How to make the deflection equation at any point along a snow ski profile?

[Skierman]

I am trying to figure out how to do a deflection profile for a snow ski profile at 50 mm increaments along the ski. Since the ski has different widths and heights at each crossection, the EI changes at each location making it a lot of 2nd order derivatives if I am not mistaken.

I am assuming that the ski thickness profile is a solid piece of wood so E is the same throughout the equation just to see if I can get to first base before I start adding in laminates and other material to the EI part of the beam equations.

The load across the beam is not quite centered so that throws another wrench in the gears. I am thinking that there will be about 33 crossections on the ski I am trying to do. The single load is 300 Newtons.

Is there there anyone that can lead me to a location that explains how I go about this or tell me how to approach this in some kind of spread sheet? Thanks

 

[erobz]

How is it supported. A diagram would be helpful?

 

[Skierman]

This would be a typical 3 point load with the load being slightly off center. The reaction points are at the
effective edge of the ski which is the wide points at each end of the ski. The load is approximately 55% from the wide point of the tip.

 

[erobz]

I suspect you are going to have to use the following:

$$ \frac{ \frac{d^2y}{dx^2} }{ \left[ 1 + \left( \frac{dy}{dx} \right)^2 \right]^{(3/2)} } = \frac{M(x)}{E I(x)} $$

When we encounter this equation in engineering, we typically can neglect $\frac{dy}{dx}$ for structural beams. However, a ski, with a 300 N ( 70 lbf ) load... I have wonder if the simplifying assumptions are no more?

Have you checked the bending stresses don't exceed the allowable stress for the desired loading?

As for the changing modulus of rigidity, unless you can write $I(x) = \frac{1}{12} B(x) h(x)^3$ , you are going to have to discretize it as you planned. I would try to write the shape functions, and solve the single resulting equation numerically, but maybe the FEA analysis isn't too bad? Its something I haven't personally performed.

 

[Skierman]

Thank you for you input