Roof Framing Geometry

Friday, February 14, 2014

Tréteaux Dit Cadet Task Model

Getting started on the Tréteaux Dit Cadet task model.
tréteau à devers

Profile Slope Angles
74.95720°
75.96376°
Dimensions
20" x 48" x 30"

SawHorse Leg Bevel Angles
Roof Surface -- 75.38684°
DP --75.61879°

Miter Saw Settings for foot of Rafter at 75.38684° face
Miter -- 14.6°
Blade Bevel -- 14.8°

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.

none

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.

none

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.

none

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.

none

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.

Sunday, February 9, 2014

Rees Hawkins Acheson

Rees Hawkins Acheson contacted me the other day and I wanted his story to be told about the development of the Hawkindale Timber Framing Angles and credit for him developing the timber framing angles on his own in 1987.

Mr. Ayers,

I tried to post the following on your blog:

http://sbebuilders.blogspot.com/2013/08/mckibben-martindale-hawkindale.html

but was unable to do so. I obtained your email and have included post

below:

I appreciate your short history of the framing angles and I am pleased that someone is taking it on. But I am also writing to point out that you got my name wrong, the name is Rees Acheson.

While I'm at it, you may find interesting the origin of the name "Hawkindale". While working at Benson Woodworking (1987-1991) I developed a set of angles to complement the Martindale angles (I had not known of Mckibben). My recollection is that all my angles would have been developed between November, 1987 and February or March, 1988. Tedd Benson suggested that these new angles be named after me as Acheson Angles. I disagreed and wanted to keep Martindale's name. So we made a compromise and choose Hawkindale - a combination of my middle name, Hawkins, and Martindale. This seemed to fit well with a concatenation of the angles, making one list.

Further, it might be interesting to note that your description seems to list Ed Levin and Rees Anderson as co-developers in the angle development process. This is misleading in that I talked to Mr. Levin once in the late spring or summer of 1988 (I remember it was green

outdoors) when Tedd brought the two of us together, and perhaps again in

1990 after the Timber Framer's Guild conference where I gave a talk. I do not remember the second meeting but the first was brief, about 5 minutes, in which I told Mr. Levin that in my coding I was describing the joints exclusively with angles. Having said that, apparently the conversation was over and, other than niceties, no more was said - I learned nothing of what he was up to. My guess is that he either thought my approach was brilliant or ludicrous. He did not say. My code comments below indicate a 2nd meeting, or at least that a communication, took place.

We were independent, and although I made every effort to share what I had done with anyone who asked, Mr. Levin and I did not collaborate.

Your blog does not suggest that most of the non-Martindale angles were mine. Perhaps you could contact Tedd Benson for clarification on that matter. tedd My involvement in compound joinery was brief, about three years, after which I went back to my profession in machine design which I feel better suits my strengths. I am unfamiliar with what occurred with compound joinery after I left, but I am quite familiar with my doings for those three years.

What follows is the computer code for calculating the Hawkindale angles from my HV computer program that I began writing in late 1987 for Benson Woodworking. The file date is 27 October, 1990, so the final modification was apparently of Mr. Levin's influence (see comment for P7). The comments, in curly brackets {}, indicate that angles A1 thru

A9 were mine and that early on I changed their names to fit in with the Martindale scheme (e.g. A1 became P4). Two angles appear to have had Ed Levin's involvement in 1990 (P7 & R6) and the rest are Martindale's.

Mr. Levin's R6 seems to have replaced my R6 for some unstated reason.

Due to dependencies, they are not necessarily in order.

(I think Martindale's are as you say: R1, R2, R3, P1, P2, P3, C1-C9).

R1 := ATan(Tand(S) * Sind(D));

R2 := ATan(Sind(S) * Cosd(S) * Cosd(D) / Tand(D));

R3 := ATan(Sind(S) * Cosd(S) * Cosd(D) / Sind(R1));

P1 := ATan(Sind(S) / Tand(D));

P2 := ATan(Cosd(S) / Tand(D));

P3 := ATan(Cosd(D) * Sind(R1) * Cosd(R1) / Cosd(S));

C1 := ATan(Sind(P1) / Tand(S));

C2 := ATan(Tand(R2) * Cosd(R3));

C3 := ATan(Cosd(S) / Tand(D));

