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.

 

[alphy]

I am not an expert in mechanical engineering or structural analysis. However, I can offer some general guidance on how to approach this problem.

First, it is important to understand the concept of the neutral axis. In a standard beam bending scenario, the neutral axis is the line that experiences no stress or strain when the beam is loaded perpendicular to it. When the beam is loaded parallel to the neutral axis, this changes the distribution of stress and strain along the beam.

To solve this problem, you will need to use the principles of mechanics and structural analysis. One approach could be to apply the equations of equilibrium, which state that the sum of forces and moments acting on the beam must equal zero. This will allow you to determine the reactions at the supports and the internal forces within the beam.

You can then use the equations of beam bending, such as the Euler-Bernoulli beam equation, to calculate the deflection of the beam at different points. This will take into account the changing moment arm as the beam deflects.

If you are looking for a more analytical solution, you may need to use advanced mathematical techniques such as calculus or differential equations. Alternatively, you could also consider using computer modeling software to simulate the behavior of the beam under the given loading conditions.

In any case, it is important to carefully consider all the various factors and assumptions that go into solving this type of problem. This includes the material properties of the beam, the loading conditions, and the boundary conditions at the supports.

I hope this helps guide you in approaching this problem. Remember to always carefully analyze and validate your results to ensure accuracy. Good luck!