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This blog is a companion to the Virgin Kiteboard Project.


Monday, May 23, 2011

Couple of useful references on simple bending

Good introduction with some worked examples:
http://courses.washington.edu/me354a/chap3.pdf


Good overview that puts the simple bending in context of more complex bending including shear forces and dynamic forces.
http://en.wikipedia.org/wiki/Bending

Another good source
http://en.wikipedia.org/wiki/Euler%E2%80%93Bernoulli_beam_equation

Sunday, May 22, 2011

A short reflection on the model so far

So far we've set up or model based on a simply implementation of the Bernoulli - Euler beam equation.

We're only considering bending due to moments created by forces applied in the vertical direction and have ignored bending due to shear forces acting on the board (which in most cases is reasonable). The model is a small deflection model (and we need to keep our eye on this) and is a based on linear elasticity ( i.e it assumes that the elastic properties can be modelled by simple springs).

According to the wikipedia source ( I have not confirmed ) the large diflection considerations only need to be applied when the radius of curvature at any point is less than 10 times the baord thickness. In our case this is going to less than 15 cm which obviously we would never reach.

Thinking about stiffness in terms of E,I and L, gives quite a usable way of thinking about designing flex. Although the model is (necessarily) simple you canhopefully see that it has already provides some great insights into the relationship between many of the board parameters that we play with when designing boards: E captures the contribution of the core materials, I the geometry including concave and channels, width thickness and L the length.

One glaring omission from this list is the laminate and layup schedule ( i.e how many layers, what orientation the fibres have, the type of resin and reinforcement etc). The answer fairly obviously lies in the the values and way you use E.


Unfortunately, its not quite as straightforward as constructing the value of 'I' by building up more complex shapes by adding and subtracting various pieces. While we do build up a piecemeal model of the 'E' structure of the core the laminate itself needs a simplifying model to make dealing with the complexities of the load transfer between the resin and reinforcement. We are of course going to choose the simplest model ( good old engineers approach!) called the Slab Model which will yield an 'effective' E for the composite material that we will be able to make use of in building up our 'jigsaw' of different 'E' materials.

Flexural Rigidity 'Stiffness'

In the last post we did a preliminary exploration of the Area Moment of Inertia which arises in beam bend theory as is colloquially referred to as a beam resistance to bending. This comes about because if you recall the Bernoulli-Euler beam equation, in our simple situation it can be integrated twice to give a second order differential equation (ie involving the second derivative of the deflection) and when you rearrange it you end up with

d^2w/dx ^2= 1/(EI) M(x)

where d^2w/dx^2 is the second derivative of the deflection w and M(x) is the moments (force times the distance to the force) that the cross-section at x are subject to.

The second derivative can be shown (for our model) to be equal to 1 over the radius of curvature at that point. That is, the radius of curvature at x is proportional to I. Therefore, the greater the value of I the larger the radius of curvature and hence the lower the amount of deflection (i.e stiffer board).

One example was give of introducing concave which added an additional positive terms to the I of a standard rectangular section and hence made the board stiffer. It can be demonstrated that when you add concave of about 95% of the thickness you double the 'stiffness'. Concave can be thought of as artificial thickness.

The resistance to bending is also proportional to E, the elastic modulus of the material. Again, stiffer material, higher E, less flex.

The product of E*I (x) is a very useful figure and is referred to as the flexural rigidity of the structure. More often than not this is what people mean when they refere to 'stiffness'. E and I are the primary parameters available for changing the stiffness of the board assuming that the length of the board is subject to other design constrains such as planing area or the like.

It is worth noting that the length of the board also makes a significant impact on the overall flex of the board.
To see this first consider the beam equation as its written above. Suppose that there is just a single point force being applied right at the tip of magnitude F and suppose that the half length of the board is L. Just for illustration purposes suppose that EI is constant along the length of the board ( that is no taper). In this case the moment at the center of the board is FL. Integrating the beam equation along the board from the middle to the tip of the board twice (remember its a second derivative involved) yields a deflection value (at the middle point) of

FL^3/(2EI)

What this shows is that the deflection is proportional to the length (in our case the half length) cubed.

