Wednesday, April 16, 2014

Math Notes #1

Roof Eave Angle = 90.00000
SS = Main Rafter Slope Angle = 36.86990
S = Adjacent Rafter Slope Angle = 45.00000
DD = Main Plan Angle = 53.13010
D = Adjacent Plan Angle = 36.86990
R1 = Hip Rafter Slope Angle = 30.96376
C5m Main Hip Rafter Backing Angle = 21.10020
C5a Adjacent Hip Rafter Backing Angle = 34.44990
Valley Sleeper Saw Blade Bevel Angle = 90° - (C5m + C5a) = 34.44990
P2m = Main Jack Rafter Side Cut Angle = 30.96376
90° - P2m = Main Roof Sheathing Angle = 59.03624
P2a = Adjacent Jack Rafter Side Cut Angle = 43.31386
90° - P2a = Adjacent Roof Sheathing Angle = 46.68614


R8-DP = Tilted Hip Rafter Slope Angle on DP Side = arctan( tan ( Hip Rafter Backing Angle) ÷ tan (Jack Rafter Side Cut Angle ))

C8 = (C5m + C5a) = 55.55010
R8-DP = arctan( tan ( C5m )  ÷  tan ( P2m ))
R8-DP = arctan( tan ( 21.10020 )  ÷  tan ( 30.96376 )) = 32.74587

R9-DP = Horizontal Plane Rotation Angle for Tilted Hip Rafter on DP Line from Footprint to Hip Rafter Run Line
R9-DP = arctan( sin ( R1 ) × tan ( C5m ))
R9-DP = arctan( sin ( 30.96376 ) × tan ( 21.10020 )) = 11.22889

D-DP = Horizontal Plane Rotation Angle for Tilted Hip Rafter on DP Line from Eave Line to Hip Rafter Footprint
D-DP = D - R9
D-DP = 36.86990 - 11.22889 = 25.64101

D-DP = arctan((cos SS ÷ tan DD) ÷ (1 ÷ cos SS))
D-DP = arctan((cos 36.86990 ÷ tan 53.13010) ÷ (1 ÷ cos 36.86990)) = 25.64101

R10-DP = Vertical Plane Rotation Angle from Plumb for Tilted Hip Rafter on DP Line
R10-DP = arctan((cos(R1) × tan(R9)) / sin(R1));
R10-DP = arctan((cos(30.96376) × tan(11.22889)) / sin(30.96376)) = 18.30847


Rotation Angle = 70°
S = Adjacent Rafter Slope Angle = 45.00000
Rotated Rafter Slope Angle = arctan(tan(Rafter Slope Angle) × sin(Rotation angle))
Rotated Rafter Slope Angle = arctan(tan(45.00°) × sin(70°)) = 43.219179°

sin 18.30847° = b = 0.314132

90° - R9-DP = C

90° - 11.22889° = C = 78.77708°

A = D -( 90° - Rotation Angle)

A = 36.86990° -( 90° - 70)= 16.86990°

B = 180° - A - C = 84.35302°

Law of Sines c = ( b × sin( C ) ) ÷ sin( B)

c = ( 0.314132 × sin( 78.77708° ) ) ÷ sin( 84.35302°) = 0.30962837

Rafter Vertical Plane Rotation = arctan(c ÷ cos R10-DP)

Rafter Vertical Plane Rotation = arctan(0.30962837 ÷ cos 18.30832° ) = 18.06258°

Rafter Upper Claw Angle = (90° - Rafter Slope Angle) - Rafter Vertical Plane Rotation

Rafter Upper Claw Angle = (90° - 43.219179°) - 18.06258° = 28.71824°




Long version of trigonometry for the upper claw angle

R10-DP = arctan((cos(30.96376) × tan(11.22889)) / sin(30.96376)) = 18.30847
90° - R10-DP = Vertical Rotate Angle
90° - R10-DP = 71.69168°
(Plan Angle + Purlin Rotation Angle) - 90° = Horizontal Rotate Angle
(36.86990° + 70°) - 90° = 16.86990°
90° - 18.30832 = 71.69168°
cos 71.69168° = 0.314132
R9-DP = 11.22889
90° - 11.22889° = A'
90° - 11.22889° = 78.77708°
 sin A' x cos Vertical Rotate Angle = a
 sin 78.77708° x 0.314132 = 0.3081257
180° -  78.77708° - 16.86990° = B'
180° -  78.77708° - 16.86990° = 84.35302°
a ÷ sin B'  = b
a ÷ sin 84.35302°  = 0.30962837
 0.3081257 ÷ sin 84.35302°  = 0.30962837
 a ÷ sin B'  = b
sin 71.69168 = c
sin 71.69168 = 0.9493798
arctan(b ÷ c) = 18.063089°
arctan(0.30962837 ÷ 0.9493798) = 18.063089°
Rafter Upper Claw Angle = (90° - 43.219179°) - 18.063089° = 28.71824°


Law of Sines Formulas

Law of Sines Formulas
C = 180° - (A + B)
a = ( c × sin( A) ) ÷ sin( C)
a = ( b × sin( A ) ) ÷ sin( B)
a = ( b ÷ sin( B ) ) × sin( A)
b = ( c × sin( B ) ) ÷ sin( C)
b = ( a × sin( B ) ) ÷ sin( A)
b = ( a ÷ sin( A ) ) × sin( B)
h = b × sin( A )
h = a × sin( B )
c = ( a × sin( C ) ) ÷ sin( A)
c = ( b × sin( C ) ) ÷ sin( B)


Purlin Rafter Foot Print

Tony sent me a drawing, showing the development of the purlin rafter foot print in plan view, that develops the purlin rafter lines on the draw down roof surface. I haven't drawn the purlin rafter foot print for about a year. So this drawing was just a refresher for the technique used to draw the purlin rafter foot print in plan view. I see the purlin rafter foot print drawn in plan view as more of an artistic type of drawing.


