- #1

Master1022

- 611

- 117

- Homework Statement
- Use the derived deflection equation to calculate the first buckling load?

- Relevant Equations
- Derived equation

Hi,

I was working through a problem about calculating buckling loads. The problem had the following geometry (I apologise for the poor drawing skills):

and the total length is ## L ##.

The boundary conditions are therefore:

1. ## y(0) = 0 ##

2. ## y(L) = 0 ##

After taking a cut and taking moments we can eventually get the following solution (which agrees with the answer):

$$ y(x) = \frac{-wEI}{P^2} cos(\beta x) + \frac{wEI}{P^2} \left(\frac{cos(\beta L) - 1}{sin(\beta L)} \right) sin(\beta x) - \frac{w}{2P} x^2 + \frac{wL}{2P}x + \frac{wEI}{P^2} $$

where ## \beta^2 = \frac{P}{EI} ##

but I really have no clue how to proceed by using this equation to find the first buckling load. Usually we find expressions for ## \beta ## that arise from the boundary conditions...

Any help is greatly appreciated. Thanks

I was working through a problem about calculating buckling loads. The problem had the following geometry (I apologise for the poor drawing skills):

The boundary conditions are therefore:

1. ## y(0) = 0 ##

2. ## y(L) = 0 ##

**My approach:**After taking a cut and taking moments we can eventually get the following solution (which agrees with the answer):

$$ y(x) = \frac{-wEI}{P^2} cos(\beta x) + \frac{wEI}{P^2} \left(\frac{cos(\beta L) - 1}{sin(\beta L)} \right) sin(\beta x) - \frac{w}{2P} x^2 + \frac{wL}{2P}x + \frac{wEI}{P^2} $$

where ## \beta^2 = \frac{P}{EI} ##

but I really have no clue how to proceed by using this equation to find the first buckling load. Usually we find expressions for ## \beta ## that arise from the boundary conditions...

Any help is greatly appreciated. Thanks