## Friday, April 11, 2014

### Rotated Rafters intersecting Rotated Hip Rafters

Joe Bartok's Study on Rotated Rafters intersecting Rotated Hip Rafters.
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Rectangular Section Calculations

Main Slope Angle = 9/12

Corner Angle = 90°

.............................

SS = 36.86990°
DD = 53.13010°
P2m = 30.96376°
C5m = 21.10020°
R1 = 30.96376°

S = 45.00000°
D = 36.86990°
P2a = 43.31386°
C5a = 34.44990°
R1 = 30.96376°
...........................
The 9/12 Side of the central (30.96376°) Hip Rafter is rotated to the roof surface
Dihedral Angle between Hip upper shoulder and Roof Surface on 12/12 Side
= 90° – (C5m + C5a) = 90° – (21.10020° + 34.44990°) = 34.44990°
.............................

Rotated Rafter Offset on 9/12 Eave = 60°
Rotated Rafter Slope Angle = arctan (tan 36.86990° sin 60°) = 33.00449° Rotated Rafter Backing
Bevel = arctan (sin 33.00449° ÷ tan 60°) = 17.45760° Rotated Rafter P2 Angle = arctan (cos
36.86990° ÷ tan 60°) = 24.79128° Rotated Rafter 90° – P2 Angle = arctan (tan 60° ÷ cos 36.86990°) =
65.20872°

Rotated Rafter Peak Calculations:
Sum of P2 Angles at Rotated Rafter Peak = 30.96376° + 24.79128° = 55.75504° Blade Bevel along Sum
of P2 Angles = 0°
Angle on Adjoining Face = 90°
Blade Bevel for Adjoining Face = 90° – 55.75504° = 34.24496° Trace Angle = arccos (cos 55.75504°
cos 90°) = 90°
Rotated Rafter Foot Calculations:
DDP = arctan (sin 33.00449° tan 17.45760°) = 9.72052° Angle on shoulder of Rotated Rafter @ DP Line
= arccos (cos 33.00449° cos 9.72052°) = 34.25020°
Blade Bevel = arctan (sin 34.25020° ÷ tan 65.20872°) = 14.57152° Angle on adjoining shoulder of
Rotated Rafter
= arctan (sin 34.25020° ÷ tan 14.57152°)
= 65.20872° = 90° – P2
Blade Bevel along 90° – P2
= DD = arctan (sin 65.20872° ÷ tan 34.25020°)
= 53.13010° = 90° – SS
Rotated Rafter Footprint Angle
= arccos (cos 65.20872° cos 34.25020°) = 69.72052° Offset Angle + DDP = 60° + 9.72052°

.............................

Rotated Rafter Offset on 12/12 Eave = 70°
Rotated Rafter Slope Angle = arctan (tan 45° sin 70°) = 43.21918°
Rotated Rafter Backing Bevel = arctan (sin 43.21918° ÷ tan 70°) = 13.99545° Rotated Rafter P2 Angle
= arctan (cos 45° ÷ tan 70°) = 14.43276°
Rotated Rafter 90° – P2 Angle = arctan (tan 70° ÷ cos 45°) = 75.56724°

Rotated Rafter Upper Claw Calculations:
Sum of P2 Angles at Rotated Rafter Peak = 43.31386° + 14.43276° = 57.74662° Blade Bevel along Sum
of P2 Angles
= C5m + C5a = 21.10020° + 34.44990° = 55.55010°
Angle on Adjoining Face = arctan (sin 57.74662° ÷ tan 55.55010°) = 30.11986° Blade Bevel for
Adjoining Face = arctan (sin 30.11986° ÷ tan 57.74662°) =
17.57093°
Trace Angle on shoulder of 30.96376° Hip Rafter arccos (cos 30.11986° cos 57.74662°) = 62.50910°

Rotated Rafter Lower Claw Calculations:
Sum of P2 Angles at Rotated Rafter Peak = 43.31386° + 14.43276° = 57.74662°
Blade Bevel along Sum of P2

