bending math

I worked out some equations last winter for my skiff design computer
program and borrowed some others and ony got around to putting them on my
webpage for other people to use today in file

www.ncf.ca/~ag384/BendingMath.txt.

Here's a copy of what's in that file.

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When the curve of a deck beam is the arc of a circle, or the bend
in the side of a plywood boat hull (eg dory, priogue, skiff,
catamaran) is the arc of a circle the equations below can be
used.

These letters are used in the equations:

"L" is the length of the deck beam or of the line between the
ends of the bent plywood panel, called the chord of the arc

"H" is the height of the crown of the beam or the width of the bend
in the plywood panel, called the height of the arc

"R" is the radius of the circle of which the deck beam or the bent
plywood is the arc, called the radius of curvature

"A" is the angle enclosing the arc at the centre of the circle

"l" is the length of the arc

"a" is the area under the arc, the area of the side of the beam or
of the space under the curve of the plywood panel.

all of the measurements have to be in the same units, i.e. all
inches, all feet, all centimeters, all cubits, all ells, or whatever.

all of the angles are measured in radians, not degrees. to get
degrees multipy by 3.14159/180 or 0.01745.


Then:


R = H + LxL L = sqrt( 8xH ) H = R - sqrt( RxR - LxL )
--- ----- (--------) ( --- )
2 8xH ( R - H ) ( 2 )
( ---)
( 2 )


A = 2 x arcsin( L ) l = RxA a = RxR x (A - sin(A))
(-----) ---
( 2xR ) 2

where "sqrt(...)" means take the square root of what's in the (...)

"arcsin(...)" means look up the arcsine of what's in the (...)
in a trigonometry book

"sin(...)" means look up the sine of what's in the (...)
in a trigonometry book

"arcsin" ameans "the angle whose sine this is" so it can
be found from a table of sines

or use a computer program like a spreadsheet which can
calculate square roots, sines, and arcsines


Some uses:

"R" can be compared to the bending radius of the plywood to see if
it will bend to the shape of the hull.

"a" can be multiplied by the thickness of the beam to find its
volume and used with the weight of the wood to get the weight of
the beam.

"l" gives the length of the plywood panel to make the skin of the boat
or to cover the deck beam.


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