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).