Roof Planes with Unequal Pitched Roofs

To check all of the Rafter Tools App results I draw geometric roof framing layouts in Google Sketchup. To make sure it's constructable and the java mathematical results are correct. Here's a situation where you have an 8:12 main roof slope angle and an adjacent 12:12 roof slope angle. To calculate the main roof common rafter lengths you just deduct 1/2 the thickness of the ridge for the main common rafter length. However, you can't deduct 1/2 the thickness of the ridge for the adjacent 12:12 king common rafter length. You will have to chamfer the end of the ridge on a unequal pitched roof to keep the ridge in the roof plane alignment.  The Rafter Tools App Irregular Hip rafter function calculates the correct king common rafter length for unequal pitched hip rafter roofs.

The correct way to calculate the minor pitch king common rafter, keeping all of the roof framing members in the roof planes,  is to use the major and minor plan angles.

 minor ridge deduction run  = (ridge width * 0.5) ÷ tan(major plan angle)

Calculate Irregular Hip Rafter Angles

Here's a couple of screen shots of the Rafter Tools for Anfroid App showing the Calculate Irregular Hip Rafters function. Hopefully, it's Erik proof.

Input variables

1. Roof Eave Angle
2. Major Pitch Angle
3. Minor Pitch Angle
4. Input Length
5. Overhang Run
6. Jack Rafter Spacing
7. Hip Rafter Width
8. Hip Rafter Depth
9. Jack Rafter Width -- Common Rafter Width
10. Major Rafter Depth
11. Seat Cut Level Cut Length
12. Ridge Width

For Input Length use any part of the hip rafter geometry as input to find the rest of the angles.

• Common Rafter Run
• Common Rafter Span
• Common Rafter Rise
• Common Rafter Length
• Hip Rafter Run
• Hip Rafter Length
• Eave Wall Length -- Minor Run

Calculated Results for the Irregular Hip Rafters function. 

1. Major Plan Angle
2. Major Roof Pitch Angle
3. Major Roof Pitch
4. Minor Plan Angle
5. Minor Roof Pitch Angle
6. Minor Roof Pitch
7. Major Rafter Run
8. Major Theoretical Rafter Length
9. Major Rafter Length To Ridge
10. Major Rafter Seatcut Length
11. Major Overhang Rafter Length
12. Minor Rafter Run
13. Minor Theoretical Rafter Length
14. Minor Rafter Length To Ridge
15. Minor Rafter Seatcut Length
16. Minor Overhang Rafter Length
17. Hip Rafter Angle
18. Hip Rafter Pitch
19. Hip Rafter Run
20. Hip Rafter Theoretical Rafter Length
21. Hip Rafter Length plumb to plumb along the center line to the ridge.
22. Hip Rafter Long Point Length to Ridge -- long point of hip rafter sidecut
23. Saw Blade Bevel Angle to cut the hip rafter sidecut angle along the hip rafter peak plumb line.
24. Hip Rafter Overhang Length
25. Major Hip Rafter Plumb Backing Mark Height
26. Minor Hip Rafter Plumb Backing Mark Height
27. Major Hip Rafter Perpendicular Backing Mark Height
28. Minor Hip Rafter Perpendicular Backing Mark Height
29. Major Pitch First Jack Rafter Length To Long point of miter cut.
30. Major Pitch Jack Rafter Length Difference @ jack rafter spacing
31. Major Pitch Jack Rafter Saw Blade Bevel Angle
32. Minor Pitch First Jack Rafter Length To Long point of miter cut.
33. Minor Pitch Jack Rafter Length Difference @ jack rafter spacing
34. Minor Pitch Jack Rafter Saw Blade Bevel Angle
35. Theoretical Common Rafte Rise
36. Rise To Top Of Ridge
37. HAP -- Height Above Plate (Heel Height)
38. Major Rafter Plumb Cut Height
39. Minor Rafter Plumb Cut Height
40. Hip Rafter Plumb Cut Height
41. Hip Rafter Drop For Roof Plane Alignment
42. Hip Rafter Backing Offset Main Perpendicular to hip rafter run line.
43. Hip Rafter Backing Offset Adjacent Perpendicular to hip rafter run line.
44. Hip Rafter Backing Offset Main along eave line.
45. Hip Rafter Backing Offset Adjacent along eave line.
46. Hip Rafter Backing Angles
47. Hip Rafter Plumb Backing Angle
48. Hip Rafter Side Cut Angles
49. Jack Rafter Side Cut Angles
50. Roof Sheathing Angles
51. Hip Rafter Purlin Housing Angles
52. Hip Rafter Diamond Post Miter & Saw Blade Bevel Angles
53. Plywood Roof Sheathing Offset Cut Line
54. Valley Sleeper Miter & Saw Blade Bevel Angles
55. Dormer Square Tail Miter & Saw Blade Bevel Angles
56. Purlin Rafter Miter & Saw Blade Bevel Angles
57. Purlin Layover cut perpendicular to roof surface plane

