Wednesday, February 12, 2014

Tréteaux Dit Cadet Saw Blade Bevel Angle

I thought I was pretty good with prisms, but the foot of the Sawhorse has a prism footprint that's an parallelogram on the base as well as one of the sides. This SawHorse Leg -- Hip Rafter is rotated into the hip rafter backing angle.
tréteau à devers


Here's the development of the Sawhorse footprint.






The prism on the bottom of the SawHorse Leg.

Prism Unfolded

Top view of Prism.



Timber Framing Angles are meant to be used with Right Angled prisms, so these don't work.

Timber Framing Angle: R5B 
R5B = Hip-Valley Rafter ridge or plumb header miter housing angle following the bottom of the hip-valley rafter.
    HawkinDale Formula
  • none
    Hip-Valley Roof Ratios Angle Formulas
  • R5B = arctan( tan( R1 ) x cos (D))
  • R5B = arctan( tan( hip rafter slope angle ) x cos (plan angle))
Test angles for R5B
Hip Slope Angle  70.16263
plan angles without footprint rotation
42.04769°
47.95231°
plan angles with footprint rotation
43.13036°
43.36857°

  • R5B = arctan( tan( 9.39285 ) x cos (43.13036)) = 6.883666
  • R5B = arctan( tan( 70.16263 ) x cos (43.13036)) = 63.6956

Timber Framing Angle: A5B 
A5B = Saw Blade Bevel Angle for R5B.
    HawkinDale Formula
  • none
    Hip-Valley Roof Ratios Angle Formulas
  • A5B = arctan( sin( R5B ) x tan (D))
  • A5B =arctan( sin( R5B ) x tan (D))
Timber Framing Angle: R5P
R5P = Hip-Valley Rafter ridge or plumb header miter housing angle following the bottom of the hip-valley rafter.
    HawkinDale Formula
  • none
    Hip-Valley Roof Ratios Angle Formulas
  • R5P = arctan(sin D x tan R1)
  • R5P = arctan(sin(plan angle) x tan( hip rafter slope angle ))
Timber Framing Angle: A5P
A5P = Saw Blade Bevel Angle for R5P.
    HawkinDale Formula
  • none
    Hip-Valley Roof Ratios Angle Formulas
  • A5P = arctan( sin( R5P ) ÷ tan (D))
  • A5P = arctan( sin( P6 ) ÷ tan (P2))
  • A5P =arctan( sin( R5P ) ÷ tan (D))
  • A5P = arctan( sin(Clip angle on the bottom of the Jack Rafter ) ÷ tan (Jack Rafter Side Cut Angle))


Time for some head scratching to find the saw blade bevel angle for the foot of the sawhorse leg.

Since the hip rafter is rotated into the hip rafter backing angle the miter angle for the foot of the hip rafter should be the roof sheathing angle. 
Main Slope Angle 76.41401°
Adjacent Slope Angle 75°

Main Plan Angle = arctan (tan Adjacent Slope Angle ÷ tan Main Slope Angle)
Hip Rafter Slope Angle = arctan( tan( Common Rafter Slope Angle ) * sin( Plan Angle ))
Hip Rafter Backing Angle  = arccos (cos Common Rafter Slope Angle ÷ cos Hip Rafter Slope Angle)
Roof Sheathing Angle = 90 - P2 = arctan (tan Hip Rafter Slope Angle ÷ sin Hip Rafter Backing Angle)

Main Plan Angle = arctan (tan 75° ÷ tan 76.41401°) = 42.04769°
Hip Rafter Slope Angle = arctan( tan( 76.41401° ) * sin( 42.04769° )) = 70.15895°
Hip Rafter Backing Angle  = arccos (cos 76.41401° ÷ cos 70.15895°) = 46.20403°
Roof Sheathing Angle = 90 - P2 = arctan (tan 70.15895° ÷ sin 46.20403°) = 75.40155°


Main Roof Sheathing Angle 75.40155°
Hip Rafter Miter Angle at Foot of Hip Rafter = 75.40155°

Now for the hard part, the Saw Blade Bevel Angle.


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