Solving Dirac Delta Function Beam Problem

[SanchezGT]

1. The Problem

Homework Statement

4 Parts to the Assignment. Finding the Displacement of a beam assuming w to be constant.
1. Cantilever beam, free at one end. Length =l, Force P applied concentrated at a point distance rl from the clamped end. Boundary Conditions y(0)=0, y'(0)=0, y"(l)=0, and y'"(l)=0.
2. Cantilever beam. Same as above only load is uniformly distributed. So, P=w$$\delta$$$$\epsilon$$ at distance $$\epsilon$$ from clamped end.
3. Simply supported beam at both ends same as number 1. Concentrated load at point rl from end, force of P, length l.
4. Simply supported beam at both ends same as number 2. Evenly distributed load.
2. Homework Equations .
This is using the Dirac Delta Function in Laplace Transforms thus. y$$^{(4)}$$(x)=(P$$\delta$$(x-rl))/EI.
The actual PDF of the assignment can be viewed http://www.math.gatech.edu/~bourbaki/math2403/pdf/BeamProblems.pdf"

The Attempt at a Solution

I got through the first part. I am pretty confident in my answer. I basically took the laplace of the equation above. Letting y"(0)=c1 and y"'(0)=c2. Then the inverse transform running through all the derivatives with respect to x and l. I got y(x)=(P/6EI)(x-rl)$$^{3}$$U(x-rl)+(lPx$$^{2}$$/4EI)-(Px$$^{3}$$/6EI).
U is the unit finction.

 

[SanchezGT]

Note: The Greek letters all look like they are in superscript for some reason. They are not meant to be. As in P=w(delta)(epsilon).