Which Beam Cross-Section is Most Weight-Efficient?

[rahshail]

Mechanics of solids prob. help!

here is a prob. i have.anybody help.
Three beams have the same length, the same allowable stress and the same B.M. The cross section of the beams are a circle, a square, and a rectangle with depth twice the width. weight wise best section in order of merit will be...??(arrange in increasing order pls)

pls tell the method by which u solve the prob.

 

[PhanthomJay]

here is a prob. i have.anybody help.
Three beams have the same length, the same allowable stress and the same B.M. The cross section of the beams are a circle, a square, and a rectangle with depth twice the width. weight wise best section in order of merit will be...??(arrange in increasing order pls)

pls tell the method by which u solve the prob.

You should please show at least some effort as to how you would approach the problem. What do you know about the geometric properties (area, moment of inertia, etc.) of each given cross section?

 

[rahshail]

Actually i have got this problem while m studying a book. i have written the problem here as it was in the book. so please try to boggle ur mind on the prob. as it is written here. i have no additional details on it. however i myself don't think it requires any further details. we can use bending equation and formulae for bending and section modulus. please try.

 

[PhanthomJay]

Actually i have got this problem while m studying a book. i have written the problem here as it was in the book. so please try to boggle ur mind on the prob. as it is written here. i have no additional details on it. however i myself don't think it requires any further details. we can use bending equation and formulae for bending and section modulus. please try.

Yes, the section modulus (and area) of each shape is very important to know in this problem, but the burden is on YOU to try, not me or any other responders. If you would please show your attempt at how your knowledge of Section Modulus can be used to solve this problem, then, and only then, can we be of further assistance. Please show some effort at an approach to the solution, however minimal it might be. Peace.

 

[rahshail]

OK fine. Here is what i have tried.

from universal bending equation, f_s= M * Z
where M= bending momet
Z= section modulus
f= allowable stress
Now if M and f are same for the all three sections then Z will also same.
For circular section Z= (π/32)d3 d--> diameter of section
For rectangular section Z= bd³/12 b-->width and d-->depth of section
For square section same as rectangular but here b=d


mass= Volume * density
= area of section * length * density
and length 'l' and density same for all beams so mass will depend upon area of section only.

now we need to solve by these two facts: 1. section modulus same
2. mass dependent on area of section
here i am unable to establis the relation between these two. now further help needed in this

 

[nvn]

rahshail: Bending stress is f = M/Z. Elastic section modulus is Z_circle = pi*(d^3)/32, Z_rect = b*(h^2)/6, and Z_square = (a^3)/6. You correctly stated that Z is the same for all three beams. Also, b = 0.5*h. Therefore, set Z_circle = Z_rect, and solve for d in terms of h. Also, set Z_square = Z_rect, and solve for "a" in terms of h. Now compute cross-sectional area, A_circle, A_rect, and A_square, in terms of h. Now arbitrarily let h = 2^0.5, to see which beam is more efficient.