Which is Less Material Given Deflection due to Force?

[_Anonymous_]

Homework Statement

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.

Homework Equations

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

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

I = ∫∫r²dA

I_circle = πR⁴/4

I_rectangle = h³/12

d = (FL³)/(3EI)

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?

 

[anathema]

Your solution is correct and you used the correct formulas. The assumption that there is a relationship between volume and surface area is valid in this case. However, to take into account that the canes are hollow, you could have used the formula for the moment of inertia of a hollow cylinder or rectangular tube instead of the solid shapes. This would give a more accurate result as the hollow canes would have a different distribution of material compared to solid canes.