Strength of materials/Max bending moment

[MechEngrStdnt]

Homework Statement

It's been a while since I've done this, it's essentially a strength of materials problem in my mechanical analysis class so I haven't worked problems like these in quite a a while. I'm fairly confident I've done it right however I was hoping someone might be able to confirm this for me? Looking for the max bending moment and ignoring sheer (taking a slice at A, essentially the base of the pole). F2 is into the page (-Z axis). The homework was assigned Friday and is due Monday and office hours are not until after it is due on Monday because that makes so much sense. Any help very much appreciated!

Homework Equations

The Attempt at a Solution

L2 is 35 feet, not inches.[/B]

 

[SteamKing]

Homework Statement

It's been a while since I've done this, it's essentially a strength of materials problem in my mechanical analysis class so I haven't worked problems like these in quite a a while. I'm fairly confident I've done it right however I was hoping someone might be able to confirm this for me? Looking for the max bending moment and ignoring sheer (taking a slice at A, essentially the base of the pole). F2 is into the page (-Z axis). The homework was assigned Friday and is due Monday and office hours are not until after it is due on Monday because that makes so much sense. Any help very much appreciated!

Homework Equations

The Attempt at a Solution

View attachment 89735

There's some dimensions which are confused here.

You indicate L₁ = 20' which I take to mean 20 feet, and you have written L₂ = 35", which I would take to mean 35 inches.
Yet, in your moment calculations, you treat both distances as feet. Which is which? Are L₁ and L₂ both measured in feet?

You also have bending occurring about two different axes. You can't just add the bending moments due to F₁ and F₂ as your calculations show.

 

[MechEngrStdnt]

35 feet. Thank you for pointing this out, I will fix it in the original question. Also I thought it was odd to add the two stresses without having to do Pythagorean, however when looking through my strength of materials book the flexure formula shows them adding stresses due to bending moments like I did in my problem.

 

[SteamKing]

Because the two bending moments are about different axes, the maximum bending stress due to each moment will occur at different points on the circular cross section. You don't add stresses together if they do not occur at the same location.

Study this diagram of unsymmetrical bending:

Sorry for the small size.

 

[MechEngrStdnt]

Looking at this picture it seems as though we have some areas adding, and others cancelling. It looks to me as though where the stress is max is at the two corners where we have the same sign, one tensile, one compressive. conceptually it makes sense, however I'm still unsure as to how I am supposed to find the magnitude... hint? lol and thank you.

 

[MechEngrStdnt]

Because they seem to be combining for the greatest stress at essentially 45° (Unit circle), can I use trig to add them together?

 

[SteamKing]

You know that the bending stress is going to be a maximum at the point the farthest from the neutral axis. All along the neutral axis, the bending stress must equal zero.

Bending stresses are either tensile or compressive, so the combined bending stress at a particular location on the cross section can be found by adding together algebraically all of the different bending stresses which occur at that location.

To solve your problem, you must figure out, for each bending moment, which stresses are compressive, which are tensile, and where each stress if located on the circular cross section, in order to find the correct maximum bending stress produced by the combined loading on this structure.

Unlike a bar which has a rectangular or square cross section, a circular cross section can have infinitely many neutral axes, because any diameter passes thru the centroid of the section. The direction in which the bending moment is applied becomes the determining factor in locating where the maximum bending stress is located.

For instance, the moment created by applying F₁ to the pole in your problem acts about the z-axis, which means the maximum stress for this moment will be found on the x-axis. The moment created by F₂ causes bending to occur about the x-axis, but the maximum stresses will be found on the z-axis. Because the two bending moments have different magnitudes, the maximum combined bending stress max occur on these axes, or somewhere in between. You can be sure until you make the calculations and analyze the bending stresses further.

You absolutely cannot just calculate bending stress due to the total moment and say that gives the maximum bending stress.

 

[MechEngrStdnt]

it seems to me that I would in fact have to use Pythagoras to add them, but would that work? sorry I have looked through my strengths book and the text for this class and cannot find how to add them.

 

[MechEngrStdnt]

can moments be added using Pythagoras?

 

[MechEngrStdnt]

or stresses for that matter? also I'm not following what situation the flexure formula is valid...

 

[SteamKing]

It's a lot easier in this case, I think, to calculate the bending stresses separately for each bending moment and then to superpose them on the cross section to find the maximum stress. You said yourself that it's been a while since you did these types of calculations. I think it's better to do a little extra calculation so that you thoroughly understand what's going on.

 

[MechEngrStdnt]

I'm very confused. I used the flexure formula for arbitrarily applied moments, it utilizes superposition. so was my approach right or wrong? I'm really not following what your getting at... I'm ok with doing a little extra calculation but I have no idea what calculations to do... the only thing I can think of from strengths was the flexure formula. if that was incorrect then I am completely lost. thanks again for your help.

 

[MechEngrStdnt]

This is what I used.