Irregular Hip Roof Valley Rafter Framing

Irregular Hip Roof  Valley Rafter Framing with a plan angle greater than 90° is probably as tough as it gets for stick framed roofs. It's easy to get the valley rafter angles, but applying the roof framing angles to the actual valley rafter material is tricky. You'll need the valley rafter plan angles, the valley rafter offset for the two roof planes, the valley rafter backing/bevel cut depth dimension , the peak and foot valley rafter side cut angles, the hip/valley rafter side cut mark for the side cut angles.

Here's a drawing of the roof. First we need to find the plan angles.

Use the Rafter Tools " Irregular Hip Roof With Equal Overhang" calculator to find the plan angles.

Mark each side of the valley rafter with the "Hip Rafter Backing Depth Mark". Then layout the "Main and Adjacent Hip Rafter Offset Mark Perpendicular To The Hip Run Line". Then make a mark along the hip rafter offset center line the length of the "Hip Rafter Sidecut Mark For Sidecut Angles". Then draw the lines that form the Hip/Valley rafter side cut angles at the foot of the valley rafter. The hip/valley side cut angles are the opposite of the hip/valley rafter side cut angles at the peak of the rafter.

Here's section view of the valley rafter with the valley rafter backing/ bevel cut angles developed from the "Hip Rafter Backing Depth Mark".

Valley Rafter backing/bevel cuts applied to the valley rafter material.

In the Rafter Tools " Irregular Hip Roof With Equal Overhang" calculator click on "View Hip Rafter Angles" to view the hip rafter backing angles, hip rafter side cut angles, jack rafter side cut angles.

Note: Edit: Need to look into the valley rafter side cut angles. Something seems wrong with these angles.

Found it. The correct formula for eave angles greater than 120° with unequal pitches is

(trade secret).... I'll have to update the Rafter Tool calculator with this formula.

Valley rafter side cut angles at the foot of the valley are the same as the side cut angles at the peak of the valley rafter.

Valley backing depth is 5/16" perpendicular to the top of the valley.

Valley rafter plumb backing angles on the end of the square cut valley rafter.

Valley rafter side cut angles at the peak of the valley rafter.

Drawing of valley rafter with plumb cut heights of the two different pitches. The red area could be wrapped in Sheetrock.

Advanced Timber Framing by Steve Chappell

Advanced Timber Framing by Steve Chappell ."Exploring the Seven Planes of Compound Joinery"

I started reading through the book Advanced Timber Framing by Steve Chappell http://chappellsquare.com/product/advanced-timber-framing/ 

and will add notes to my roof framing geometry blog on concepts covered in the book.


First up is the valley rafter foot to principal rafter joinery. Steve takes you through the process of laying out the geometry on the valley rafter foot and discuses the working plane. This got me to thinking about all the roof framing angles rotated into different planes.

Rotated Angle = arctan(tan (of angle to be rotated) * cos( rotation angle))
or
Rotated Angle = arctan(tan (of angle to be rotated) * sin( rotation angle))

These examples use an 8:12 pitch and a plan angle of 45°

Common Rafter Angle rotated by the plan angle
Hip/Valley Rafter Angle = arctan(tan (common rafter angle ) * cos(rotation angle))
Hip/Valley Rafter Angle = arctan(tan (33.69007° ) * cos(45°)) = 25.2394°


Hip/Valley Rafter Angle rotated by the plan angle
Hip/Valley Housing Angle = arctan(tan (hip rafter angle ) * cos(rotation angle))
Hip/Valley Housing Angle = arctan(tan (25.2394° ) * cos(45°)) = 18.43495°


Hip/Valley Rafter Plumb Backing Angle rotated by the Hip/Valley Rafter Angle
Hip/Valley Backing Angle = arctan(tan (hip rafter angle ) * cos(hip rafter angle))
Hip/Valley Backing Angle = arctan(tan (25.2394° ) * cos(25.2394°)) = 23.09347°

