Tuesday, April 22, 2014

When Hips Collide #3

Drawings showing the secret line. I'm calling it a secret line, because it took me a month to find it. This secret line defines the lower claw angles on rotated hip rafters. This line is the "Intersection of the Hip Rafter Run Line & Hip Rafter Claw Line". Drawing a perpendicular from the DP Line defines the intersection point on the profile drawing of the rotated hip rafter. Drawing a line from the intersection of the plan view DP Lines to the intersection point  draws the line representing the lower claw angle on the rotated hip rafter.




Drawing showing the plane that defines the lower claw angle on the rotated hip rafter.


Drawing showing the hip rafter head cut unfolded. It also shows the 2 parallelogram planes that represent the head cut on the hip rafter.  To layout the head cut on the timber you need 4 different angles and 1 dimension. The 2 upper claw angles that form one of the parallelogram cutting planes and the 2 lower claw angles of the rotated hip rafter that form the other parallelogram cutting plane. It's important to remember that the cutting planes are always  parallelograms. Once you layout 2 of the angles of the parallelogram then the other 2 sides of the parallelogram are automatically developed by using the same 2 angles.



Drawing showing  the geometry for the 2 upper claw angles and the 2 lower claw angles.
Drawing showing the intersection of the DP Lines.



Saturday, April 19, 2014

Hip rafters twisted into the roof surface

Robert Simpson sent me an email with a picture of a drawing of a lectern in Sevilla, Spain. Just about every medieval cathedral has an Octagonal baptismal  and the base of the lectern is Octagonal with the legs of the lectern rotated into the roof surface. Or as Robert stated... twisted into the roof surface. This lectern  looks like some of the bases holding up the masterpieces of the European Trade Guilds dating back to the 13th century.

Possible base of an masterpiece with Saint Andrew's Cross.




Email from Robert Simpson
_____________________________________________________________


Sim,
Earlier this month we were in Sevilla, Spain. In the massive cathedral there was a lectern under repair. Its name in Spanish is Facistol del Coro.You could not see it as it was protected by boards, but on the boards was some explanation, including this illustration.You can see in photo Rafters 1 that the lectern consists of a heavy base, a wooden support structure and a decorative cover with a cupola. Photo Rafters 2 show the support structure a bit bigger, and what do we see?  Hip rafters twisted into the "roof" surface . . .

Recently the construction St Andrew's Cross has come up a lot in your blog.
In the third photo I thought you might like to see its gruesome origins (by Juan de las Roelas, in Fine Art Museum, Seville). de las Roelas, like many artists of 16th century religious work in Spain, was born in Flanders, the area of Belgium where I live.

__________________________________________________________________________



tréteau à devers... Trestles with tilt.




Thursday, April 17, 2014

Rotated Rafters Upper and Lower Claw Angles

In the previous study on Upper and Lower Claw Angles, with plumb rafters intersecting rotated hip rafters the key to finding the upper claw angle was the intersection of the DP line with the rafter run line. With  a rotated rafter intersecting a rotated hip rafter the key to the upper claw angle is the intersection of the Hip Rafter DP Line and the Rotated Rafter DP line.

I was trying to draw out the rotated rafters using Michel Verdon's art of line method when I saw the connection of the intersecting DP lines.

Michel's blog sites
Apprendre la charpente
Le forum exclusivement dédié à la charpente bois.


Drawing with intersecting DP lines.



Here's the art of line drawing I was trying to draw.




I drew this drawing a couple of months ago and now it's all coming together for the upper & lower claw angles on any rotated rafter. It's all based on the foot print of the rafters.


Pentahedron and Non-Rectangular Sections

Pentahedron and Non-Rectangular Sections Study by Joe Bartok
__________________________________________________
When I compare the Golden Rhombus and the current Offset Rotated Rafters this Compound Angle Formula works all around:
μ = arctan (sin Angle at Rafter Peak ÷ tan Blade Bevel Angle)

Golden Rhombus (my study half-split the roof angle but the formula works just as well for your equal width Hips solution):
μ = arctan (sin (90° – 27.73230°) ÷ tan (90° – 47.05622°) = 43.56300°
β = 90° – R1 = 90° – 27.73230° = 62.26770°
α = C5 = 16.04506°

70° Offset 12/12 Side Rafter ... Claw Angle Version:
μ = arctan (sin 28.71825° ÷ tan 5.36467°) = 78.94194° (Angle on Upper Shoulder of Offset Rafter)
β = 28.71825° (Upper Claw Angle)
α = 13.99545° (Rotated Rafter Backing Angle)

70° Offset 12/12 Side Rafter ... Plumb Line Version:
μ = arctan (sin (90° – 43.21918°) ÷ tan 16.86990°) = 67.40640° (Angle on Upper Shoulder of Offset Rafter)
β = 90° – R1 = 90° – 43.21918° = 46.78082°
α = 13.99545° (Rotated Rafter Backing Angle)

Looking good, but a long way to go yet.
______________________________________________








Development of a Pentahedron describing a Non-Rectangular Section, given μ, β and α. After the first few steps the rest of the drawing falls into place.

