 # 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)

#### 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

#### 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 ## σ = ρyg

### 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.

#### 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 ### I hope you like this section. Please share with friends and like my Facebook page and never miss an update.

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