Irregular Hip Roof Example
8:12 main pitch
10:12 adjacent pitch


Common Rafter Angle rotated by the plane rotation angle
Hip/Valley Rafter Angle = arctan(tan (common rafter angle ) * cos( plane rotation angle))
Hip/Valley Rafter Angle = arctan(tan (33.69007° ) * cos(38.65981°)) = 27.50055°


Hip Rafter Angle rotated by the plane rotation angle
Hip/Valley Housing Angle = arctan(tan (hip rafter angle ) * cos( plane rotation angle))
Hip/Valley Housing Angle = arctan(tan (27.50055° ) * cos(51.34019°)) = 18.01470°
or
Hip/Valley Housing Angle = arctan(tan (27.50055° ) * sin(38.65981°)) = 18.01470°


Hip Rafter Angle rotated by the plane rotation angle
Hip/Valley Housing Angle = arctan(tan (hip rafter angle ) * cos( plane rotation angle))
Hip/Valley Housing Angle = arctan(tan (27.50055° ) * cos(38.65981°)) = 22.12194°
or
Hip/Valley Housing Angle = arctan(tan (27.50055° ) * sin(51.34019°)) = 22.12194°

Hip/Valley Rafter Plumb Backing Angle rotated by the Hip/Valley Rafter Angle
Hip/Valley Backing Angle = arctan(tan (hip rafter angle ) * cos(hip rafter angle))
Hip/Valley Backing Angle = arctan(tan (22.60994° ) * cos(27.50055°)) = 20.27452°

Steve Chappell uses axioms to guide you thru the timber framing angles layout on the actual timbers. When I first looked at his drawing of the Bisected Foot Print Angle, I wondered why he didn't use the word plan angle. However, after reading through the book I realized he was using the adjacent plan angle to calculate some of the angles that were rotated into different planes. The BFA is worth exploring and seems to be the key to laying out the correct rotated angles on the timbers.

Steve does not use tetrahedrons in his book. Using a tetrahedron is just my way of checking the axioms based on a unit circle.

Can't wait to find the time to actually layout out the valley to principal rafter joinery on real timbers, following Steve's examples in his book.

Alhambra Granada Ad Quadratum Ground Plan Part 2

Geometry has been passed down through the centuries.So, my best guest is the Islamic Moors used regulating lines of geometry to develop the roof plan and the hip rafter head cuts for the roof at Alhambra, Granada Spain over the Hall of  Abencerrajes. The Islamic Moors most likely used a Hexadecagon King Post, Ad Quadratum King Post 4 - Squares Rotated in Circle. 16 sided polygon king post for the 8 hip rafters and 8 valley rafters.

Using mitered- beveled  hip rafter and valley rafter head cut, would have never work structurally.

Using an purlin structural ring might have worked, but the purlin would have to an V in the middle of the purlin to follow the roof planes of the valley rafters. 

Plan view for developing the square butt head cuts of the hip rafters without a king post.

Roof surface development for the Ad Quadratum ground plan with the hip rafters developed from a 4 sided polygon and the valley rafters developed from am 8 sided polygon. The jack rafter side cut angles are developed on the real roof surface. As well as the roof sheathing angles. I wonder if they called the roof sheathing angles hopper cuts, like the English carpenters did/do? Did they use roof sheathing angles developed from geometry to cut their roof sheathing material? Or did they scribe and cut the roof sheathing material? 

The Alhambra Granada Ad Quadratum Ground Plan is now a calculator in Rafter Tools.

Rafter Tools help file link for the Alhambra Granada Ad Quadratum Ground Plan Roof Rafters.

Pythagorean Theorem and Angry Old Men

I haven't used the Pythagorean Theorem  since the 1970's, as a apprentice carpenter to calculate rafter lengths. Roof framing is all about right triangles, but the Pythagorean Theorem is not widely used in calculating the lengths of the roof framing members and calculators use trigonometric ratios to calculate the lengths of the right triangles, not the Pythagorean Theorem . So when I hear or read about the Pythagorean Theorem as the basis of roof framing it brings out the Angry Old Men/Man in me. The Greeks, 1500 years ago, used full scale drawings of the roofs to develop the rafter lengths  and bevels, and used the Pythagorean Theorem to square up the strings they used for the geometric layout of the roof. Full scale geometric layout was used by all carpenters up until the 18th or 19th century and is still used today to scribe the roof framing members. 

