Two concentric tubes, where the inner tube is made of aluminum and outer tube is steel, are assembled vertically with a mass on top. Which tube will see more deflection?

Mechanical Engineering Cantilever Beams
1

Two tubes (one solid, one hollow) concentrically arranged as shown attached to the ground with a known mass on top. Both have equal areas of cross section.

Screenshot 2023-02-07 at 7.10.30 PM
Screenshot 2023-02-07 at 7.10.30 PM1751×990 21.1 KB

Which tube sees more load? More deflection? Calculate stresses and strains in both. What if the mass on top was rigid? The mass is not attached, just placed.

2

Key things to recognize:

  1. Displacement will be the same for both the steel and aluminum.
  2. The lengths and areas are the same.
  3. Force will be shared across both materials, but not equally.

d = FL/AE (where d is displacement, F is the force imparted by the block, A is the cross-sectional area of the axially loaded body, E is the elastic modulus of the axially loaded body.)

Since d_al = d_st,
(F_al * L) / (A * E_al) = (F_st * L) / (A * E_st)

A and L can be canceled to get,
(F_al) / (E_al) = (F_st) / (E_st)

We recognize that E_al < E_st, thus F_st > F_al.

stress_st = F_st/A
stress_al = F_al/A
strain = (d_0 - d)/d_0

Summary:
The steel tube sees more load.
Deflection is the same in both tubes.
(not sure what is meant by what if the mass on top was rigid, I assumed it was rigid)

3

Hello,

I think the question is just focused on two steel tubes. One hollow and one solid, hence I feel the difference will be related to the moment of inertia. What do you think? @Mark_Washington