# Homework Help: Beams and bending

1. Dec 27, 2009

### 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

σmax=Ymax$$\frac{M}{I}$$

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 σmax is minimal

since the rectangle is contained in the circle

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

σmax=Ymax$$\frac{M}{I}$$

σmax=(h/2)*(12M/(bh2)
σmax=6M/(b*h2)
σmax=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 cant get to the ratio, i feel i am so close but just not getting it

Last edited: Dec 27, 2009
2. Dec 27, 2009

### 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.

Last edited: Dec 27, 2009
3. Dec 27, 2009

### 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??

4. Dec 27, 2009

### nvn

Use a straightedge to see what d is.

5. Dec 27, 2009

### 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 cant get to the ratio??

i supose that b*d2 will be maximum when d/b-->infinity
but that doesnt seem right at all

6. Dec 27, 2009

### 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)

7. Dec 27, 2009

### 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.

8. Dec 27, 2009

### Dell

thanks but i got it already sqrt(2)