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

## Elongation in Taper Rod

#### 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)

#### Stress = E * Strain

Putting values and we reach this expression as follows

#### 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.

## σ = ρyg

### By hook’s law

##### 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.

#### 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.

##### 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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