Octagon Roof Framing Layover Valley

After looking at the pictures I took of a house last week, that I framed 15 years ago, I thought it would be fun to see how I would now calculate the California Layover Valley Rafters on the equal sided Octagon Roof and the unequal sided polygon roofs with 12 sides. Fifteen years ago I remember calculating the California Layover Sleeper that's parallel to the front plate, but the rest of the hip rafters and sleepers were scribed and cut to fit.

The California sleeper - Layover board are similar to the   discussion of the at JCL - Online Rough Frame forum about 7 years ago.
Round Roof into Gable

and the calculator I developed for that thread.
Sloped Frustum of Pyramid Calculations for Polygons or Cones Roof Framing Geometry based on Complex Roof Geometry 

Here's my view on the basic steps for Octagon Roof Framing without using one of my RafterTools app calculators.

Octagon Eave Angle

(135.00°)

Octagon Plan Angle

(67.50°)

Octagon Roof Slope Angle, Profile Rafter Slope Angle

12:12 Pitch (45.00°)

Main Roof Slope Angle 10:12 Pitch (39.80557°)

Comm Rafter Run

72.4264"

Jack Rafter Spacing = 16" O.C.

Hip Rafter Width = 1.5"

Calculate the king common rafter length for the roof

72.4264" ÷ cos (45.00) = 102.426397"

Calculate the mitered king common rafter length for the roof

0.75 ÷ sin(22.5°) = 1.95984"
1.95984" ÷ cos( 45.00°) = 2.771638"
Mitered King Common Rafter Length = 102.426397" - 2.771638" = 99.654758"

Calculate the rise for the roof
72.4264" x tan (45.00) = 72.4264"

Calculate the hip rafter run

72.4264" ÷ cos (90° - 67.50°) = 78.39377"

Calculate the hip rafter slope angle

arctan(72.4264 ÷ 78.39377) = 42.73420°

or use arctan (tan Profile Rafter Slope Angle x sin Plan Angle)

arctan (tan 45.00° x sin 67.50°) = 42.73420°

Calculate the roof sheathing angle

arctan (tan Plan Angle ÷ cos Profile Rafter Slope Angle)

arctan (tan 67.50° ÷ cos 45.00°) = 73.67505°

Calculate the jack rafter length difference based on the On Center spacing

tan 73.67505° x 16" = 54.62742"

I like to calculate all of my Jack Rafter Lengths using the Roof Sheathing Angle. First you need to calculate the Hip Rafter Offset Along the Eave Line. The trigonometry below only works on equal pitched roofs.

1/2 Hip Width ÷ sin (67.50°) = Hip Offset Along Eave Line
0.75" ÷ sin (67.50°) = 0.811794"
Hip Offset Along Eave Line = 0.811794"
Jack Rafter Length Deduction for Hip Rafter
Hip Offset Along Eave Line x tan(Roof Sheathing Angle) = Jack Rafter Length Deduction for Hip Rafter
0.811794" x tan(73.67505°) = 2.77164"
Jack Rafter Length Deduction for Hip Rafter =  2.77164"

Calculate the Jack Rafter Length for Descending Jacks from the King Common Rafter.

Center of King Common Rafter = 30"

First Jack  15 1/4" from center of King Common Rafter

First Jack Spacing From Corner = 30 - 15 1/4 = 14 3/4"

First Jack Rafter Length = (14 3/4" × tan(73.67505°)) - 2.77164" = 47 9/16"

Calculate the Frieze Block miter angle & Saw Blade Bevel Angles

Frieze Miter Angle = arctan( sin ( Pitch Angle ) ÷ tan( Plan Angle ))
Frieze Miter Angle = arctan( sin ( 45.00° ) ÷ tan( 67.50° ))= 16.32494°
Frieze Saw Bevel Angle = arcsin( cos ( Pitch Angle ) x cos( Plan Angle ))
Frieze Saw Bevel Angle = arcsin( cos ( 45.00° ) x cos( 67.50° )) = 15.69985° 

Calculate the Hip Rafter Plumb Line Shift at foot of hip rafter for Equal Pitched Roofs.

1/2 hip rafter width × tan( 90° - Plan Angle) = hip plumb line shift at foot of hip rafter

0.75" x tan (22.50°) = 0.31066" = or 5/16"

Calculate the Jack Rafter Plumb Line Shift for the Jack Rafter Side Cut Angle.

jack rafter width × tan( 67.50°) =Jack Rafter Plumb Line Shift  = 3.62132 = 3 5/8"

Drawing with Basic Hip Rafter Shift for Hip Rafter Plumb Line Shift at foot of hip rafter for Roof Plane Alignment.