Just a couple of lines in plan view develops the purlin rafter on the draw down roof surface.



Sunday, April 13, 2014

European Championship for Carpenters task model 2014.

This current European Championship for Carpenters task model is extremely complex. I wonder if the apprentice carpenters had the drawings before the competition? Or how many of them actually completed the task model? 

Here's a couple of drawings of the European Championship for Carpenters task model 2014. I'm still not done, but you can get an general idea of how much geometric roof framing knowledge each of the apprentice carpenters must know to complete the task model.

3D drawing of the European Championship for Carpenters task model. I'll use this drawing to verify the geometry I develop is correct.

Basic ground plan of the roof.


Technique to develop the eave line that is level in plan view.

3D drawing of the hip rafters and raking/sloping plate lines.

3D drawing with plumb lines to establish the location of the jack rafters and purlin rafters on the roof surface.

Drawing of the base that holds up the roof.

Front sloped legs of the base.


Drawing with the legs and plates of the base.


Trying to establish the rotation of the hip rafter rotated into the roof surface.

Developing the profile rafter slope angle on the sides of the task model.

Raccord drawing of the roof surface.



Raccord drawing of the roof surface with the rafters on the roof surface.

Trying to develop the correct location of the rafters on the roof surface.
Drawing showing the technique I used to establish the miter line on the roof surface using the hip rafter backing triangle and the depth of the profile rafter. 


Drawing with the hip rafter foot print developed from the hip rafter material block rotated to the hip rafter backing angle.

Drawing showing the development of the Devers De Pas line for the DP side of the hip rafter rotated into the roof surface. Sure am glad I studied the Treteaux angles.

Transferring the lines to the sides of the timbers on the rof surface.



Drawing showing the miter and claw lines on the roof surface. I got the miter and claw lines dimensions from my RafterTools app for iPhone or Android.

Tréteaux Angles
tréteau à devers

Roof Eave Angle = 126.58678
SS = Main Rafter Slope Angle = 38.65981
S = Adjacent Rafter Slope Angle = 37.05220
__________________________________________________
DD = Main Plan Angle = 60.00000
D = Adjacent Plan Angle = 66.58678

R1 = Hip Rafter Slope Angle = 34.71501
C5m Main Hip Rafter Backing Angle = 18.20076
C5a Adjacent Hip Rafter Backing Angle = 13.85266
Valley Sleeper Saw Blade Bevel Angle = 90° - (C5m + C5a) = 57.94657

P2m = Main Jack Rafter Side Cut Angle = 24.26751
90° - P2m = Main Roof Sheathing Angle = 65.73249
P2a = Adjacent Jack Rafter Side Cut Angle = 19.06421
90° - P2a = Adjacent Roof Sheathing Angle = 70.93579

Purlin Depth = 70
Hip Rafter Depth = 60
Hip Rafter Width = 60

DP Side Claw Line Dimensions
Roof Sheathing Angle
70.93579°
Seat Line for Rafters in Ground Plan Dimension
116.17440
Seat Line for Rafters on Roof Surface Dimension
92.71727
Claw Line #1 Dimension
0.00000
Claw Line #2 Dimension
113.05613
Claw Line #3 Dimension
43.83167
Claw Line #4 Dimension
1.26481



Treteaux Angles
  
R8-DP
Tilted Hip Rafter Slope Angle on DP Side of Hip Rafter.

36.10362°

R9-DP
Horizontal Plane Rotation Angle for Tilted Hip Rafter on DP Line from Footprint to Hip Rafter Run Line.

10.60575°

D-DP
Horizontal Plane Rotation Angle for Tilted Hip Rafter on DP Line from Eave Line to Hip Rafter Footprint

55.98103°

R10-DP
Vertical Plane Rotation Angle from Plumb for Tilted Hip Rafter on DP Line

15.12401°

PSBm-DP
Prism Footprint Saw Blade Bevel Angle Along side of Hip Rafter Rotated into the Roof Surface Plane at Foot Of Hip Rafter

51.34019°

PSBa-DP
Prism Footprint Saw Blade Bevel Angle Along side of Hip Rafter on DP Side at Foot of Hip Rafter

14.87719°

SR4Bm-DP
Prism Footprint Angle in Plan View at Eave Line Intersection

70.60575°

SR4Ba-DP
Prism Footprint Angle in Plan View at Intersection of DP Line and Profile Rafter Footprint Line in Plan View

109.39425°

Drawings with claw line development using the draw down technique.








Friday, April 11, 2014

Rotated Rafters intersecting Rotated Hip Rafters

Joe Bartok's Study on Rotated Rafters intersecting Rotated Hip Rafters.

______________________________________________________



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:
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°

____________________________________________________________
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).
.............................
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° (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)
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° (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)
Blade Bevel for ZDIH = 10.15094° (Blade Bevel for Rectangular Section