Of course, if you do the same exercise for a point further down the board the effective value of L reduces (it is the distance from the point your considering to the tip)  and so the amount of deflection experienced by the slither of cross-section your looking at reduces and so the local radius of curvature tends to increase (i.e get flatter towards tips and hence why we need to thin the tips of the board out to make sure they flex and don't just transfer the moments they experience back to the center section of the board).

Monday, May 16, 2011

'I' - the Area Moment of Inertia of the Board Cross-section

The Area Moment of Inertia is a quantity that is used to calculate resistance to bending that is due to the geometry of the beams cross-section. I incorporates information about everything related to the cross-section shape including the stringers, laminate thickness, concave, board width and thickness and the rail material. The remarkable thing is that for any given structure I is a single number that can be calculated in a fairly straight forward way but encapsulates everything about the boards cross-section geometry at any given point that we will need to model the flex.

It is defined as the integral of the distance to the neutral axis squared over the area of the cross section at a given point on the axis. In our blog notation (and limited to the symmetrical cross-section we are looking at:

I(x) = Int (y^2.dA)
 
where Int is the surface integral over the cross-section at x, x is the location of the cross-section were considering, y is the vertical distance from the centroid to the infinitesimal area element dA. 
 
Another way to think about it is the sum of the vertical distance squared to every point on the face of the cross section. 

Remember that the neutral axis runs through the point in the cross-section where the stress is zero ( where the stress transitions from tension to compression). For a homogeneous material this point coincides with the geometric center of the cross-section which is also referred to as the centroid. 

Why is the fact that it’s the sum of the squared distances important? 

It relates to how much the board needs to flex in order to store the amount of energy that the external forces impart on the board. Stiffer boards will store more energy per unit of strain than flexible boards (remember strain is the change in a unit length of material). If you think about this as how much the top and bottom skins are stretches or contracts then you can see that this corresponds to how much the board needs to flex in order to suck up the energy imparted on it by the external forces. 

When you deal with elastic materials you frequently end up modeling things by using points connected by springs. These springs model the elastic behavior of the material.  The fundamental equation used to model a spring is Hooke’s law which says that the force that attempts to restore a compressed spring is equal to a constant (the spring constant for that spring) time the distance it has been compressed. 

The energy that is stored in that spring is equal to the amount of work you do to compress it. Work in this case is equal to the force by the distance that it moves through. Using Hooke's law this 

int (F dx) = int ( k.x dx) = ½ k x^2

where k is the spring constant. Thus the energy stored in the spring is proportional to the square of the amount its compressed. 

If you think about the face of a cross section as being a vertical bed of springs and suppose that the neutral axis is in the middle of that bed then the extent to which a spring at any point on the face of the cross section stretches or compresses depends on how far it is from the neutral axis. Summing up the squares of the this vertical distance then gives a quantity that is directly proportion to the energy stored in that slither of cross section. Integrate that throughout the entire board and you have the amount of energy stored in the board as a result of the external forces bending it. 

For example, for a rectangle centered on (0,0) of height t and width w

 I=t^3w/12

So if you have a wood core board where the cross-section is close to rectangular then I is proportional to the thickness cubed. That is a 10% change in the thickness of the core results in a 33% increase in I. 

Keeping in mind that the amount of energy that a thin cross-sectional element stores is determined by I, then if instead of a solid rectangular cross-section you had just a thin rectangular shell such as formed by the laminate around the core of a board, then you can calculate the I of this shell by simply subtracting the I of the inner dimension rectangle from the out dimension rectangle. Effectively what you are doing is subtracting off the energy storing capacity of the material you are removing and ending up with the energy storing capacity of the shell of material that remains. 

This is powerful because you can then calculate the I for more complex shapes by adding and subtracting pieces to get the shape you want. In the case of a kite board you can add a curved section to the top of a rectangle and subtract the same curved section from the bottom to get the I for a cross-section that has concave in it!!!!!