Setting a different reference length = 1 has produced a few new relationships:
ρ = (90° – BEV) – arctan (cos YDIH ÷ tan β)
Blade Bevel for YDIH = arcsin (tan β ÷ tan (BEV + ρ))
MIT = arctan (sin μ tan β ÷ (cos α cos μ tan β + sin μ sin α))

Test firing the formulas on real Hip roof studies:

Golden Rhombus ... Half-split Roof Angle
μ = arctan (sin (90° – 27.73230°) ÷ tan (90° – 47.05622°) = 43.56300°
β = 90° – R1 = 90° – 27.73230° = 62.26770°
α = C5 = 16.04506°
ρ = Offset Rafter Slope Angle – arctan (tan Main Hip Slope Angle sin Blade Bevel
ρ = 30.79574° – arctan (tan 27.73230° sin (90° – 47.05622°) = 11.08972°
MIT = arctan (sin 43.56300° tan 62.26770° ÷ (cos 16.04506° cos 43.56300° tan 62.26770° + sin 43.56300° sin 16.04506°)) = 40.86615°

70° Offset 12/12 Side Rafter ... Claw Angle Version:
μ = arctan (sin 28.71825° ÷ tan 5.36467°) = 78.94194° (Angle on Upper Shoulder of Offset Rafter)
β = 28.71825° (Upper Claw Angle)
α = 13.99545° (Offset Rafter Backing Angle)
ρ = (90° – Trace Angle on Main Hip) – arctan (tan (90° – Upper Claw Angle) sin Blade Bevel)
ρ = (90° – 55.50766°) – arctan (tan (90° – 28.71825°) sin 5.36467°) = 24.80855°
MIT = arctan (sin 78.94194 tan 28.71825 ÷ (cos 13.99545 cos 78.94194 tan 28.71825 + sin 78.94194 sin 13.99545)) = 57.74662°

70° Offset 12/12 Side Rafter ... Plumb Line Version:
μ = arctan (sin (90° – 43.21918°) ÷ tan 16.86990°) = 67.40640° (Angle on Upper Shoulder of Offset Rafter)
β = 90° – R1 = 90° – 43.21918° = 46.78082°
α = 13.99545° (Offset Rafter Backing Angle)
ρ = Main Hip Slope Angle – arctan (tan Offset Rafter Slope Angle sin Blade Bevel)
ρ = 30.96374° – arctan (tan 43.21918 sin 16.86990°) = 15.71018°
MIT = arctan (sin 67.40640° tan 46.78082 ÷ (cos 13.99545 cos 67.40640° tan 46.78082 + sin 67.40640° sin 13.99545)) = 57.74661°

There doesn’t appear to be a pattern for the formulas for ρ in terms of the roof angles, but:
... 30.79574° and 30.96374° are really 90° – Trace Angles for their respective rafters
... Main Hip Slope Angle and Offset Rafter Slope Angle are 90° – Upper Claw Angle are the complements of the angles at their respective rafter peaks.

ρ = 90° – Trace Angle – arctan( sin Blade Bevel ÷ tan Angle at Rafter Peak)???

Penathedron and Non-Rectangular Section Test
Intersecting Hip Rafters – Warlock Cut, Upper Shoulder Hip B
Angle on Hip B Upper Left Shoulder = 37.65287°
Blade Bevel @ 37.65287° = 55.39851°

μ = arctan (sin 37.65287° ÷ tan 55.39851°) = 22.85244° = Angle on Hip B Upper Right Shoulder ... expected that, the rafter section is rectangular
β = 37.65287° = Angle on Hip B Upper Left Shoulder
α = 0° ... at first glance it might seem like this should be 45°, but there is no backing or rotation angle here

MIT =  37.65287°
BEV = 43.14862° (Trace Angle on Hip A)
ρ = 0°
Projected Right Angle = 136.85138°
Supplementary Angle = 43.14862° (= Trace Angle on Hip A)
Blade Bevel for XDIH = 55.39850°
Blade Bevel for YDIH = 26.71762° (Blade Bevel @ 22.85244°)
Blade Bevel for ZDIH = 55.39850° (= Blade Bevel for XDIH)

The code returned angles for a tetrahedron (which we already know works here). This isn't telling me anything new or lending insight as to an easier means of drawing the intersection.



Wednesday, April 16, 2014

Upper and Lower Claw Angles

Here's the geometric layout for the Upper & Lower Claw Anges for the European Championship for Carpenters task model for the hip rafter that's plumb to the earth. This method also works for Joe Bartok's Study on Rotated Rafters intersecting Rotated Hip Rafters.

The key points to this method are the "Intersection Point of Hip Rafter Claw Line & Rafter Run Line" and the perpendicular line to the rafter run line. A line drawn from the "Intersection point of Foot Print Line & Rafter Run Line" thru the  perpendicular line to the rafter run line develops the lower claw angle on the rafter. The upper claw angle drawn from the "Intersection of Rafter Run Line and DP Line" to the peak of the rafter in profile is fairly easy to draw and understand.





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)