C4 := ATan(Sind(R1) / Tand(D));

C5 := C4;

C6 := ATan(Tand(D) / Sind(R1));

P4 := P2 - Atan(Tand(R2) * Tand(C5) * Cosd(C5));

{Formerly A1}

P5 := ATan (Sind(D) *(Cosd(S) /Cosd(D)) ); {5/6/88 }

{formerly A2}

P6 := Atan (Tand(C5) * Cosd(90-p2)); {1/22/89} {c5 rotated by 90-p2}

{P6 formerly A3} {OK 1/12/90} {JP seat cut}

P7 := P1 - (ArcCos(Cosd(R3) / Cosd(C1)) ); {Ed Levin 10/26/90}

R4 := Atan (Tand(p2) /Cosd(c5) );

{Formerly A4}

R5 := ATan ( Tand(r1) * Cosd(D) );

{Formerly A5}

{r1 rotated by D}

{R6 := Atan (Tand (P6) * Cosd (P2) );} {Obsolete}

{Formerly A8}

R6 := Atan (Tand(90-R5) / Cosd(D)) - (90-R1); {Ed Levin 10/26/90}

{ R6 = (the angle from the edge of rafter to the edge of bottom }

{ housing of VR foot

[p6]). }

{ It exists on as the seat cut on bottom of VR Ft }

A7 := ATan ( Tand(S) * Cosd(D) ); { S rotated by D or C5 rotated by r1.}

{ Same as c5 but a plum section of VR}

{ Angle of Roofs in a plum

section }

{ normal to the Valley

Line. }

A8 := Atan (Tand(r2) * Tand(c5) * Cosd(c5));

A9 := Atan (Tand(R1) * Tand(C5) * Cosd(C5));

February 2, 2014

Rees Hawkins Acheson

Sim,

To answer your questions:

"Does HV programming refer to Cobol computer programming?"

All programming was done in pascal, a compiled language that was very fast, both to compile and to execute. It's too bad the language never really caught on.

"HV" (stands for "Hip & Valley") was a computer program I wrote to create drawing files for AutoCAD to read. These files contained all the necessary views for the shop to cut the member. A valley Rafter Peak Tenon might have 11 views. The program would take about 5 seconds to run and create the drawing - with more than four of those seconds doing the relatively slow disk output. This was in 1987 when the CPU speeds were only 40MHz or so. The program was very fast. If it could be done at all, doing the same task in AutoCAD's AutoLisp scripting language would probably have taken an hour or two to run.

"1: How did you prove or test your formula's in 1980's? Especially the formula for angle P4."

To find the angles I worked with a pencil on a drafting table. Angles were worked out on E size sheets of paper. One angle lead to the next along with its formula and derivation for each stage written there on the paper until the desired angle was obtained. As proof, once a desired angle was worked out, I would work it out from another perspective - using different givens or different proven angles - coming at the angle from another direction. The results of the two must match. To clarify: I would find at least a 2nd formula for the same angle (sometimes a 3rd or 4th) and their results must match.

I think there were finally seven or eight such work sheets, each completely filled up with calculations, notes and drawings. I do not know what eventually became of these sheets, they may still exist somewhere and if so, I would like to see them again. These sheets were very valuable to me.

I do not remember P4 as being a particularly difficult one. Each one of them brought their own struggle in which I would grapple with with their complexities until I figured it out. And I did struggle.

I only remember one angle as being real a problem, A2, and it makes for an interesting story (A2 later became P5). In fact, I could not get it - the formula would not work, always pretty close but no cigar.

Looking at my notes it seems that the Y and Z components of a test point were correct but X was always off by a small amount (in what plane I do not now know). This should have been a clue, but I did not yet understand how. Finally, I had spent so much time on this angle that I needed to move on. I wrote an iterative solution (computer) that worked out the angle buy narrowing in on it using two methods of calculating the X component (working at it from two different sides) until the error was small enough (>0.00001") and then calculating the angle from the now correct X, Y & Z. This worked well enough that I found no need to fix it (it ran the hundred or so iterations necessary in about 200ms, so waiting for it was not an issue). But in early 1990, two years later, Tedd Benson asked that I give a talk on what I had been up to for that summer's TFG conference. This terrified me. But it also meant that I needed to figure out A2.

