Rotated Rafters intersecting Rotated Hip Rafters

Joe Bartok's Study on Rotated Rafters intersecting Rotated Hip Rafters.
Download Rafters Rotated To Roof Surface PDF file
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Rectangular Section Calculations

Main Slope Angle = 9/12

Adjacent Slope = 12/12

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:

D_DP = 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°

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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:

D_DP = 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 + D_DP = 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°)

Blade Bevel for ZDIH = 34.24496° (Blade Bevel for Rectangular Section)

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

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° (D_DP )

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)

Blade Bevel for ZDIH = 14.57152° (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)

Blade Bevel for XDIH = 5.36467° (Blade Bevel along 28.71825°)

Blade Bevel for YDIH = 55.55010° (Blade Bevel remains the same)

Blade Bevel for ZDIH = 17.57092° (Blade Bevel 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 XDIH = 34.32007° (Blade Bevel along 57.60620°)

Blade Bevel for YDIH = 34.44991° (Saw Blade Bevel remains the same)

Blade Bevel for ZDIH = 26.10840° (Blade Bevel for Rectangular Section)

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

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° (D_DP)

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)

Blade Bevel for ZDIH = 10.15094° (Blade Bevel for Rectangular Section