Elastic Deformation of an Axially Loaded Member

[user12323567]

Homework Statement

Collar A can slide freely along the smooth vertical guide. If the vertical
displacement of the collar is 1 mm and the supporting 20-mm-diameter
rod AB is made of 304 stainless steel (E = 200 GPa), determine the
magnitude of P.

Relevant Equations

ẟ = FL/AE

Sum of forces in the y-direction = 0 and downwards is +ve
P + Fab,y = 0
P + Fab (4/5) = 0
Fab = -1.25P
ẟ = FL/AE -> ẟab = FabLab/AabE

ẟab = (-1.25P*.75)/(pi*(.01)^2*(200*10^3)) = -0.0149P
After this step, I am uncertain of how I can relate the vertical elongation with AB's elongation to find Force P. Please assist.

 

[Lnewqban]

Under the action of load P, the shown dimmensions change to be 599 mm vertical and 450 mm horizontal.
Link AB is loaded on compression and its length is reduced some, while its internal stress increases.

 

[user12323567]

Under the action of load P, the shown dimensions change to be 599 mm vertical and 450 mm horizontal.
Link AB is loaded on compression and its length is reduced some, while its internal stress increases.

So, are you saying that the length of the supporting rod, AB would now change to 749.20 mm?

 

[Lnewqban]

So, are you saying that the length of the supporting rod, AB would now change to 749.20 mm?

I don’t know exactly by how much (it depends on geometry), but AB becomes shorter under its axial load (which magnitude is greater than P, think mechanical advantage).

Link AB must deform less than 1 mm because the angle it forms with the vertical.

You need to calculate how much force is needed to deform link AB that much; then, you can calculate the value of P.

Please, see:
https://www.engineeringtoolbox.com/young-modulus-d_417.html
:)

 

[Chestermiller]

So, are you saying that the length of the supporting rod, AB would now change to 749.20 mm?

A change in length of AB equal to 0.8 mm is correct.