I began again using my original sheets on the drafting board. It's difficult to believe now, but I spent several days working on this one angle. Finally one day I stopped, disgusted, and decided to go for a walk to clear my mind. On that walk I realized something and when I got back, sure enough, I had an incorrect initial assumption. It was so near the beginning, it had been a given, that I had overlooked it each time through. Fixing this made the formula work fine. This became one of those learning experiences that one never forgets. I cannot remember exactly what the bad assumption was but it was pretty basic - something like the starting angle not being in the plane of the paper but very close to it. I knew better than that and was amazed that I had not noticed.

"2: Did you draw out roof framing kernels? Like the French?"

Sorry, don't know what this is.

"3: Did you use tetrahedrons in the 1980's?"

No, I don't think so. But I'm not sure what the method is, either.

Looking up tetrahedrons on your blog, a cursory look suggests the use of geometry to unfold views.

As I recall, most of what I was doing, was after seeing that an angle was needed, I would begin rotating other known angles until I got to the desired plane of measurement. This was in my head at first, then once I found a way I would begin the real rotations on paper. Obviously, after thinking through several rotations I would be in danger of becoming "lost", necessitating starting again. But this was less restricting than the interference created by having to leave one's mind to write things down. The series of rotations would be the formula (e.g. to pick a few angles at random: R1 by A7 by C1, etc.). Of course, I also needed

to keep track of rotation axes and the like. There might be a great

many rotations involved and the formula long. Once found, though, the formula would be simplified. It was often impossible to deduct the roots of the formula once simplified. Those original sheets of paper were therefore very valuable to me. They told the story and I would use them later to prove other things. So, as to your curiosity regarding R4, I do not remember - it's on those sheets. Right now I can't even remember what R4 is used for.

As I mentioned previously, as a proof, I would work out at least one other way to get to the same angle (producing a different formula). It was always best to use few angles that were common to the formulae being tested. It was also best to obtain the angle from a completely different starting place. The more independence the higher the degree of confidence I could have. If each component was unique, just two formulae would suffice.

To check that these formulae were similar I had written a computer test jig. In it the list of formulae to be compared would be run with a series of data to test a spectrum of possible input comparing the output. The output of each formulae would have to agree within a rather small "CloseNuf" constant. At the time I was using 80 bit floating point numbers (called "Extended" in pascal) and so the wandering due to processor error was very small. Errors were probably mine.

To help with angle development I wrote a short computer program that would rotate numerical angles. In use, after entering test roof pitches and dormer angle it would calculate the Hawkindales. Then it was ready to do test rotations. I had worked out a simple language for data entry and it would spit back a resultant numerical angle. This result could be further rotated again and again. The result could be checked for reasonableness and became a valuable tool for me to see if I was on the right track. If desired, data entry could use the Hawkindale angles for rotation. For example a rotation instruction might be "r4,a" which would rotate the currently displayed angle by Hawkindale R4 around the "a" side of the angle as the axis of rotation. Entries such as (90-R4)+C1,a were also possible. This was very much quicker than using a manual calculator and entering the numbers.

Looking through my notes I see that the change in "A" angle names occurred on Jan 16, 1990. Presumably this was because of the upcoming TFG talk I was to give that summer.

I hope that this adequately answers your questions. However, I fear that my response to your R4 query has been disappointing. But somehow I also suspect that I may have been going about angle development differently than most people and so how I arrived at R4 might not answer your question anyway. I have no formal training and so I developed the methods myself.

Rees

I would like to try answering question 3 once more. What I said was not clear and while thinking it over some more I have remembered more of it. So I will try again:

"3: Did you use tetrahedrons in the 1980's?"

No, I don't think so. But I'm not sure what the method is, either.

Looking up tetrahedrons on your blog, a cursory look suggests the use of geometry to unfold views. Although this may have been what I was actually doing, it was not in such a formalized fashion.

As I recall, upon realizing the need for a new angle, I would begin rotating planes or other angles until they were in the plane of the angle I wanted. Sometimes this would mean starting with the two lines of the angle I desired and perform rotations until it was in a plane I could deal with. Other times I would start with a plane I was comfortable in and perform rotations until I was in the plane of the angle I wanted. Either way, I would end up with an angle calculation in its own or some other plane followed by a series of rotations along various axies. The rotations were usually by a Hawkindale angle, but, of course, all angles had to be in terms of the original angle input - the roof pitches (S angles) and dormer angles (D angles).