Saillies De Toit

I thought this might be a good book.


La CHARPENTE en bois de Gilbert Emery ( RELI - 1 avril 2005)

I draw out page 148, .Saillies De Toit.

Roof projections
With regular projection,  = equal overhang

The ground plan is a trapezoidal ground plan with equal overhang. The book does not explain how to calculate the hip offset from the corner of the ground plan. Disappointing!!!
Back to the drawing board to see how to draw or calculate the hip offset for roof eave angles that are not at 90° and have unequal roof slope angles.

This drawing was generated by my Google SketchUp plugin using the Trapezoidal Ground Plan function.

Irregular Hip Rafter Ground Plan With Equal Overhang

It easy to draw out the tetrahedrons at the hip overhang when the roof eave angle is 90°.

It's also, just a simple math equation to find the hip offset from the corner of the fascia line.

Major Pitch Overhang Run / tan(minor plan angle) = Offset Run on fascia line

24 ÷ tan(36.8699) = 32

Here's a drawing with the math for the trapezoidal ground plan with equal overhang to find the hip rafter offset from the corner of the building plate line.

Unconstructable Irregular Roof Framing and My Android App

This is an example of an unconstructable irregular roof with an 135° eave angle. There aren't very many roofs in construction that have this set of angles, but I see a lot of carpenters trying to calculate roof framing angles that are unconstructable.

The common rafter pitch on the right side is an 4:12 pitched roof with roof angle of 18.43495°. The common rafter pitch on the left side was suppose to be an 8:12 pitched roof with an roof angle of 33.69007°. However, the 8:12 pitch on the right side has to be at 90° to the eave line. Which is impossible with an 135° eave angle. In order to make the 135° eave angle work with the major 4:12 roof pitch the minor roof pitch needs to be changed to some where around an 5:12 pitch.

Here's a screen shot of my Android Frieze Block Miter Angle & Saw Blade Bevel Angle app showing  Minor Tetra Angle = 22.99236°. This is another good example of the different math functions on different platforms and different devices calculating different angles when the roof is unconstructable to begin with. The Minor Tetra Angle is from an function that I wrote that tries to draw the 2 tetrahedrons for the 2 different pitches. If the 2 tetrahedrons don't align with the hip rafter than the roof is unconstructable.

This image shows an screen shot of the Android Frieze Block Miter Angle & Saw Blade Bevel Angle app where I've changed the minor pitch to an 5:12 pitch, so we can view the results for the Frieze Block Miter Angle & Saw Blade Bevel Angle for both the major roof pitch and minor roof pitch.

This image shows an screen shot of the Android Frieze Block Miter Angle & Saw Blade Bevel Angle app when the View Hip Rafter Angles button is pressed. The hip rafter or valley angles are not needed to cut the frieze blocks, but are shown for reference. 

In this screen shot of the Android Frieze Block Miter Angle & Saw Blade Bevel Angle app you can see that the Major & Minor plan angles are set to 0.00 when you enter an Roof Eave Angle of 1356°, which is an unconstructable roof.

