# Buckling shape

1. Jun 19, 2011

### dirk_mec1

1. The problem statement, all variables and given/known data
A uniform beam with stiffness EI, cross section A and length L is subjected to a force P. The beam is an ideal column. When the force P reaches a critical value and moves distance a downwards, the first buckling mode has the shape:

$$v(x) = C \sin \left( \frac{ \pi x}{L} \right)$$ for $$0 \leq x \leq L$$

Determine the constant C.

http://img710.imageshack.us/img710/6766/81286000.png [Broken]

2. Relevant equations
Use the assumption of small deflections and small rotations.

3. The attempt at a solution
I assume (as a first approximation) that the beam can't be shortened (e.g. no PL/(EA)). I can setup an integral from 0 to L-a equalling L relating the distance a to C but this integral cannot be expressed in elementary functions, am I doing something wrong?

Last edited by a moderator: May 5, 2017