I've been doing a bit more upgrading of the design workbook that I've used for board #2. Previously I modelled the flex in the board by ignoring the concave and just modeling it as a flat rectangular cross section with the laminate and the core modelled separately because the elastic modulus of the two are orders of magnitude different.
The latest version of the worksheet incorporates the concave profile along the length of the board.
The key quantity that you need to calculate to determine the flex profile is the Second Moment of Area which, because the cross-section of the board is symmetrical, is equal to the average value of y^2 over area of the cross section at a point along the board. The Flexural Rigidity is defined as the Elastic modulus of the material times the second moment of area. To work out the radius of curvature at a point on the board due to a force applies at (say) the tip, you divide the value of the moment ( force x distance from the point of interest to the force) divided by the Flexural Rigidity. the formula is
1/ R = M/EI
where R - radius of curvature at the point x
M - is the moment at x
EI - flexural rigidity ( E- elastic modulus, I - SMOA)
You can use this to plot out the curve that the board makes when it bends under various forces.
The SMOA for a rectangular section is
I = t^3*w/12 - (1)
where t-is the thickness, w is the width. The equations in the DIAB Sandwich Handbook for a closed beam cross section comes straight from this.
After a lot of algebra and googling around beam bending theory sites, the correction term that I came up with for the SMOA when the rectangular section has concave of 'c' is
4/9*t*c^2*w - (2)
To get an idea about the size of this term, when concave is about 50% of the core thickness, this term equals the value of (1). That is, it doubles the flexural rigidity. Bearing in mind that the concave is not constant throughout the length of the board and typically tappers of at the tips you need to do the full calcs to see just what it does to the total flex of the tip.
When I loaded this term into Board-Off (the design worksheet) for the current board, with 7mm concave tapering to zero at the tip then it reduce the amount the tip flexed when the board is bent from the middle was about 22% compared to no concave.
The flex of the tip (from the footpad to the tip) was negligible which corresponds with concave being just 1-2mm at that point.
So, if these calcs are correct then the mid section stiffness is quite sensitive to concave. I've been told that recently the trend is to no concave in the boards and I wonder if it might be that the increasing use of carbon which incredibly stiff and strong negates the need to have concave for strength reasons. ...
To really make any use of the flex calcs the model still needs to be calibrates with real figures for the elastic modulus of the laminate. I'm using a figure that I have no real point of reference so once the board is done I'll be able to put the theory to work in designing number 3.
Below is the flex profile for 20mm concave. The thin black line shows the radius of curvature from the tip to the middle of the board.
The bulge in the curvature line towards the left hand side means that board is getting flatter ( big radius = flatter section). One possible reason for this is that with 20mm concave and a board thickness of just 12mm the neutral line ( the centroid) which is where the stress in the board changes from compression to tension. When the board has no rocker this line is about in the middle of the board cross section. In this situation you have one side of the board in tension and the other in compression and the characteristics of the laminate are different in these two situations. However, when the centroid lies outside the board it means that board sides will be tension at the same time when you try to bend it in the middle.This means a lot more stiffness and strength. As you progress to the tip and the concave reduces to nothing the centroid as some point falls inside the cross-section and you sides start working together in a complementary way rather than the co-operative way when board are in tension simultaneously.