It's been a slow day on the oldtools list, so I had to fill out
   my dinner break with _something_... :-)

   So far we have a concrete estimate by the ISO of acceptable standards
   for plane sole flatness.  As far as I know, the derivation of that
   standard did not take into account sole flexure, i.e. bending of the
   sole of the plane under downward pressure applied to the plane,
   through the tote, while planing.

   It's not hard to get a reasonable idea of what effect that would
   have, for a cast iron bench plane, using standard structural
   analysis foo.  The exact analysis allowing for precise shape and
   so forth is tedious, but we can get a reasonable ballpark figure.

   So, consider a Stanley bench plane.  The principal elements to
   consider are the sole ( natch, ) and the sides.  The other stuff
   is more or less just floppin' around in there.  We'll assume the
   sole is concave from end to end by some measure, and that during
   use to sole is initially supported at the front and back ends,
   and moreover that the downwards force applied through the tote
   is centrally located, i.e. roughly in a spot equidistant from the
   front and back edges.  I'm thinking in particular of a Stanley #8,
   which will likely be the most extreme case where we could get
   flexure, and will constitute an upper limit of the amount of
   flexure we could get, for equivalent dimensions.

   All figures for cast iron.

   The Young's modulus, or modulus of elasticity, of cast iron, is:
      E = 152.3 GPa = 22.1x10^6 psi

   The rigidity modulus is similarly:
      G =  60.0 GPa =  8.7x10^6 psi

   The deflection of the sole plate alone will be:

      d(sole)  =  (WL)/(bh) * ( L^2 / (4E h^2 ) + 3/(10G) )

   and the deflection of the sides alone will be:

      d(sides) = (2WL)/(BH) * ( L^2 / (4E H^2 ) + 3/(10G) )

   where:

      W = force applied, in pounds.
      L = length of plane, in inches.
      b = width of sole, in inches.
      h = thickness of sole, in inches.
      B = thickness of a side, in inches.
     
   Also, H = a "typical height" of the side of the plane.  One might
   choose a value for this which is equal to the minimum value of the
   side height across the length of the plane.  OTOH, one might choose
   the maximum side height, which wouldn't be very realistic.  We'll
   compromise and define this to be just a little more than the 
   average side height out where it's not so high - we want to see
   what's the maximal flexure we can get realistically.

   Plugging in the values for E and G, we get:

      d(sole)  =  (WL)/(bh) * ( 1.13x10^-8 (L/h)^2 + 3.45x10^-8 ) inches,

      d(sides) = (2WL)/(BH) * ( 1.13x10^-8 (L/H)^2 + 3.45x10^-8 ) inches.

   Now consider our Stanley #8, for which we have the following data:

      L =   24 inches,
      b =    3 inches,
      h = 3/16 inches,
      B =  1/8 inches,
      H =    1 inch.

   Plugging in that data, we arrive at:

      d(sole)  = ( 7.9x10-3 W ) inches,  or 7.9W thou.
      d(sides) = ( 2.5x10-3 W ) inches,  or 2.5W thou.

   Remember that W is numerically equal to the downwards force in
   pounds, applied to the plane in use, i.e. whenever this _matters_
   for want of a better way of putting it.

   The sole and sides operate in unison, rather than independently,
   of course, so I'd tend to think we can safely estimate the total
   sole flexure at around 2W thou.

   So, we'd be led to believe that a downwards force of 10 pounds
   during planing, not totally unrealistic, would result in a
   downwards mouth deflection of around 20 thou, or around 1/64 inch.

   Make of this what you will.  It always has to be taken into
   account in considerations of sole flatness and its significance.

   For metal bench planes, this is likely gonna be a maximal figure,
   i.e. I doubt it would get any higher.  The #8 is a _big_ plane,
   and it's _long_.  OTOH, it's pretty sturdily built.  Nevertheless,
   shorter planes will have the point of applied pressure more and
   more further back from the centre point of the plane.  E.g., for
   a #3 or #4, you get relatively little pressure in the middle of
   the plane counteracting the natural concavity of the sole, and
   flexure there will be _significantly_ less, much less so than
   the above numbers and formulas would indicate methinks.

   I've given the formulas above, and if it amuses you, you can
   plug in numbers for other planes, or modify the reasoning at
   your will.  Again I emphasize, this is just a back-of-the-envelope
   calculation there, but anyone here is welcome to make it more
   precise, or certainly check my figures ( I was never very good
   at sums, let alone noticing typos. )

   The generalization to wooden bench planes, or more exotica, is
   left as an excercise for the reader.

   Doug Dawson
   dawson@p...