This would be done in my head first, then once I found a probable solution I would begin the real rotations on paper. Obviously, after thinking through several rotations I would be in danger of becoming "lost", necessitating starting again. But this was less restricting than the disturbance created by having to leave one's mind to write things down.

There might be a great many rotations involved and the formula long.

Once found, though, the formula would be simplified. It was often impossible to deduct the roots of the formula once simplified. Those original sheets of paper were therefore very valuable to me. They told the story and I would use them later to prove other things. So, as to your curiosity regarding R4, I do not remember - it's on those sheets.

Of course, the final proof was when the HV program used these formulae to create the 3D points of a joint. If a formula were incorrect at least one of the ordinates of a point would be measurably and probably very noticeably incorrect.

Rees

Saturday, February 8, 2014

Louis Mazerolle's Planche 28 Tréteaux

Tréteaux Dit Cadet
Trestle or sawhorse for the apprentice

This is from Louis Mazerolle's book, Traite Theorique Et Pratique De Charpente, planche 27-28.

I had developed this drawing, only to realize it was wrong. I didn't get the hip rafters, sawhorse legs,correct in Plan View. So the geometric development of the sawhorse legs on the roof surface is wrong.

Here's a couple of drawings with the correct sawhorse leg development in Plan View.

The correct geometric development for the sawhorse leg on the roof surface.

After Developing a 3D model of the Tréteaux Dit Cadet I see that my original hip rafter foot print was off slightly, but it affects the entire geometric layout of the model. Back to the drawing board.

Finally got it right. The hip rafter foot print is now correct. Next, I developed the profile rafter slope just like a regular roof. Swing some arcs to a line parallel with plan view. Draw the Tréteaux,Trestle,sawhorse leg, hip rafter on the real roof surface and transfer the lines to the timber just like the drawdown method.

Checking the bevel angles at the foot of the sawhorse/hip rafter.

Here I took the developed 3D legs of the sawhorse and rotated them into position to make sure the 2D geometric development worked.

Next up, are the side crossing timbers of the sawhorse.

For the side cross rafter I used the sawhorse bevel angle and the sawhorse footprint angle to develop the cross piece. It's not precise, but it's a lot easier than Mazerolle's method. The barbe angle should be 3.38° instead of the 3.50° angle I used, but this is less than the thickness of pencil lead.

Here I develop the 3D view of the barbe angle to check it against the footprint angle I used.

3D view of the cross rafter after geometric development.

Next up, are the front crossing timbers of the sawhorse.

The crossing timbers of the sawhorse can be laid on top of the elevation view for geometric layout.

3D view of the pieces fitted together from the geometric layout for the Tréteaux Dit Cadet.

Side elevation view with arcs transferring the lines to the roof surface view.

Thursday, February 6, 2014

Octagon Framing With the iPhone

How to use RafterTools+ for iPhone to frame an Equal Sided Octagon Roof

Enter an eave angle of 135 and enter the same roof pitch for the Main & Adjacent Rafter Slope Angles. The default is Imperial English, like 8:12 or enter the roof slope angle in degrees and select degrees from the drop down menu box at the top of the screen. Enter the width of the Octagon hip rafter, like 3.5" or enter it in millimeters.

Then press the calculate button. Press the mm button if you entered the width of Octagon Hip Rafter in millimeters.

Rafter Tools will display the following results for the Octagon roof framing.

