Non-standard Beam Bending Question - Parallel to Neutral Axis

[TerryTate]

Hi there, first post on these forums. I have a seemingly simply question about beam bending.

We (mechanical engineers) are all very familiar with the standard beam bending scenario. The beams are always shown being loaded perpendicular to the neutral axis, regardless of the type of support. My question is, what is the method of solving a beam loaded parallel to the neutral axis? (See attachment)

The image depicts a beam being pulled by a rope, let's say. The interesting part of it is that as the beam deflects, the moment arm of the load begins to grow with respect to the support.

Any suggestions as to how to solve such a thing without resorting to computer modeling? I'm really just looking for the beginnings of an approach so that I can work it out for myself, but I'm on the fence as to where to begin.

Any thoughts appreciated!

 

[TerryTate]

Bump, because why not! Still interested in an approach.

 

[SteamKing]

IMO, you won't be able to do the calculation in one step. An iterative approach will be called for, whether by hand or by computer. One method to research is called the 'P-delta' method, which is used a lot for columns loaded eccentrically.

Obviously, if you want to do a few iterations by hand, replace the tension on the eyebolt with an equivalent axial load and end moment at the free end. Calculate deflections using regular beam methods. Use resulting deflection to recalculate the end moment. Check deflections to see how much of an increase results. Rinse and repeat until the change in deflection becomes small enough to ignore.

 

[AlephZero]

You can formulate a linearized model that can be solved "in one step". The effect of the load acting about a new position is the same as an extra stiffness term, which can be negative or destabilizing, unlike the stiffness due to the material properties.

This extra term is called the "load stiffness", or sometimes a "follower force". Google will find the math details for you.

This linearized formulation is often a good enough approximation, if the final displacements of the structure are small. Otherwise, as SteamKing said, you have to do an iterative numerical solution.