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
Image Courtesy: Google
We will see another important topic i.e.
Elongation of uniformly tapering circular rod in the category of strength of
material.
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