# Homework Help: Which is Less Material Given Deflection due to Force?

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1. Nov 10, 2014

### _Anonymous_

1. The problem statement, all variables and given/known data
You are considering two different designs for a walking cane. Both designs use hollow metal tubes, but design A has a circular cross section (outer diameter h) whereas design B has a square cross section (outer width and length h). Both canes are designed to withstand an end deflection of exactly 5mm with 2kN of force applied to one end, normal to the force, while the other end is held fixed. Show which design will require less material to meet these specifications.

2. Relevant equations

Deflection with a circular cross section = (4FL^3)/(3Epir^4)

Deflection with a rectangular cross section = (4FL^3)/Eh^3B

I = ∫∫r2dA

Icircle = πR4/4

Irectangle = h3/12

d = (FL3)/(3EI)

3. The attempt at a solution

Deflection with a rectangular cross section = (4FL^3)/Eh^3B

5000 = (4(2000)L^3)/(Eh^4)

L^3 = (5000Eh^4)/8000

Deflection with a circular cross section = (4FL^3)/(3Epir^4)

5000 = (4(2000)L^3)/(3Epi(h/2)^4)

((5000)(3Epi)(1/16)h^4)/(8000) = L^3

Therefore ((L_r)^3)(3pi/16) = ((L_c)^3), so the circular cross section will require less material.

My solution assumes that there is a relationship between volume and surface area. Is this assumption okay? Did I use the correct formulas? Should I have done something different to take into account that the canes are hollow?

2. Nov 15, 2014