# Homework Help: Closed form solution for second order beam deflection?

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1. Apr 1, 2017

### veganfox

1. The problem statement, all variables and given/known data
A beam of length L is fixed on one end and roller supported on the other end. An axial force P is applied on the ends of the beam. The beam is loaded with a uniform distributed load (q) along its entire length. The beam has constant EI. Find an expression for the maximum second order moment, the maximum deflection and its location along the beam.

2. Relevant equations
The general solution for the problem is given as w''''(x)+k^2 w''(x) = q/EI
Where w(x) is the deflection of the beam
Axial force, P=k^2 EI
Moment, M=-EI w''(x)

3. The attempt at a solution
I have tried solving for w(x) = Asinkx + Bcoskx + Cx + D + p x^2 /(2P)
End conditions are:
For the pinned end, w(0)=0, w''(0)=0
For the fixed end, w(L)=0, w'(L)=0

I could not find a closed from solution for this problem. After eliminating A, B, C and D, I end up with a mess of sines, cosines and x. Software works if I substitute in arbitrary values for k and L, but I require an expression to answer the question.