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.(adsbygoogle = window.adsbygoogle || []).push({});

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.

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Analysis of beam bending against constraint

**Physics Forums | Science Articles, Homework Help, Discussion**