and also we have discussed the different types of stress and also different types of strain as well as concept of Poisson ratio in our previous posts. We have also seen the concept of Hook’s Law , types of modulus of elasticity in our recent post.

Now we are going further to start our discussion to understand “Elongation of uniformly tapering circular rod”, in subject of strength of material, with the help of this post.

Let us see here the elongation of uniformly tapering circular rod

First we will understand here, what is a uniformly tapering circular rod?
 
A circular rod is basically taper uniformly from one end to another end throughout the length and therefore its one end will be of larger diameter and other end will be of smaller diameter.

Let us consider the uniformly tapering circular rod as shown in figure, length of the uniformly tapering circular rod is L and larger diameter of the rod is D1 at one end and as we have discussed that circular rod will be uniformly tapered and hence other end diameter of the circular rod will be smaller and let us assume that diameter of other end is D2.
Let us consider that uniformly tapering circular rod is subjected with an axial tensile load P and it is displayed in above figure.

Let us consider one infinitesimal smaller element of length dx and its diameter will be at a distance x from its larger diameter end as displayed in above figure.

Let us consider that diameter of infinitesimal smaller element is Dx
Dx = D1-[(D1-D2)/L] X
Dx = D1- KX
Where we have assumed that K= (D1-D2)/L

Let us consider that area of cross section of circular bar at a distance x from its larger diameter end is Ax and we will determine area as mentioned here.
Ax = (П/4) Dx2
Ax = (П/4) (D1- KX) 2

Stress

Let us consider that stress induced in circular bar at a distance x from its larger diameter end is σx and we will determine stress as mentioned here.
σx = P/ Ax
σx = P/ [(П/4) (D1- KX) 2]
σx = 4P/ [П (D1- KX) 2]

Strain

 
Let us consider that strain induced in circular bar at a distance x from its larger diameter end is Ԑx and we will determine strain as mentioned here.

Strain = Stress / Young’s modulus of elasticity
Ԑx = σx /E
Ԑx = 4P/ [П E (D1- KX) 2]

Change in length of infinitesimal smaller element

Change in length of infinitesimal smaller element will be determined by recalling the concept of strain.

Δ dx = Ԑx. dx
Where, Ԑx = 4P/ [П E (D1- KX) 2]

Now we will determine the total change in length of the uniformly tapering circular rod by integrating the above equation from 0 to L.
 
And we can say that elongation of uniformly tapering circular rod will be calculated with the help of following result.

Do you have any suggestions? Please write in comment box

Reference:

Strength of material, By R. K. Bansal
Image Courtesy: Google
We will see another important topic i.e. Stress analysis of bars of varying sections in the category of strength of material.

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