Elongation in taper Rod, Volumetric Strain (Stress and Strain Part-3)

Elongation in Taper Rod

Let us consider a circular taper rod of length l whose diameter is changing uniformly from d to D.

Let us consider a small element of a thickness of dx at x distance from one end. The relationship between dx,d, D,l is given by applying the similar triangle property and we get 

Let us call that as equation (1)

δx is the elongation in the element dx. So by Hook’s law.

Stress = E * Strain

Putting values and we reach this expression as follows

Now we want to find the elongation in whole tapered rod. So all we have to do is integrate the above expression from 0 to l and we will finally get the expression Elongation in  tapered rod.

Remember above expression for elongation in a taper rod.

Let us double check the formula that we have arrived with so many calculations and integrations.

This formula should be valid for uniform diameter rod of d.

With this Elongation in Taper rod is over and write down that expression so that it can come handy in exams.

Elongation in an object due to slef weight

When an object remains hanging from one end. It’s length increases due to self-weight. Let’s find out how much length increases and what is expression for it.

Let us assume a bar of length l having density ρ is Fixed at one end vertically.

In these type of problems we have to start with assuming an element of thickness dy at some distance y as shown in the figure

If we make a free body diagram of this hanging bar then we see that only Mg force will be there downwards, where M is mass of the whole bar.

Similarly, we calculate the force up to small section dy that will be  = m*g.

Where m = mass of small element up to y having area of cross-section Ac

Force up to section y = ρ Ac y g

Stress up to section y = Force/Area = ρ Ac y g/Ac

σ = ρyg

This is the expression we have to remember

Let’s see special cases 

    • Stress at the bottom     At bottom, y = 0  ⇒ σ = 0

    • Stress at top   At top  y=l ⇒ σ = ρlg, 

↑ This maximum stress in the bar

Now we see the elongation in the element dy due to above calculated stress.

By hook’s law

We can easily calculate total elongation with integrating above expression from limits 0 to l
Remember above expression for elongation in a bar due to self-weight.

Now in exams suppose we are given to calculate elongation in tapered rod due to self weight?How will you solve it?  Please let me know in comments

Volumetric Strain

Well in exams you won’t see much questions from this part. But sometimes there may be a multiple choice question from this part.
Don’t ignore this topic just make one reading and you won’t forget it.

To have a better understanding of volumetric strain. Let’s assume a cuboid of length l, breadth b, height h.

Volume of that cuboid is given by V = lbh

If due to stresses, its dimensions changes in length or breadth or height or any of them ( all of them). Then it will affect the volume i.e. volume will also change.

Change in volume dV = δ(lbh)

dV = bh(δl) + lh(δb) + lb(δh)

These are the volumetric strain in x,y,z directions respectively.

Since the stresses are in all three directions so both lateral and longitudinal strain will be there.

Due to σx change in length 

Similarly there will be a change in length due to σy, σz.

Which will be calculated with the help of Poisson’s Ratio formula

Let’s calculate the change in length due to stresses in y & z direction.

Net change in length = δLx + δLy + δLz

So finally we get expression

So remember these final expressions these are very important.

Let’s see some special cases, from which directly conceptual question comes to exams
If there is uni-axial loading i.e. Force is only in one direction let’s say x , then stress will be only in one direction σx

In above expressions put σy = σz = 0, we get 

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