Analysis of beam bending against constraint

[jernst]

I have been trying to analyze the deflection of the free end of a cantilevered beam with a point load P at the end. The trick is, the beam is supported underneath by a surface described by the arbitrary function g(x). So let's say that g(x)= 0.001*x^2, a very shallow parabola. As the beam deflects, some portion of the beam begins to rest on the constraint surface. So the active length of the beam is changing as a function of P.

I have come up with an algorithm that seems to work but is not terribly elegant. This has piqued my curiosity, since I have not found this topic treated in any text or paper that I looked at. So I wonder if anybody out there has a better way to analyze this. Is there a reasonable analytic solution for such a problem?

My algorithm involves using vector X that contains equally spaced points x1, x2 etc. along the length of the beam, and solving the deflection equation for P, setting the next point in X equal to the constraint surface g(x) (basically asking what force P will bend the beam so the next point x in my vector will touch the constraint function g). Once I have that force, I recast the problem at the next point x with new axes t,n tangent and normal to the constraint function, and do the analysis over again with a shorter beam, translating and rotating all curves and results appropriately between the x,y and t,n reference frames. As expected the beam gets stiffer as it gets shorter, hence for deflection F, dF/dP approaches zero as P gets larger.

But I start each new bending problem as if it were a simple cantilever with no initial internal forces. But obviously, if I have bent the beam around a constraint surface, my beam has internal forces that I am ignoring.

Can anybody think of a more accurate and/or elegant way to do this.

 

[Mapes]

Hi jernst, welcome to PF. This seems like a fine start, but I think you need to iterate the solution to accommodate the new loads. Each additional discretized vertical load will change the shape of the beam to its left also, so the earlier loads will have to be recalculated.

The more elegant way, I suppose, is to cast the problem as one of variational calculus, where you would minimize the strain energy subject to the constraint of contact with the support. This approach could be easier or harder, depending on your experience with math vs. programming.

And the most accurate way would be to numerically simulate it, but assume a frictionless surface. Then the applied loads from the contact situation won't be perfectly vertical; they'll have a horizontal component. You'll need to incorporate stretching of the beam in your calculations. This solution will be more accurate than even the beam theory solution, which assumes small displacements and thus ignores stretching.

 

[jernst]

Hi Mapes, thanks for the warm welcome and prompt reply.

Each additional discretized vertical load will change the shape of the beam to its left also, so the earlier loads will have to be recalculated.

By discretized vertical load, do you refer to the point load, P, which I am increasing incrementally, or are you thinking of modelling the contact as discretized vertical loads?

The more elegant way, I suppose, is to cast the problem as one of variational calculus, where you would minimize the strain energy subject to the constraint of contact with the support. This approach could be easier or harder, depending on your experience with math vs. programming.

The words "variational calculus" do not strike fear, but the "subject to the constraint of" is where I run into trouble.

And the most accurate way would be to numerically simulate it, but assume a frictionless surface. Then the applied loads from the contact situation won't be perfectly vertical; they'll have a horizontal component. You'll need to incorporate stretching of the beam in your calculations. This solution will be more accurate than even the beam theory solution, which assumes small displacements and thus ignores stretching.

Yes, in each new problem I was throwing away the component of P tangent to the constraint. I will work that in as well. Presumably this is in conjunction with the iterative solution you suggest above.

Thanks.

 

[Mapes]

I think it's got to feature discretized vertical (nearly vertical to be more exact) loads from the constraint surface where the beam touches. These will obviously correspond to the pressure against that surface. The last point of compressive load will correspond to the contact point.

The analytical approach does indeed sound tricky!