What is the failure mode for a vertical pole with guy ropes?

[SJB]

Hi,

Thank you for taking the time to look at my problem.

I am considering a vertical pole fixed in the ground at the bottom and experiencing a horizontal point load at the top. This pole is fixed with guy ropes from the top to the ground at a certain angle. See attached image.

Assuming that the guy rope/cable is of infinite strength then the mode of failure for this arrangement will be that the pole will buckle (correct?). As the tip force P increases as will the force in the guy rope and the resultant vertical force on the pole itself will reach a point where the euler buckling condition is not satisfied...

However it's what happens next that I need to understand. In a normal euler buckling scenario the vertical load is fixed and once the beam has buckled it will collapse. In this case though as the beam begins to buckle (as per image attached) the angle of the guy rope increases and therefore the vertical component of this force gets less. However once the beam has started to buckle it is only able to take a fraction of the vertical load it could pre-buckling.

So, where to go from here? Once the initial buckling load has been satisfied will the beam always collapse? Or, in the right conditions will the pole reach a stable equilibrium with a certain bend? If the latter then how do I go about calculating the actual failure load?

Thank you very much for reading this and any guidance or advise would be very much appreciated.

Thanks

Simon

 

[Studiot]

Although the vertical load is fixed in the Euler analysis, his method could be applied to your situation. You still equate the second order differential elastic equation of the pole to the moment, in terms of the vertical force, V . You just have a different differential equation to solve since V is now a function.

 

[nvn]

SJB: If this is a long column, once the pole reaches one half of the Euler buckling load, assume it will completely collapse. Assume the cross section buckles locally, and the structure continues to fall.

Or if you do not want to assume it completely collapses, and you have a way to compute or estimate the tip horizontal deflection after buckling (such as, compute the maximum deflection prior to yielding), then to estimate the post-buckling strength, perhaps analyze the deflected pole as a beam-column (?). If you choose this latter option, I would refer you to your favorite textbooks for beam-column theory.

 

[stewartcs]

Hi Simon,

Assuming that the guy rope/cable is of infinite strength then the mode of failure for this arrangement will be that the pole will buckle (correct?).

Not necessarily, it could also fail due to stress (i.e. yielding).

However it's what happens next that I need to understand. In a normal euler buckling scenario the vertical load is fixed and once the beam has buckled it will collapse. In this case though as the beam begins to buckle (as per image attached) the angle of the guy rope increases and therefore the vertical component of this force gets less. However once the beam has started to buckle it is only able to take a fraction of the vertical load it could pre-buckling.

The beam doesn't necessarily collapse once it buckles. Buckling is just an alternate state of equilibrium. Depending on the factors involved the column may collapse due to buckling or yielding or it may simply stay in the alternate equilibrium state and not collapse. The big taboo with buckling is that it is no longer known what load the column can support structurally using standard calculations that most civil and structural engineers use. A lot more effort is involved to find the post buckling solution. It is generally more conservative (safer) to limit the design such that it does not buckle.

So, where to go from here? Once the initial buckling load has been satisfied will the beam always collapse?

No.

Or, in the right conditions will the pole reach a stable equilibrium with a certain bend? If the latter then how do I go about calculating the actual failure load?

Once the column buckles it is not stable based on the definition of stability (i.e. it doesn't return to it's original shape once the load is removed). However, it can find an equilibrium state after buckling depending on the variables involved.

The solution is found with post buckling methods. Most journals on non-linear mechanics will have quite a bit of papers on these topics. But be warned, they are not easy to solve and will require numerical solutions to the differential equations.

As a starting point try:

"Complete solution of the stability problem for elastica of Euler's column" by V.V. Kuznetsov and S.V. Levyakov. International Journal of Non-Linear Mechanics.

That should give you an idea of what all is involved.

Hope this helps.

CS

 

[SJB]

Hello Studiot, nvn, CS,

Thank you for taking the time to read this and your comments are much appreciated!

It's interesting that there is at least a possibility that the tube can initially buckle but then reach an equilibrium. I'm intrigued by this and in the long-term I'll try and find a mathematical solution to my particular case by following Eulers method but with the vertical force as a function. Time is not on my side with this so in the short-term I'll design the beam so that it doesn't buckle in the first instance.

Thanks again.

Simon

 

[Studiot]

I'm guessing that the vertical column carries some kind of dish at the top and the horizontal force is wind loading.

I don't know which is the more valuable component the dish or the column.
However I presume that a buckled, but not collapsed, column is no more valuable than a completely failed one. It would still have to be replaced.

So the possiblility exists of deliberately introducing a 'weak section' near the top that would fail in preference, leaving the coulmn standing unharmed, but with the loss of the top section plus load.

 

[vargasjc]

Non-linear mechanics is always a pain because there are so many variables involved that we are only beginning to see and understand and there is still a long way to go.

In the mean time we are bound to do things in the way that Studiot suggests: by designing for linearity and making it easy to isolate and replace the things that are likely to behave differently.

 

[SJB]

Hi All,

Thanks for the continued input.

Yes, and it gets even more complicated as in actuality the pole is a telescopic tube so reducing diameters as it goes up.

From a stress point of view the pole could actually be self standing as the horizontal tip load is so small. The problem is that there is a deflection criteria which can't be achieved self-standing hence the guy ropes.

What I've decided to do is drop the guys down to about 80% mast height. This is still enough to contain the tip deflection but the drop in height makes a massive difference to the buckling issue which is proportional to unsupported length squared...

Thanks for your help all...

Simon