Ratio of d/b to minimize maximum stress in a beam cut from a circular log

[Dell]

find the ratio of d/b such that the largest stress in the beam will be minimum

i know that the maximum stress is

σ_max=Y_max$$\frac{M}{I}$$

and i know that for a rectangle I=bh³/12

now in the question i am asked to find the ration of the (diameter of the log)/(the width of the rectangle) such that σ_max is minimal

since the rectangle is contained in the circle

d²=b²+h²
where h is the height of the rectangle

σ_max=Y_max$$\frac{M}{I}$$

σ_max=(h/2)*(12M/(bh²)
σ_max=6M/(b*h²)
σ_max=6M/(b*(d²-b²)

basically from here i need to find the ratio d/b so that (b*(d²-b²) is maximum,

but how can i do this??

d/b=K

(b*(d²-b²)
=(bd²-b³)

=d/b*(d*b₂-b⁴/d)

but i can't get to the ratio, i feel i am so close but just not getting it

 

[nvn]

That is not shear stress. Your comment does not make sense. [EDIT: OK, you edited your post in response to my comment, without letting anyone know, thereby making my above comment appear as nonsense.]

You seem to not understand what d is, as if you have not looked at the diagram.

 

[Dell]

at first i thought that d was the height of the rectangle, but thatdidnt make sense to me, if that is so what is the circular log given for?? it is not 100% clear to me in the diagram what they mean by d.

as far as i know that is the equation for stress,

how would you have gone about solving it??

 

[nvn]

Use a straightedge to see what d is.

 

[Dell]

so you say that d is the height of the beam, if so why do they need to tell me that the beam is taken from a log,

if i take the height as d

I=bd³/12

σmax=(h/2)*(12M/(bd³)
σmax=6M/(b*d²)

now i need b*d² to be maximum

but i still can't get to the ratio??

i supose that b*d² will be maximum when d/b-->infinity
but that doesn't seem right at all

 

[Dell]

never mind, i think i got it

b²+d²=D²

i differentiate the stress equation adn compare to 0 and i get sqrt(2)

 

[pongo38]

They don't tell you the size of the log, but the ratio r= d/b is constrained by the circular boundary. In my experience, if a problem says to find a ratio, then the first line should be: Let r be the ratio. Then work everything towards equations in r.

 

[Dell]

thanks but i got it already sqrt(2)