Hip Rafter Layout @ Foot of Hip Rafter

Hip Rafter Slope Angle

31.62973°

Main HR Backing Angle

12.25568°

Adjacent HR Backing Angle

12.25568°

Hip Rafter Width

3 1/2''in

Hip Rafter Backing Depth Dimension

3/8''in

Hip Rafter Plumb Line Shift Dimension

3/4''in

Main HR Shift Dimension Perpendicular to HR Run Line

1 3/4''in

Adjacent HR Shift Dimension Perpendicular to HR Run Line

1 3/4''in

Main HR Saw Blade Bevel Angle @ foot of HR 22.50000°

Adjacent HR Saw Blade Bevel Angle @ foot of HR 22.50000°

Hip Rafter Layout @ Peak of Hip Rafter

Main HR Back Bevel Angle 19.42694°

Adjacent HR Back Bevel Angle 19.42694°

Main HR Saw Blade Bevel Angle @ peak of HR 67.50000°

Adjacent HR Saw Blade Bevel Angle @ peak of HR 67.50000°

Main HR Plumb Line Shift Dimension

4 1/4''in

Adjacent HR Plumb Line Shift Dimension

4 1/4''in

Hip Rafter Square Tail Fascia Layout

Hexenschnitt The Witches Cut

Main HR Miter Angle

85.81571°

Adjacent HR Miter Angle

85.81571°

90° - Main HR Miter Angle

4.18429°

90° - Adjacent HR Miter Angle

4.18429°

Main HR Back Bevel Angle

71.38688°

Adjacent HR Back Bevel Angle

71.38688°

Main HR Saw Blade Bevel Angle

18.56691°

Adjacent HR Saw Blade Bevel Angle

18.56691°

Main HR Miter Line Shift Dimension @ tail of HR 9/16''in

Adjacent HR Miter Line Shift Dimension @ tail of HR 9/16''in

Sent from my iPhone

Press the Lengths Button to calculate the Octagon Hip Rafter Length.

Here I entered a common rafter run of 120" and set the Ridge Width to 0.

Main Rafter Run

120 ''in

Main Rafter TL

144 1/4''in

Main Rafter Length to Ridge

144 1/4''in

Main Rafter OverHang Run

24 ''in

Main Rafter OverHang Rafter Length

28 7/8''in

Main Rafter Seatcut Length

3 1/2''in

Main Rafter HAP

9 1/16''in

Main Rafter Plumb Height

11 7/16''in

Plate Height Difference

Hip Rafter Offset From Corner

Adjacent Rafter Run

120 ''in

Adjacent King Rafter TL

144 1/4''in

Adjacent King Rafter Length to Ridge

144 1/4''in

Adjacent Rafter OverHang Run

24 ''in

Adjacent Rafter OverHang Rafter Length

28 7/8''in

Adjacent Rafter Seatcut Length

3 1/2''in

Adjacent Rafter HAP

9 1/16''in

Adjacent Rafter Plumb Height

11 7/16''in

Hip Rafter Run

129 7/8''in

Hip Rafter TL

152 9/16''in

Hip Rafter Length to Ridge

152 9/16''in

Hip Rafter Length to Long Point

153 3/8''in

Hip Rafter OverHang Run

26 ''in

Hip Rafter OverHang Rafter Length

30 1/2''in

Ridge Height

89 1/16''in

Sent from my iPhone

Press the Jacks Button to calculate the Octagon Jack Rafter Lengths.

Here I entered a jack rafter spacing of 12" and set the Jack Rafter Material Thickness to 1.5".

Main Jack Rafter Miter Angle

33.69007°

Main Jack Rafter Bevel Angle

19.01633°

Main Jack Rafter Saw Blade Bevel Angle

67.50000°

Main Jack Rafter Shiften

3 5/8''in

Adjacent Jack Rafter Miter Angle

33.69007°

Adjacent Jack Rafter Bevel Angle

19.01633°

Adjacent Jack Rafter Saw Blade Bevel Angle 67.50000° Adjacent Jack Rafter Shiften

3 5/8''in

Main Jack Rafter Length Difference

34 13/16''in

#1 O.C.=12.00

31 1/2''in

#2 O.C.=24.00

66 5/16''in

#3 O.C.=36.00

101 1/8''in

#4 O.C.=48.00

135 15/16''in

#5 O.C.=60.00

170 3/4''in

#6 O.C.=72.00

205 9/16''in

#7 O.C.=84.00

240 7/16''in

#8 O.C.=96.00

275 1/4''in

Main Special Jack Rafter PL Run

60 ''in

Main Special Jack Rafter Length

168 5/8''in

Adjacent Jack Rafter Length Difference

34 13/16''in

#1 O.C.=12.00

31 1/2''in

#2 O.C.=24.00

66 5/16''in

#3 O.C.=36.00

101 1/8''in

#4 O.C.=48.00

135 15/16''in

#5 O.C.=60.00

170 3/4''in

#6 O.C.=72.00

205 9/16''in

#7 O.C.=84.00

240 7/16''in

#8 O.C.=96.00

275 1/4''in

Adjacent Special Jack Rafter PL Run

60 ''in

Adjacent Special Jack Rafter Length

168 5/8''in

Main Side HR Layout First Jack Rafter

34 3/16''in

Main Hip Rafter Layout JR Difference

36 13/16''in

Adjacent Side HR Layout First Jack Rafter

34 3/16''in

Adjacent Hip Rafter Layout JR Difference

36 13/16''in

Sent from my iPhone

For Advanced Timber Framing Angles for the Octagon roof press the Timber Framing Angles Button.