Jack Rafter Length Deduction for Hip Rafter drawing.

Jack Rafter Plumb Line Shift for the Jack Rafter Side Cut Angle drawing.

First Jack Rafter Length drawing.

3D drawing of the Octagon Roof Framing

3D drawing of the Octagon Roof Framing with Plan View and Elevation View

Drawing with the Octagon Roof in Plan View, with the Roof Elevation drawn above the Plan View. Draw lines perpendicular from Plan View from the Hip Rafter Run lines to the base of the Elevation view to establish the location of the hip rafters in elevation. Draw in the main roof slope angle in elevation view starting from the roof intercept point of the two roofs. Then drop perpendiculars back to plan view from the intersection of the main roof slope line and the hip rafters in elevation. The perpendicular intersect the hip rafter run lines in plan view and locate the valley layover sleepers in plan view. Draw out the fold down roof surface for sides of the roof that have layover sleepers and then draw perpendicular lines from the plan view layover sleeper to the roof surface hip rafter. These lines will locate the valley layover sleepers on the roof surface. It will also establish the length of the hip rafters that tie into the valley layover sleepers.

You can also draw the hip rafter profile and draw perpendicular from the layover sleepers in plan view to establish the true length of the hip rafters that tie into the valley layover sleepers.

Using the Law of Sines to calculate the layover hip rafter lengths and layover sleepers.

Law of Sines ---( 48.4264 × sin( 39.80557°) ) ÷ sin( 95.19443°) = 31.1297" = intercept calculation

31.1297" × cos( 45.00°°) = 22.01199" = intercept run B

72.4264" - 22.01199 = 50.4144" = king common run

50.4144" ÷ cos(45.00°) = 71.29672" king common length

50.4144" ÷ cos(22.50°) = 54.56815" = hip rafter run

54.56815" ÷ cos( 42.73421°) = 74.29195" = hip rafter length

 (50.4144" × tan(22.50°)) × 2 = 41.76465" = sleeper length

60.00" × sin(45.00°) =  42.4264"

72.4264" - 24.00" - 42.4264" = 6" 

arctan(72.4264" ÷ 30") = 67.50° = hip slope elevation view

180° - 39.80577° - 67.50° = 72.69423° = C

Law of Sines -- ( 6" × sin( 39.80577°) ) ÷ sin(72.69423°) = 4.023234" = intercept calculation

4.023234" × cos( 67.50°) = 1.539625" = intercept run B

1.539625" × tan(67.50°) = 3.716984" = intercept calculation

72.4264" - 3.716984" = 68.709415" = king common run

68.709415" ÷ cos(45.00°) = 97.169787" king common length

68.709415" ÷ cos(22.50°) = 74.370535" = hip rafter run

 74.370535" ÷ cos( 42.73421°) = 101.25196" = hip rafter length

This next drawing develops the Length of the King Common Rafter that ties into the layover sleeper and the  Layover Cut on King Common Rafter

Octagon Rafter Slope Angle + Main Roof Slope Angle =Layover Cut on King Common Rafter

Here's a list of all of the layover Cut Angles

Layover Cut Angle on King Common Rafter (84.80557°)
Layover Cut Angle on Hip Rafter  (80.32687°)
Layover Cut Angle on King Common Rafter  (75.50896°)
Layover Cut Angle on Jack Rafter  (68.85377°)
Layover Cut Angle on Hip Rafter  (60.42184°)

Layover Cut Angle = Layover Rafter Slope Angle + arctan (tan Main Roof Slope Angle x sin Layover Rotation Angle in Plan View)

Layover Cut Angle on King Common Rafter (84.80557°)

Layover Cut Angle = 45.00° + arctan (tan 39.80557° x sin 90.00°) = 84.80557°

Layover Cut Angle on Hip Rafter  (80.32687°)

Layover Cut Angle = 42.73420° + arctan (tan 39.80557° x sin 67.5°) = 80.32687°

Layover Cut Angle on King Common Rafter  (75.50896°)

Layover Cut Angle = 45.00° + arctan (tan 39.80557° x sin 45.00°) = 75.50895°

Layover Cut Angle on Jack Rafter  (68.85377°)

Layover Cut Angle = 45.00° + arctan (tan 39.80557° x sin 33.10957° ?) = 69.47510° ?

Layover Cut Angle on Hip Rafter  (60.42184°)

Layover Cut Angle = 42.73420° + arctan (tan 39.80557° x sin 22.5°) = 60.42184°

I need to work on this for the layover saw blade bevel angles

saw blade bevel = arcsin (cos(polygon roof slope angle) × cos(main roof angle) - sin(polygon roof slope angle) × sin(main roof angle) × cos(theta))...