Designing without curvature

The tricky thing is that actual sheet material boats are often made with non-developable surfaces. For example consider a 4 plank sailing dinghy [1]; these generally include some twist from vertical stem to flared sides. It is easy to model a ruled surface that is not necessarily developable (see my post #28), but harder to be sure that real materials can be forced into that shape. Potentially there is a rule of thumb that can be applied to say whether a shape is close enough to developable, for example:

One could find the rate of twist along each ruled line [2] and set a maximum value for it, though I cannot think how to automatically generate this over the whole length of a surface.

[1] National 12 kit prototype / Build Progress Logs / Fyne Boat Kits Forum https://forum.fyneboatkits.co.uk/viewtopic.php?id=47
[2] this twist would be 0 for a developable surface, hence the normals falling on a plane.

 

Agreed. Conical surfaces are produced by connecting a focus (fixed point in space) with points on a chosen curve. Cylindrical surfaces are produced by applying the same chosen slope to all points on a defined curve. In the first case, calculations involve finding the coordinates of a third point given two points on a line (which will define its slope). In the second case, calculations involve finding another point on a line by projection given one point and a slope.

 

When a sheet of material of thickness t is envisaged for the developable surface, we have also to consider the material maxi elongation of the fibers : at each point there is a single flexure due to the local unique curvature radius R (as there is no orthogonal curvature by definition), and the material fiber maxi elongation is then :
dl/l= (t/2) / R (that comes from sigma = E (dl/l)=M/(I/(t/2)) and MR =EI)
- If R has steps (e.g. combination of 2 cylinders R1 and R2, or 2 cones, one cone and one cylinder, ... sharing the same ruled line at the frontier), the surface, although developable, cannot be followed exactly by the sheet of material without local cracks (or need to locally heat the wood to « fall » down the module d’Young E).
- if R function is continuous everywhere but with sharp peaks of max or min ( i.e.not derivable everywhere), that introduces overtensions. R Peak max = a local flatness between two curvatures. R Peak min : a local high curvature between two softer curvatures. >>> Acceptable but not ideal.
- if R function is smooth enough everywhere (i.e. continuous derivable), the real tension is no more than the one above from a single flexure. It is optimal surfaces to look for.

And in any cases, it is necessary to check also that you stay everywhere in the elastic limit for the considered material (Young modulus E, yield stress sigma y, thickness t)
sigma = E (t/2) / R << yield stress sigma y
>>> so in conclusion for curvature radius R : to be a smooth function everywhere (at least continuous) and no smaller that the value the material can support.
+ when the material is not isotrope : to use the easiest direction for flexure.

A last point, the edges of the panels under a flexure look like a bit « bevelled » due to the Poisson coefficient effect, i.e. the difference between the fibers in traction (usually the external face of the panel) which are shortened transversaly and the fibers in compression (usually the internal face of the panel) which are stretched transversaly.
>>> anyway practically, better to plan a slight oversized cutting of the panels and to adjust in situ the edges.

 

Looks like you are assuming a line in the material normal to the mid-surface stays normal to the mid-surface when the thin sheet bends - a fundamental assumption of basic beam and plate theory.

In a real material there will be local shearing of the material near the curvature discontinuity so that lines originally normal to the mid-surface don't stay normal to the mid-surface to accomodate the difference in strain due to the difference in curvature. The strain will be continuous but will not exactly follow the "normal lines remain normal" simplifying assumption. No need for cracks or plastic deformation.

Continuous curvature is desirable but lots of boats have been built with design surfaces which are do not have continuous curvature.

One approach is to determine an acceptable minimum radius based on material properties, fabrication methods, usage, etc. and then ensure the design surface curvature is no where less than that minimum.

 

Or just use Rhino to develop the shape using only "developable surfaces", then analyse stability in a different package.
Rhino will allow you to "unroll" the surfaces so that they can be cut from flat sheets.

 

As will DelftShip and (IIRC) Carlson Hull Designer or Hullform 9 and they're free.