Blade Bevel for Adjoining Face = arctan (sin 50.95245° ÷ tan 57.74662°) =
26.10841°°
Trace Angle on shoulder of 30.96376° Hip Rafter arccos (cos 50.95245° cos 57.74662°) = 70.35508°
Rotated Rafter Foot Calculations:
DDP = arctan (sin 43.21918° tan 13.99545°) = 9.68589° Angle on shoulder of Rotated Rafter @ DP Line
= arccos (cos 43.21918° cos 9.68590°) = 44.08148°
Blade Bevel = arctan (sin 44.08148° ÷ tan 75.56724°) = 10.15094° Angle on adjoining shoulder of
Rotated Rafter
= arctan (sin 44.08148° ÷ tan 10.15094°)
= 75.56724° = 90° – P2
Blade Bevel along 90° – P2
= DD = arctan (sin 75.56724° ÷ tan 44.08148°)
= 45.00000° = 90° – S
Rotated Rafter Footprint Angle
= arccos (cos 75.56724° cos 44.08148°) = 79.68589°
= Offset Angle + DDP = 70° + 9.68589°

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NR Section Calculator
The rafters are treated as if the bevel were applied to the side face (the face not set
in the roof surface).
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9/12 Side Rotated Rafter Peak
μ  = 90° (Angle on Rectangular Section)
ß  = 30.96376° + 24.79128° = 55.75504° (Sum of P2)
α  = 17.45760° (Backing Bevel)
MIT = 78.45754° (Angle on Plumb Face of Rotated Rafter)
BEV = 69.17126° (Trace Angle on face of 30.96376° Hip)
= 20.82874°
Projected Right Angle = 90° (Trace Angle for Rectangular Section)
Supplementary Angle = 90°
Blade Bevel for XDIH = 32.46690° (90° – 55.75504° ... Blade Bevel remains the
same)
Blade Bevel for YDIH = 0° (Dihedral Angle between faces of 30.96376° Hip =
90°)
.............................
9/12 Side Rotated Rafter Foot
μ  = 34.25020° (Angle on Rectangular Section)
ß = 65.20872° (90° – P2)
α  = 17.45760° (Backing Bevel)
MIT = 33.00449° (Rotated Rafter Slope Angle)
BEV = 60.00000° (Rotated Rafter Footprint)
= 9.72052° (DDP )
Projected Right Angle = 110.27948°
Supplementary Angle = 69.72052° (BEV + = 60° + DDP)
Blade Bevel for XDIH = 0.00000°
Blade Bevel for YDIH = 53.13011° (90° – SS ... Blade Bevel for Rectangular
Section)
.............................
12/12 Side Rotated Rafter Peak – Upper Claw
μ  = 30.11986° (Angle on Rectangular Section)
ß = 43.31386° + 14.43276° = 57.74662° (Sum of P2)
α  = 13.99544° (Backing Bevel)
MIT = 28.71825° (Upper Claw Angle on Plumb Face of )
BEV = 55.50766° (Trace Angle on face of 30.96376° )
= 7.00143°
Projected Right Angle = 117.49090°
Supplementary Angle = 62.50910° (Trace Angle for Rectangular Section)
.............................
12/12 Side Rotated Rafter Peak – Lower Claw
μ = 180 – 50.95244° = 129.04756° (Angle on Rectangular Section)
ß = 43.31386° + 14.43276° = 57.74662° (Sum of P2)
α  = 13.99544° (Backing Bevel)
180° – MIT = 57.60620° (Lower Claw Angle on Plumb Face of )
BEV = 96.49999° (Trace Angle on face of 30.96376° )
= 13.14492°
Projected Right Angle = 70.35508° (Trace Angle for Rectangular Section)
Supplementary Angle = 109.64491°
Blade Bevel for YDIH = 34.44991° (Saw Blade Bevel remains the same)
.............................
12/12 Side Rotated Rafter Foot
μ = 44.08148° (Angle on Rectangular Section)
ß = 75.56724° (90° – P2)
α = 13.99544° (Backing Bevel)
MIT = 43.21918° (Rotated Rafter Slope Angle)
BEV = 70.00000° (Rotated Rafter Footprint)
= 9.68589° (DDP)
Projected Right Angle = 100.31411°
Supplementary Angle = 79.68589° (BEV + = 60° + DDP)
Blade Bevel for XDIH = 0.00000°
Blade Bevel for YDIH = 45.00000° (90° – S ... ... Blade Bevel for Rectangular
Section)