## Sunday, 15 January 2017

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 rectangular rod”, in subject of strength of material, with the help of this post.

### Let us see here the elongation of uniformly tapering rectangular rod

First we will understand here, what is a uniformly tapering rectangular rod?

A rectangular rod is basically taper uniformly from one end to another end throughout the length and therefore its one end will be of larger width and other end will be of smaller width. However thickness of the rectangular rod will be constant throughout the length of rod.

Let us consider the uniformly tapering rectangular rod as shown in figure, length of the uniformly tapering rectangular rod is L. Width of larger end of the rod is a and as we have discussed that rectangular rod will be uniformly tapered and hence other end width of the rectangular rod will be smaller and let us assume that width of other end is b. Let us assume that thickness of the rectangular rod is t.

Let us consider that uniformly tapering rectangular rod is subjected with an axial tensile load P and it is displayed in following figure.
Let us consider one infinitesimal smaller element of length dx and its width will be at a distance x from its larger diameter end as displayed in above figure.

Let us consider that width of infinitesimal smaller element is Cx
Cx = a-[(a-b)/L] X
Cx = a- KX
Where we have assumed that K= (a-b)/L

Let us consider that area of cross section of rectangular bar at a distance x from its larger diameter end is Ax and we will determine area as mentioned here.
Ax = Width x Thickness
Ax = (a- KX) t

### Stress

Let us consider that stress induced in rectangular bar at a distance x from its bigger width end is σx and we will determine stress as mentioned here.
σx = P/ Ax
σx = P/ [(a- KX) t]

### Strain

Let us consider that strain induced in rectangular bar at a distance x from its bigger width end is Ԑx and we will determine strain as mentioned here.
Strain = Stress / Young’s modulus of elasticity
Ԑx = P/ [(a- KX) t. E]

### 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 = P/ [E (a- KX) t]

Now we will determine the total change in length of the uniformly tapering rectangular rod by integrating the above equation from 0 to L.

And we can say that elongation of uniformly tapering rectangular 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
1. 