Note: the screen shots of my Android Frieze Block Miter Angle & Saw Blade Bevel Angle app are just about the same size of my Google Nexus 7 tablet. 

Apprentice Carpentry Roof Framing Geometry Part 2

In part 2 of  Apprentice Carpentry Roof Framing Geometry we'll discuss the hip rafter backing angle and the dihedral angle. The hip rafter backing angle aligns the top of the hip rafter with the roof surface and the plan angle.


This picture of an 8:12 & 10:12 irregular hip roof model shows how the hip rafter backing angle aligns the hip rafter with the square tail fascia.

This picture shows how the hip rafter backing angle aligns the hip with the roof surface.

This picture shows the valley rafter backing angle aligning with the plan angle.

This picture shows how the hip rafter backing angle aligns the hip rafter with the plan angle.

Valley rafters backed out to keep the top of the valley rafters aligned with the roof surface.

The hip rafter backing angle is developed from the dihedral angle triangle. The dihedral angle is the angle between two planes. In roof framing geometry the dihedral angle is the angle between the two roof surfaces on each side of the hip rafter.  


This drawing shows the dihedral angle drawn in plan view as well as a 3D view of the dihedral angle.

In this drawing , in plan view, the dihedral angle triangle is drawn from the eave line (AB) to the opposing eave line (AF). Note, the eave line is also referenced as the gutter line. To draw the dihedral angle triangle start with a line at point W. The point W can be located anywhere on the eave line. The line WX is always drawn perpendicular to the hip rafter run line AE. At point X draw a line to point Y that is perpendicular to the hip rafter length line AG. Then swing an arc from point X with the radius equal to the line length XY until it intersects the hip rafter run line AE. Angle XWZ is the hip rafter backing angle and angle XFZ is the hip rafter backing angle for the other side of the hip rafter. For an equal pitched roof these angles will be the same. For irregular pitched roofs the hip rafter backing angle will not be the same angle.

The dihedral angle is angle WZF. The angle between the roof planes.

Here are a couple more drawings showing the dihedral angle in 3D.

Here's a link to the Google SketchUp files showing the Basic Roof Framing Kernel with the hip rafter backing angle and the location of the dihedral angle.


Hip Rafter Backing Angle-1.skp

Hip Rafter Backing Angle-3.skp

Hip Rafter Backing Angle-4.skp

Hip Rafter Backing Angle-6.skp

Apprentice Carpentry Roof Framing Geometry Part 1

Tony McGartland,  who teaches carpentry at a college in N.Ireland and who is also responsible for training the better apprentices for carpentry competitions in the UK, sent me a couple of Google SketchUp models the other day. From past carpentry competitions in the UK. These Google SketchUp roof framing models made me realize how poor the American Union Apprentice program is compared to the apprentice programs in Europe. France?,Switzerland, Austria and Germany have specialized training for their apprentices for these apprentice competitions and are light years ahead of the United States of America apprentice program in roof framing construction and geometry. If we sent American apprentices to these carpentry compensation, not only would they not medal they would most likely embarrass themselves. By the time I completed my 4 years of the Union Carpentry apprenticeship program I had already cut and stacked (framed) about 100+ track home roofs, but I didn't know anything about roof framing geometry. Yes, I used a framing square to mark and layout the rafters, but if anyone had asked me to draw out the roof framing geometry of the roofs I was cutting, I would have said no I can't draw out the roof framing geometry to the roofs that I've cut.


This is part 1 in Apprentice Carpentry Roof Framing Geometry for the Americans and maybe the Canadians. 


Roof framing geometry is empirical-type knowledge.
Information gained by means of observation, experience, or experiment.

It will take more than just reading to understand roof framing geometry. You'll need to draw out the roof framing geometry. Preferably on 4x8 sheets of plywood or mdf  to gain the experience you will need to compete in the apprentice carpentry competition. You'll need a compass, straight edge and framing square. Yes, you can use your calculator in the apprentice carpentry competition, but you'll also have to draw out the roof framing geometry as well. Just like they did 100+ years ago.

