# How to solve Euler-Bernoulli beam for point load?

I am student and learning beam theory. I am trying to solve Euler-Bernoulli 4th order differential equation to find the deflection of a beam. The equation is

EI v''''(x) = w

where v is the transverse deflection and w is the distributed uniform load on the beam.

But how would you solve the above if the beam is fixed on both ends, and instead of w, I have a point load P at say distance 'a' from the left end? For a beam fixed on both ends, I use the following boundary conditions

v'(0)=0; v(0)=0; v'(L)=0; v(L)=0; assuming the beam has length L.

But the problem for me, is that the above equation is meant to be used when the load is distributed over the beam. So, when the load is only a point load, what should I do? The wiki page http://en.wikipedia.org/wiki/Euler%E2%80%93Bernoulli_beam_theory here seems to talk a little about this, but I could not understand it. I am only a first year student.

Any one can explain in simple terms what I need to do for point load? (assuming there is no w, just a point load somewhere on the beam, both ends fixed). Sorry that my question is very basic for this advanced forum, but I did not know where else to ask this.

thanks.

Robert

## A couple of ideas

1) You showed the wikipedia article, start with the bending moment equation M=EIv". I assume you can do statics, sums of forces and moments. First solve the static equations for forces and moments at the two end points. Then cut the beam in two places, one side of the point where point load is applied, write the static equilibrium using assumed internal moment and internal shear force--you will get an equation for "M" as a function of "x", the distance along the beam. Do the same for a cut of the beam on the other side of the load application point. I think you have everything you need.

2) for a more general approach, visualize the point load "P" as a load distributed over a very small area, width "epsilon", but the distributed load has magnitude of P/epsilon. Now you can start to do a straight integration of the original differential equation: EIv''''(x)=w. If you integrate the right hand side, "w", remember that "w" is zero everywhere except that small width "epsilon" so the integral of "w" has limits only for that small epsilon. Finish the integrations, enforce the boundary conditions, I think then you can make "epsilon" go to zero. I did this procedure (what I remember of it) a long time ago, but I think the principle is as I have remembered it. There might be some integration tricks I am forgetting, but I think that generally this is how it can be done.