I of complex shapes

The example above of a rectangle about the origin glossed over the fact that the origin is the centroid of this shape. So to be strictly correct I should have said that the area moment of inertia of a rectangle about its centroid is …

However, what if we are constructing a more complex shape by putting together the pieces where the centroid is not at the origin? For example suppose that we have a strip of balsa wood laminate on the top of the board some distance d away from the origin. What would the Area moment of inertia be? 

Using our intuitive idea of I as representative of the energy storing capacity where the energy is stored in the stretched springs that we use to represent elastic material, then the balsa at a distance d from the origin should store more energy for amount of bending because being further from the neutral axis  is the more it will stretch more compared to a piece close to the origin. So we would expect that I should increase as we move the piece further from the neutral axis.

Indeed this is the case. Suppose that the translated object is location so that the centroid of the shape is a distance d from the origin. By replacing y->y-d and adjusting the integration limits (which will then be around the origin) you get the area moment of inertia of the shape whose centroid is located a distance d from the origin is

I(displaced) = Int(y^2 dA) - 2 d Int(y dA) + d^2 Int (dA)    where Int is the surface integral over the cross-section.

The first term is just the definition of the area moment of inertia about the x-axis. The second term is the definition of the centroid when the area A is symmetrical about the vertical axis. Because the co-ordinate transformation effectively relocates the centroid to the origin then this term is zero. The last term is just  d^2*A where A is the area of the cross section. This result is known as the parallel axis theorem.

So we get the result that the area moment of inertia about a point a distance d from the centroid of the shape equals the area moment of inertia about the x-axis plus a term equal to the distance square times the area of the shape. A corollary to this is that the area moment of inertia will have its minimum value calculated with about the centroid.

Constructing a shape using pieces that are not symmetrical about their x-axis (eg. A triangle, or semi circle or parabola)

In the above constructive approach we used the origin as the starting point and simply translated pieces up and down and either added them or subtracted them to get the geometry we wanted. By using the parallel axis theorem a value for I can be computed.

However, the Bernoulli-Euler equation requires us to know the area moment of inertia about the centroid of the object. So , we  must know whether the construction of the complex shape has led to the centroid moving away from the origin. If it has then we have to use the parallel axis theorem to take off the amount from the value of I we calculate that is due to the change in the location of the centroid.

So has it moved? If the cross-section is not symmetrical about some horizontal line drawn through the cross section’s centroid then the answer is probably yes.

Fortunately calculating the new centroid location is also straight forward. Remember that the centroid is calculated as

y(av) = Int (y dA)

That is, the average y value over the cross-section (remember we are only dealing with cross-sections that are symmetrical about the vertical axis).

The centroid of the new shape is calculated by taking the area weighted average of the centroid  of each piece. 

Concave Example

If we model the concave of a board with a parabola (that is ignore the third order term) the centroid of parabola that fits the width of the board and has a height (concave) of c turns out to be   

y(av) = 2/5 c.  

When you calculate the centroid location for each piece in the cross-section that represents a board with concave (parabolic concave) you end up with the centroid location being equal to 

Ytotal(ave) = 2/3 c. 

If the board has width w, thickness t and concave c then using the parallel axis theorem to construct the more complex shape and adjust for the change in the centroid’s location we get that the change in the area moment of inertia change due to the concave is:

4/45 c^2 t w  provided that c<t

Thus the for the area moment of inertia of the original rectangular cross-section is doubled when you add concave that is equal to about 96% of the board thickness.

The derivations for this calculation are useful in demonstrating the constructive approach we use for building up the cross section and will be shown in a later post.

The centroid and neutral axis of the cross-section


The centroid is the geometric center of a shape. If the material has a uniform density then the centroid will also be the center of mass as it is this point about which all the moments due to the weight of the material at each point net out to zero.

In our model we are only going to be dealing with cross-sections that are composed of uniform density material (the overall section we are dealing with may not be uniform but the component parts we use to build it up are uniform) and so the two concepts will coincide. 

As our cross-sections are going to be symmetrical about the vertical axis then the centroid will lay on the y-axis ( the vertical axis) and can be calculated as:

y(av) =1/A* Int(y dA)   

where ‘Int’ means integral with respect to area dA, y is the y-value of a point on the cross-section and A is the cross-section.