I suggest getting a 6" and 16" compass like these compasses from Lee Valley.

http://www.leevalley.com/US/wood/page.aspx?p=32625&cat=1,42936,42937

While your at Lee Valley you might as well order a Chappell Framing Square

http://www.leevalley.com/US/wood/page.aspx?p=69569&cat=1,42936

First up, The Basic Roof Framing Kernel

This is a plan view of a roof framing kernel with the section view drawn above the plan view.The plan view is also referenced as the ground plan. The real roof surface angles and lengths are located by drawing an arc (radius=common rafter length) at the apex of the common rafter run (point D) in plan view. The common rafter is also called the principal rafter.


F-E = Common Rafter Plan View Run
A-E = Hip Rafter Plan View Run
P-R = Jack Rafter Run
E-H = Common Rafter Real Surface Rise 
F-H = Common Rafter Real Surface Length
A-G = Hip Rafter Real Surface Length
R-S = Jack Rafter Length


Angle FAB = Eave Angle
Angle EAB = Plan Angle
Angle EFH = Common Rafter Slope Angle
Angle GAE = Hip Rafter Slope Angle
Angle NCD = Roof Sheathing Angle
Angle DNC = Jack Rafter Side Cut Angle

The next 2 drawings are  the basic roof framing kernel drawn in Google SketchUp. The basic roof framing kernel is drawn unfolded and folded up into 3D roof surfaces of the roof framing kernel. This is also a good example of swing an arc to establish the points of the real roof surfaces.

Here's a link to the Google SketchUp file Basic Roof Framing Kernel.

Basic Roof Framing Kernel.skp

In part two of Apprentice Carpentry Roof Framing Geometry we'll add the hip rafter backing angle to the basic roof framing kernel.

Chappell Master Framing Square and the Purlin Lip Cut Angle

 What is the purlin lip cut angle? The purlin lip cut angle is on the upper shoulder of a roof purlin that allows the roof purlin to project under the hip or valley rafter at a  plane that is perpendicular to the hip or valley rafter plane.

In this post I'm using the  Chappell Master Framing Square to demonstrate how to lay out the rafters of an unequal pitched roof (bastard roof), major pitch 12:12 and a minor pitch of 9:12 with an eave angle of 90° and how to layout and cut the purlin lip cut angle.

The first thing we need to do is layout the hip rafter offset. The drawing below, that dates back to the 1800s or earlier, is the easiest way to lay out the hip rafter offset. The hip rafter in this drawing is 3.5" in thickness. Mark off a line that is perpendicular to the hip rafter run and the width of the hip rafter. Then draw a line that is parallel with the adjacent eave line to obtain the hip rafter offset. In this drawing the hip rafter offset for the 12:12 side is 1.26" and the hip rafter offset for the 9:12 side is 2.24"

Next lay out the hip rafter. The hip rafter angle is on the first line of the Chappell Master Framing Square under the 9:12 on the Unequal Pitched 12/12 Main Pitch A side of the Chappell Master Framing Square. 30.964°. To layout the hip rafter plumb line use the .6 on the square. Multiply it by 10, which gives us 6.0, and use 6" on the tongue and 10" on the blade. (6 ÷ 10 = arctan (0.6)= 30.96376°).

 Next mark off the hip rafter backing depth on each side of the hip rafter. The equal pitch rafter tables on the Chappell Master Framing Square on line 4, DEPTH OF BACKING & BEVEL CUT IN INCHES PER 1" OF HIP OR VALLEY WIDTH (Multiply 1/2 Beam Width by Factor Given for Total Depth) are the tangents of the Hip or Valley rafter backing angle. For the Hip rafter in our example the hip rafter backing angle for the 12:12 side is 34.450°. The tan of 34.450° is = 0.685997.  Multiply 0.685997 by 1.24", the hip offset, to find the depth of the hip rafter backing line along the side of the hip rafter.  0.685997 x 1.24 = 0.850636 or 7/8".  On the 9:12 side we'll take the tan of 21.10° (0.385868) then multiply it by  2.24. 0.385868 x 2.24 = 0.864344 or 7/8".

