Rearrange to make "d" the subject

[spikeybrummy]

Hi,

Firstly, as this is my first post, apologies if this is not in the correct area.

I am struggling to rearrange the following equation to make "d" the subject.

40*10^6 = (M(d/2))/((pi*d^4)/64)

This is an engineering question about finding the required diameter (d) of a cylindrical beam, based on a maximum bending stress (40Mpa)

The equation for bending stress is M*Y/I

M = Max Bending Moment (which I have the figure for)
Y = distance from neutral axis (which is the radius of the spar - hence d/2)
I = second moment of area which is: (pi*d^4)/64

If given the diameter, I can easily find the bending stress, but I am struggling to work it out the other way around.

Any help is most appreciated.

 

[HallsofIvy]

The only complicating part that I can see (and it is not very complicating!) is that you have "d" in the numerator and "\(\displaystyle d^4\)" in the denominator. So you should immediately use \(\displaystyle \frac{d}{d^4}= \frac{1}{d^3}\). Then multiply both sides of the equation by \(\displaystyle d^3\) to get \(\displaystyle 40*10^6*d^3= \frac{M/2}{\pi/64}= \frac{M}{2}\frac{64}{\pi}= \frac{32M}{\pi}\).

Divide both sides by "\(\displaystyle 40*10^6\)" to get \(\displaystyle d^3= \frac{32M}{40(10^6)\pi}= \frac{4M}{5\pi} \times 10^{-6}\).

Finally, of course, take the cube root of both sides: \(\displaystyle d= \sqrt[3]{\frac{4M}{5\pi}}\times 10^{-2}\).

 

[spikeybrummy]

Many thanks

Thank you HallsofIvy.

I knew I was missing something relatively simple. It was the d^3 I couldn't see.
I definitely need to brush up on my algebraic fractions.

Much appreciated