The centroid is the 'average y value' of the surface of the cross-section.

Intuitively, the top layer of a board (between the footpads) will typically be in tension and the bottom skin in compression (ie water pushing up between your stance). So, it follows that there must be a point somewhere between the two where the tension is zero. It turns out that for the types of materials and geometry we’ll be limiting ourselves to, this so called ‘neutral axis’ coincides with the centroid.

http://emweb.unl.edu/negahban/em325/11-Bending/Bending.htm

The neutral axis is important for our model because it is the point about which all moments due to the stresses in the cross section will attempt to rotate that cross section until these moments all net out to zero - that it will bend until the moments net out to zero as the moments due to the tension above the neutral axis net out with the opposing moments due to the compression of the material below the neutral axis. These internal moments arise as a reaction the externally applied moments . This is the main mechanism for bending that we shall model (pure bending).

We will see that the value of the moment of inertia (‘I’ in the Bernoulli Euler Beam question) depends on the location of the centroid and in fact that as we introduce concave the centroid of the cross-section moves and gives rise to increased stiffness.

Sunday, May 15, 2011

Starting point - Bernoulli-Euler Equation

These pages are retrospective and bring together the bits of theory I have picked up so far in the construction of BoardOff (the design spreadsheet for creating twin tips).

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.

Key Concepts

The typical approach to engineering analyses of just about anything is to divide the material (or space) into infinitesimally small units, typically cubes, and analyse the forces that the faces of these cubes are subjected to and then integrate over the entire volume to get the net effect of the infinitesimal forces. In beam bending problems you often introduce infinitesimally small spring to hold things together and again integrate up the stretch or compression of the springs or the stored energy in the spring to calculate quantities connected with the elastic deformation of the material.


Forces are vector valued functions meaning that they are defined by both the direction in which they act and their magnitude. In analyzing beam problems this directionality can become confusing because the forces are often named with reference to the particular plane in the beam that you are considering. This reference plance can change without notice!

Shear forces are forces that are act parallel to the plane you are considerind and direct or normal forces  are those acting perpendicular to the plane you are considering. This can get confusing if you lose track of the plane you are considering. 

For example, suppose you have a board laying horizontally. If you place your foot on the middle of the board and stand on it the force acts directly down on the board. In this case the force is a shear force relative to the vertical cross section of the board. However, if you were able to set it up so that you applied a force to the top surface of the board that pulled it to the right and a force to the bottom that pulled it to the left these opposing forces would also be applying shear forces but this time they are shear forces relative to horizontal slice of the board. 

The message is to always check the plane to which the force is being referred or it will become very confusing.

Stress - stress is defined as the force per unit area at a point in or on the material. Its units are force over area. That is N/m^2 (newtons per metre squared) which are also the units of pressure which is calculated as the force per unit area that one substance exerts on another.  N/m^2 is also referred to as a Pascal (Pa). Typical stresses in kite boards are in the order of 10^6 Pa or MPa.

Strain is the fractional change in the length of a material that is subject to a stress. It has no dimensions because it is a ratio of length over length.

Moment – a moment at a point is a quantity equal to the product of the external forces applied to the object by the distance from that force to the point. That is the force times the distance to it. Moments are important because in the same way that all forces must net out to zero if an object is stationery, so too must all the moments net out to zero if the object is not to be rotating (different way of saying the same thing). 

So if an external force is applied at a distance d from a point on the board then we know that to counter that force the board itself must oppose that moment with a moment of equal and opposite value. 

It does this by bending which in turn causes the board material to compress or stretch (ie store energy). These restoring forces at every point in the cross-section of the board create moments about the neutral axis (the axis where the stresses in teh cross-sectino transition from compressive to tensile) which eventually become equal but opposite to the moment bending it. At this point the board stops bending because the internal/ external forces are all balanced. Being able to calculate the moments created in the board as a reaction to the external forces will allow us to determine how much the board will bend.  

Moments have the units of force times distance or Nm (newton-metres)

Elastic Modulus  is an intrinsic property of a material that determines the strain that results from the stress applied to a material. It is defined as the ratio of the stress to the strain and typically represented by the symbol E. E is only a constant over a limited range of stress for any a particular material.  Beyond this ‘elastic limit’ the material undergoes other permanent deformation or failure. Eg snaps of permanently stretches.

