Comparing structural bending strength

[nivek0078]

Hello,

Not sure how to go about solving this problem:

What outer diameter must a 2mm thick hollow rod have in order to have the same structural bending strength as a solid rod with diameter of 7mm? Also compare the weight per unit length of the two rods. Assume that material is stainless steel (E=180MPa) having weight density of 7.78 x 10^4N/m^3.

Does the 2mm thick refer to the length of the hollow rod? Where does weight density play into the equation?

Given equations:
stress = Mc/I

Bending rigidity -> ymax= cFl^3/EI

Thank you in advance for any information you can provide.

 

[nivek0078]

Ok so this is what I have so far:

for the soild bar
find area
A=1/4(pie)(dia)^2 -> =38.48mm^2

find moment
I=Pie/64(D^4) -> = 117.85mm^4

Find stress
stress=My/I where M=bending moment, y=neutral axis, I=moment
-> =M(3.5)/117.85 -> stress=.0297(M)
for this equation I don't know how to find M

So to have the outer diameter of the hollow rod
find the area
A=pie((Do^2)-(Di^2)) where Do is outer dia and Di is inner dia

solution for outer diameter is Do=4.03mm

Is this the correct approach for that part of the problem??

 

[nivek0078]

Still having some issues with how to compare the weight per unit length of the two rods. I'm given E=108x10^3MPa, weight density= 7.78x10^4N/m^3 for both rods, calculated I=117.85 for solid bar. Using the equation Ymax= cFl^3/EI where c=constant, l=1.

So please advise on how to proceed here.

Also a fix to the above post.
using the I value from the solid and plugging that value into this equation for the hollow bar I=(pi/64)((Do^4)-(Di^4)) Do=7.0115mm^4 instead of the 4.03 value, is this correct?

 

[haruspex]

You have that for a solid rod diameter D the 2nd moment of area is $\frac{\pi D^4}{64}$. For a hollow rod, just treat it as the difference of two solid rods. I.e. subtract the 2d MoA of the missing part from what you would get if the rod were solid.

 

[Nickie]

Hello,

Thank you for reaching out. To solve this problem, we need to use the equations provided and take into consideration the properties of the materials involved. The 2mm thickness refers to the thickness of the hollow rod, not the length. The weight density plays into the equation because it is used to calculate the weight of the rods, which will affect their structural bending strength.

To find the outer diameter of the hollow rod, we can use the equation for bending rigidity (ymax = cFl^3/EI) and set it equal to the solid rod's bending rigidity. We can then solve for the outer diameter of the hollow rod. The weight per unit length can be calculated by multiplying the weight density by the length of the rod.

Once we have the outer diameter and weight per unit length for both rods, we can compare them to determine which one is stronger and lighter. Keep in mind that the material properties, such as the modulus of elasticity (E) and weight density, will also play a role in the structural bending strength of the rods.

I hope this helps. Let me know if you have any further questions. Good luck with your calculations!