To double check the hip rafter backing depth marked on each side of the hip rafter you can draw out the hip rafter plumb backing angles on the end of the rafter. 

9:12 side
Hip Rafter Backing Angle = arctan( sin( Hip Rafter Pitch Angle) ÷ tan( Plan Angle ) )
Hip Rafter Backing Angle = arctan( sin( 30.9637579168711) ÷ tan( 53.1301 ) ) = 21.1001984464996°
Hip Rafter Plumb Backing Angle = arctan( tan( Pitch Angle ) * cos( Plan Angle ))
Hip Rafter Plumb Backing Angle = arctan( tan( 36.8699 ) * cos( 53.1301 )) = 24.2277483279458°


12:12 side
Hip Rafter Backing Angle = arctan( sin( Hip Rafter Pitch Angle) ÷ tan( Plan Angle ) ) 
Hip Rafter Backing Angle = arctan( sin( 30.9637579168711) ÷ tan( 36.8699 ) ) = 34.4499007767502°
Hip Rafter Plumb Backing Angle = arctan( tan( Pitch Angle ) * cos( Plan Angle ))
Hip Rafter Plumb Backing Angle = arctan( tan( 45 ) * cos( 36.8699 )) = 38.6598073928135°

In this picture we can see that the hip rafter backing mark on the 9:12 side of the hip rafter aligns with the hip rafter plumb backing  angles on the end of the hip rafter plumb cut.

Next draw out the hip rafter side cut angles at the foot of the hip rafter.  The hip rafter side cut angles are the opposite of the hip rafter side cut angles at the peak of the hip rafter. In this drawing you can see how the hip rafter side cut angles at the foot of the hip rafter align with the eave lines and the hip rafter backing depth mark.

Hip Rafter Side Cut Angle at Peak  = arctan( cos( Hip Rafter Pitch Angle ) ÷ tan( Plan Angle ))

Hip Rafter Side Cut Angle at Foot  = arctan( cos( Hip Rafter Pitch Angle ) ÷ tan( Adjacent Plan Angle ))

12:12 side

Hip Rafter Side Cut Angle at Peak = arctan( cos( 30.9637579168711 ) ÷ tan( 36.8699 )) = 48.825665948811°Hip Rafter Side Cut Angle at Foot = arctan( cos( 30.9637579168711 ) ÷ tan( 53.1301)) = 32.7458710613747°

9:12 side

Hip Rafter Side Cut Angle at Peak = arctan( cos( 30.9637579168711 ) ÷ tan( 53.1301 )) = 32.7458710613747°Hip Rafter Side Cut Angle at Foot = arctan( cos( 30.9637579168711 ) ÷ tan( 36.8699)) = 48.825665948811°

The length of the hip rafter is calculated by multiplying the decimal on line 3 of the square by the run of the common rafter. In this model the run of the 12:12 common rafter is 16.3125". So we multiply 16.3125 x 1.944 = 31.7058" or 31 11/16" for the plumb to plumb length of the hip rafter.

The length of the jack rafters are calculated by multiplying the decimal on line 4 of the square by the length of the eave line of the common rafters. In this model the eave line length from the hip rafter offset line for the  12:12 common rafter side  is 19.65". So we multiply 19.65 x 1.061 = 20.8420" or 20 13/16" for the jack rafter length on the 12:12 side. In this model the eave line length from the hip rafter offset line for the  9:12 common rafter side  is 13.5125". So we multiply 13.5125 x 1.667 = 22.5208" or 22 1/2" for the jack rafter length on the 9:12 side. 

Note: The values on line 4 of the Chappell Master Framing Square are the tangents of the roof sheathing angles. The tan of the roof sheathing angle times the jack rafter spacing, equals the difference in length of the jack rafters.  If the jack rafters were spaced 24" O.C. for the 12:12 side of the roof, then we would use 24" x 1.061 = 25.45589" for the difference in length of the jack rafters for the 12:12 side of the unequal pitched roof.