The units of E are N/m^2 or Pa, the same as  (stress over strain). 
 
By way of example, e-glass has an E of around 70 GPa (giga pascals or 10^9 Pascals),  PVC foam for cores comes in at around 30 MPa and carbon fibre can be up to 200 GPa. The larger E the smaller the strain (ie the amount it stretches) when subject to a stress. Carbon’s vey high E value means that it does not flex much and so appears to crack without warning. 

It’s important to note that Elastic modulus can vary in different directions in the material. Isotropic materials are ones where E is not dependant on direction and Anisotropic is where is does depend on the direction.

Most of the materials we’ll deal with in board making are isotropic materials.  Even unidirection fiberglass. The actual glass material itself is isotropic but the way that the fibres are held together means that the cloth has properties that depend on direction but this relates to the contruction and not the material properties. 

The elastic modulus, as we shall see, is fundamental to our efforts to model the boards flex. This is because the model of bending (due to moments) that unpins the Bernoulli-Euler equation is one where we think of the face of each cross-section as being covered in infinitely small springs that are either compressed or stretched by the bending action. The elastic modulus tells us how much these springs will stretch or compress when the board is subject to a bending force.

The concept of the elastic modulus is straight forward but there are some complexities introduced when we look for the elastic modulus of structures that use the materials in complex ways. The example of uni-direction glass was mentioned. However, even the case of normal bi-directional glass which has glass fibres running in a woven cloth at 90 degrees to each other makes for some more challenges as the off-axis properties of the structure must be calculated by resolving teh forces into components in the direction of the fibres and also taking account of the shear forces that may be created as a result of these perpendicular forces acting simultaneously.

It should be noted that the elastic modulus is an intrisic property and is not affected by the shape of the cross section. The influence of the shape of the cross section is captured in the Area Moment of Inertia which is often refered to as a 'the measure of a beams resistance to bending'. It takes account of the shape ( demensions, concave etc) and captures it in a single figure. The Area Moment of Inertia (which is sometimes erroneously refered to as the moment of inertia but should not be mixed up with the moment of inertia used in dynamic problems) requires some detailed treatment and so will be defered until a subsequent post.

Wednesday, May 11, 2011

What's this all about?

The purpose of this blog is to record what I learn about how to model a kiteboards mechanical properties so that better design decisions can be made when DIY'ing a kiteboard. The end outputs from this will ideally be:

  • the development and refinement of a model to be able to put some numbers around design questions in the vein of  'all else being equal, if I change this but that much, what happens to this other thing'.
  • the development of builders rules of thumb: 'if I take 1.5mm of core thickness off I should increase the quantity of glass by one third'.
  • (aspirational) predictions of board behaviour in the wild.

More than anything, modelling the properties of the board is just plain fascinating. It draws on theory developed for beam bending, composite materials, numerical mathematics, continuum mechanics and of course kiteboarding technique.

My hope is that this will evolve into a useful tool for designing kiteboards and that in the hands of other more experience than me in the dark art of backyard board building ('BBB') will be able to add some practice 'margins of safety' to take the theory off the page and make it a tool that helps make better design decisions and avoid breaking the boards or building a dud.

I have already started creating a spreadsheet that models the flex and the stresses in the outer skins. Its accuracy is still to be tested but the vast amount of google searches I've done to pull this together warranted recording for posterity and hopefully for the benefit of other BBB with an interest in theory (do they exist......big question)

The models are necessarily starting out simple because... well that's much easier than doing a not-simple model. Whether it evolves into something more complex or remains an FYI project remains to be seen. However, I'm pretty pumped about the snippets of insight that might come out of it on the way.

Initially the model will focus on modelling the flex of the board and predictions about its performance under test conditions.  The aspects of handling, the stuff of prototypes and underpaid pro's, will hopefully come out of the boards I build and document on the companion site to this Virgin Kiteboard Project.

You you do find this blog interesting or useful please let me know!!!! I hope you  enjoy!