For Advanced Hip Rafter Framing Angles for the Octagon roof press the Hip Angles Button.

You can select any of the lines in the Hip Angles display to show the Framing Square usage for the selected angle.

The easiest way to frame an Octagon roof iis with the 4-Butt method.

The Octagon Roof Framing 4 Butt Hip Rafter Method.

#1 Main 2 Butt Hip Rafter Length
Cut 2 hip rafters at the TL length. These hip rafters will be installed first. The hip rafter head cut will have the Octagon Hip Rafter Slope Angle for the plumb cut angle and will not be beveled.
#2 Side 2 Butt Hip Rafter Length
Cut 2 hip rafters minus 1/2 the thickness of the Octagon Hip Rafters. These hip rafters will be installed second, against the first two hip rafters. The hip rafter head cut will have the Octagon Hip Rafter Slope Angle for the plumb cut angle and will not be beveled.
#3 Mitered Hip Rafter Length
Cut 4 hip rafters that are mitered, see below for length. Set the saw Blade Bevel Angle at 45.00°. The hip rafter head cut will have the Octagon Hip Rafter Slope Angle for the plumb cut angle and be beveled.
#4 Mitered King Common Rafter Length
Cut 8 king common rafters, see below for length. These king common rafters will be beveled. Set the saw Blade Bevel Angle at 67.5°. The king common rafter head cut will have the Octagon Rafter Slope Angle for the plumb cut angle.

For the mitered Octagon Hip Rafter use

Hip Rafter Slope Angle = 31.62973°

Half Thickness of Octagon Hip Rafter = 1.75"

Hip Rafter TL = 152 9/16''in = 159.5625"

1.75 ÷ cos(45°) = 2.47487

2.47487 ÷ cos(31.62973°) = 2.90664"

Mitered Octagon Hip Rafter Length = 159.5625 - 2.90664" = 156.65585"

For the mitered King Common Rafter use

Common Rafter Slope Angle = 33.69007°

Common Rafter Length =144 1/4''in

Half Thickness of Octagon Hip Rafter = 1.75"

1.75" ÷ sin(22.5°) = 4.5730

4.5730" ÷ cos( Roof Slope Angle)

4.5730" ÷ cos(33.69007°) = 5.4960"

144.25 - 5.4960 = 138.7539"

Mitered King Common Rafter Length = 138.7539"

RafterTools for Android has the Octagon 4 butt method lengths built into the calculator app.

Sunday, February 2, 2014

Louis Mazerolle's Planche 7 dévers

Back to square one with Louis Mazerolle's book, Traite Theorique Et Pratique De Charpente for dévers.

dévers --> Rafter Tilted/Canted/Rotated from Plumb

In this drawing I've developed the DP & TC lines for dévers de pas. I don't think the TC line is necessary for the dévers de pas line of the hip rafter. The hip rafter in this drawing is Tilted/Canted/Rotated from Plumb. It's rotated in plan view to the hip rafter backing triangle, H1-H-H2. Once the hip rafter foot print is developed in plan view the DP line, dévers de pas line, follows the footprint of the hip rafter. Then it's a matter of swinging arc's to develop the point Q that defines the hip rafter slope when it's tilted from plumb.

Angle K-D-F' = Hip Rafter Slope Angle for a plumb hip rafter.

Angle K-F-P = Hip Rafter Tilt from plumb.

Point Q is developed by swinging an arc with radius D-F' with center point D and an arc with radius P-F with center point P. The intersection of the two arcs define point Q that defines the Hip Rafter Slope Angle for dévers/canted/tilted/rotated rafter out of plumb.

It's best to click on the images below and open them in a new tab for clarity in the drawings.

Point 'P' is the intersection of the Line Perpendicular to the Hip Rafter Run Line and the DP Line.
That Follows the Footprint of the Hip Rafter.