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

• I
• Skierman
In summary, the engineer is 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. The load across the beam is not quite centered so that throws another wrench in the gears. They are thinking that there will be about 33 crossections on the ski and the single load is 300 Newtons. Is there anyone that can lead him to a location that explains how he goes about this or tell him how to approach this in some kind of spread sheet? Thanks
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

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How is it supported. A diagram would be helpful?

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.

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• MSP 99 GRAPHICS OUTLINE 2018-19 TERMINOLOGY.pdf
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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.

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Thank you for you input

berkeman

## 1. How do you calculate the deflection equation at any point along a snow ski profile?

The deflection equation at any point along a snow ski profile can be calculated by using the formula: D = (E * I * w * L^3) / (48 * y), where D is the deflection, E is the modulus of elasticity, I is the moment of inertia, w is the distributed load, L is the length of the ski, and y is the distance from the neutral axis to the point of interest.

## 2. What is the significance of the moment of inertia in the deflection equation for a snow ski profile?

The moment of inertia is a measure of an object's resistance to changes in rotation. In the deflection equation for a snow ski profile, the moment of inertia represents the ski's cross-sectional shape and size, and it affects the ski's stiffness and ability to resist bending under load.

## 3. How does the distributed load affect the deflection of a snow ski?

The distributed load, which refers to the weight of the skier and any additional equipment on the ski, directly affects the deflection of a snow ski. The greater the distributed load, the greater the deflection will be at any given point along the ski's profile.

## 4. Can the deflection equation be used to determine the stiffness of a snow ski?

Yes, the deflection equation can be used to determine the stiffness of a snow ski. The stiffer the ski, the less it will deflect under load, and the higher the modulus of elasticity and moment of inertia will be in the equation.

## 5. Are there any other factors that should be considered when calculating the deflection equation for a snow ski profile?

Yes, there are other factors that should be considered when calculating the deflection equation for a snow ski profile. These include the ski's material properties, such as density and Young's modulus, as well as external factors like temperature and moisture, which can affect the ski's stiffness and deflection behavior.

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