Setting up the problem
In order to keep things simple I am only going to have a test set up where
i) the board is laid horizontally (along the x axis)
ii) the test loads (which are either a point load at the tip or a distributed load) acts only in the vertical direction (the + y axis) and the board is symmetrical about its center line when viewed from above (in the x-z plane) and each cross-section when viewed from the end is symmetrical (in the x-y plane)
These assumptions help keep the complexity of the governing equations down by reducing the number of dimensions that need to be separately dealt with.
As I'll only be looking at loading in the vertical direction this means that model won't be useful for torsion calculations. This will have to be dealt with in a future version once this simpler model has been worked through.
The Starting Point - The Bernoulli-Euler Beam equation.
A twin tip board is basically a thin plank of material wrapped in a composite material. In engineering terms it is nothing more fancy than a flat beam and this has been studied systematically from around 1750 when two brains of the time, Bernoulli and Euler, wrote down the first set of equations to model small deflections of a beam due to bending moments. The good news is that all the theory we could every wish for is out there and our job is largely to simplify it without violating too many assumptions and tailor it to our setups.
The Beam Equation is based on simplifying assumptions of linear elasticity theory and only applies when the forces are work are purely bending moments. It does not take account of bending due to large shear forces. Further, because it is based on linear elasticity it only applies for small difflections of the board. In our case this is sufficient for our current modelling effort as our test bed is the bending of a board by a force applied at the tip and we are concernced with how much it flexes at the center and the tip.
The assumption of small difflections is a bit harder to verify. It appears to arise from what called the small angle approximation in which the tan of an angle approximately equals the angle. One can use Taylor Series expansions ( a polynomial that approximates the function) to show that the first 2 terms of the approximation ( which when you include an infinite number of terms is exact) is
tan x = x+x^3/3+..
If x is small x^3 is really small and if x is small enough x^3 terms an higher can be ignored without introducting too much error.
In our board design we are typically dealing with very large radii of curvature along the length of the baord - several meters. Thus the angles through which our boards will bend are relatively small. However, this means that the model will be of limit use in determining failure which would typically occur when local radius of curvatures are quite small and so the angles they deflect through are large and so we can no longer ignore the higher order terms.
The Bernoulli-Euler beam equation is great starting point for our model as it brings together the material properties, geometry of the cross-section and dimensions of the board into a single equation that allows us to compute the deflection, stress, strain and shear forces at any point along the board. It not only has a static form (corresponding to applying a test load and holding it there) but there is a dynamic version which allows us to model the response of the board to time varying forces such as choppy water or a hard slap down from a terrifyingly big unhooked sent jump with an out of control landing. Ta da!!! Our work here is done!
Add a bit of the theory of composite materials and we are well on our way.
The Bernoulli Euler beam equation.
The diflection of a beam w that is subject only to loads perpendicular to the neutral axis (ie no axial loading) is given by
where
E- is the elastic modulus of the material (assumed to only varying in the x-direction) units N/m^2 or Pa
I - the area moment of inertia of the cross-section of the board at x: units m^4
w - the displacement of the board away from the x-axis - units m
q - the loading force applied in the y-direction. This may be a function of x in when it is a distributed load. i.e q= q(x) : units N
The elastic modulus E is an intrinsic property of a material (that is doesn't depend on the geometry). It is the ratio of the stress in the material ( the force per unit area the material is subject to) over the strain in the material ( the ratio of the change in length over the original length). E is a constant for any homogeneous sample of material over a range of stresses. However, beyond these elastic limits E is not constant. For example when a material permanently stretches and does return to its original length. This means we must must be aware that some of the test scenarios we run may push the material into non-elastic behaviour. Practically how do you do this? Theoretically the values of the elastic limits are known an we will have to rely on these in the absence of any test pieces to calibrate it.
The geometry of the cross-section is taken into account in the 'I' factor or area moment of inertia. The moment of inertia in our case (for homogeneous materials symmetric about the y axis) about a given point is the sum of the squares of the vertical distance from that point to every other point on the cross-section.
I(p) = INT (y^2 ) dA
where INT () is the integral with respect over the cross-section area A. More will be explained on I in later posts. For the moment it is sufficient to know that quantity colloquially represents the resistance of a beam to bending.
A related concept is that of the neutral axis. When you bend a board in the middle (so bend the tip up) the bottom skin of the board stretches (tension) and the top is compressed so at some point in the cross section the tension drops to zero. The plane along which the tension is zero is referred to at the neutral axis and is important in the calculation of stress and strain in the boards laminate as its the distance from this plane that will determine how much outer layers will stretch or compress as the board and bends.
There a few assumptions that are built into the derivation of the Beam Equation that seem fairly academic but may have practical implications. They relate to how the behaviour of the normals to the neutral axis as the beam bends i.e the lines that lie in the cross-sectional planes of the board ( y-z). Here's a paraphrased version
i) the normals remain normal to the neutral axis as the neutral axis bends
ii) the normals do not stretch (no distortion in the direction normal to the neutral axis)
iii) the normals remains straight ( the cross-sections remain flat despite rotating the same amount as the neutral axis)
These assumptions derive from the way that bending is modelled. Each infinitesimal rectangular section (length dx and height equal to the core thickness) distorts into a trapezoidal shape.
Assumption (i) is of particular interest because it requires that there is negligible net force in the y-direction at any one point. That is to say that the shear force (the force acting parallel to the faces of the cross-section) must be negligible for the equation to be accurate. If the shear force was large the effect would be to cause the vertical faces of these infinitesimal rectangular pieces to be displaced in the vertical direction relative to each other (become a diamond shape). This leads to displacement in the beam that is not taken into account in the B-E equation. However, because we are operating in the elastic region of the material, it is possible to superpose the displacement affect of the shear force on to that due to the B-E displacement by summing the results.
The magnitude of the shear forces that a kiteboard is exposed to will need to be explored further in the future to see if it should be included in the model.
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