Next cut the purlin's using Line 7 on the Chappell Master Framing Square for the purlin miter angle on the side of the purlin. This is where the Chappell Master Framing Squares are worth the hefty price of the square. It's one thing to have the purlin miter angle and use a speed square to layout the purlin miter angle, but by using the decimal on line 7 of  the square, .9428 in this example, all you have to do is multiply it by 10 and use 10" on the blade of the square to layout angles accurately. No more calculator calculations to transfer the angles to a framing square for accuracy.  For the cut on top of the purlin you can use line 5 on the Chappell Master Framing Square and the hip rafter backing bevel on line 7. 

The purlin lip cut angle is P3 of the Hawkindale angles.
P3 = Arctan (Cos D x Sin R1 x Cos R1 ÷ Cos S)

http://www.tfguild.org/tools/hipart.html


Calculations for the purlin lip cut angle for the 12:12 side of roof
Hip Rafter Square Tail Side Cut Angle (Foot) = 52.13635
R3 = 90° - Hip Rafter Square Tail Side Cut Angle (Foot) = 37.86365
C2 = Hip Rafter Square Tail Side Cut Bevel Angle (Foot) = 22.83365

R3 equals P2 rotated by C5.
R3 equals Jack Rafter Side Cut Angle rotated by the Hip Rafter Backing Angle
R3 = arctan( cos ( Hip Rafter Backing Angle ) * tan( Jack Rafter Side Cut Angle ))
R3 = arctan( cos ( 34.4499007767502 ) * tan( 43.3138542102836 )) = 37.8636443877934°

Tetrahedron showing the relationships of the Hip Rafter Back Angle, Jack Rafter Side Cut Angle, 90° - Hip Rafter Square Tail Side Cut Angle (Foot), Hip Rafter Square Tail Side Cut Bevel Angle (Foot)  and the Purlin Lip Cut Angle.

Purlin Lip Cut Angle = P3
P3 = arccos( sin (  Jack Rafter Side Cut Angle rotated by the Hip Rafter Backing Angle  ) ÷ sin ( Jack Rafter Side Cut Angle ))
P3 = arccos( sin ( R3 ) ÷ sin ( P2 ))
P3 = arccos( sin ( 37.86373 ) ÷ sin ( 43.3139 ))
Purlin Lip Cut Angle = arccos( sin (  90° - Hip Rafter Square Tail Side Cut Angle (Foot)  ) ÷ sin ( Jack Rafter Side Cut Angle ))

An easier  Purlin Lip Cut Angle Formula by Joe Bartok.
Purlin Lip Cut Angle = arctan (tan Backing Angle × cos Jack Rafter Side Cut Angle)

Purlin with lip cut angle.

You can draw the purlin lip cut angle on the side of the purlin. However, all you need to do is run your skill saw down the face cut of the purlin to produce the purlin lip cut angle. The purlin lip cut angle follows the bottom shoulder of the hip rafter. If you draw a line perpendicular to the hip rafter triangle, it will trace the purlin lip cut angle on the purlin, because the purlin face cut, once installed, is a vertical plane that matches the hip rafter triangle plane .

Purlin with Lip Cut Angle Drawn on the side of the Purlin.

12:12 & 9:12  purlin's with purlin lip cut angle.

Here's a better picture, by Joe Bartok,  of the Purlin Lip Cut Angle (P3) showing the square cut along the face cut of the purlin, with zero blade bevel, to produce the purlin lip cut angle that is perpendicular to the hip rafter triangle plane.

Here's two more drawings by Joe Bartok showing the relationship of P3 , P2 and C5 in a tetrahedron.

sin(90° - P2) = cos(P2)

C5 = Hip Rafter Backing Angle

P2 = Jack Rafter Side Cut Angle

Angle C = arctan( tan ( A ) × cos ( D ))

or

Angle P3 = arctan( tan ( C5 ) × cos ( P2 ))

Purlin Lip Cut Angle = arctan( tan ( Hip Rafter Backing Angle ) × cos ( Jack Rafter Side Cut Angle  ))

Roof Framing kernel with geometric layout for the purlin miter angle and purlin lip cut angle.