Pythagorean Theorem, with sides 3, 4 and 5, a sq x b sq = c sq

Quote from the Stanford Encyclopedia of Philosophy
Stanford Encyclopedia of Philosophy Pythagoras
Proclus does not ascribe a proof of the theorem to Pythagoras but rather goes on to contrast Pythagoras as one of those “knowing the truth of the theorem” with Euclid who not only gave the proof found in Elements I.47 but also a more general proof in VI. 31. 

Speaking of Angry Old Men, it amazing how this article at This is Carpentry brought out the Angry Old Men on Common Rafter Framing.

Common Rafter Framing by Mike Sloggatt @ This is Carpentry

Using trigonometric ratios to calculate the lengths of the right triangles begins with the unit circle.

Trigonometric functions of the angle θ can be constructed geometrically in terms of a unit circle. In this next drawing the unit circle has an radius equal to 1 and the angle θ  equals the roof slope angle of an 8:12 pitched roof.

In this next drawing the unit circle has an radius equal to 1 and the angle θ  equals the hip rafter slope angle of an 8:12 pitched roof.

Alhambra Granada Ad Quadratum Ground Plan

Connecting the eight points of intersection of the circle and the sides of the square form an octagon. 

Roof over the Hall of  Abencerrajes @ Alhambra

Interior view of the Hall of  Abencerrajes @ Alhambra

 Just about every medieval cathedral has an octagonal baptismal and the Alhambra in Granada Spain has several octagonal ground plans. The roof over the Hall of the Abencerrajes is unique because the roof actually follows the Ad Quadratum ground plan that is used to develop the octagonal ground plan. Squares rotated in the circle.

Ad Quadratum ground plan geometrically developed for hip and valley rafters.

Developing the roof framing plan is pretty straight forward if you use polygon roof framing geometry. The hip rafter is base on a polygon with 4 sides and the valley rafter is based on a polygon with 8 sides.  

Using the Rafter Tools for Android app select the Polygon Rafter Angles calculator and enter 4 for the Number of Sides of Polygon to calculate the angles for the hip rafter and use number of sides = 8 for the valley rafter slope angle. These angles can also be found on the Chappell Master Framing Square. 

The adjacent radius can be calculated using the law of sines.

Octagon 8 Sides

Ground Plan -- Ad Quadratum

2 - Squares Rotated in Circle

Adjacent Radius = ( Main Radius × sin( B ) ) ÷ sin( C) 

Adjacent Radius = ( 144 × sin(45 ) ) ÷ sin(112.5) 

110.2128 = ( 144 × sin(45 ) ) ÷ sin(112.5) 

Dodecagon - 12 sides

Ground Plan -- Ad Quadratum

3 - Squares Rotated in Circle

Adjacent Radius = ( Main Radius × sin( B ) ) ÷ sin( C)
Adjacent Radius = ( 144 × sin( 45 ) ) ÷ sin( 120)
117.5755 = ( 144 × sin( 45 ) ) ÷ sin( 120) 

Hexadecagon - 16 sides

Ground Plan -- Ad Quadratum

4 - Squares Rotated in Circle

Adjacent Radius = ( Main Radius × sin( B ) ) ÷ sin( C) 

Adjacent Radius = ( 144 × sin( 45 ) ) ÷ sin( 124.08620)

122.4619 = ( 144 × sin( 45 ) ) ÷ sin( 124.08620)

I'll post more information on the hip rafter and valley rafter head cuts in the next article on the Ad Quadratum ground plan.

Tetrahedron Slice -- Hucks jig for Roof Framing Angles

I've seen Hucks jig for roof framing angles and always knew it was a tetrahedron slice from the jack rafter side cut. I had some left over 6x6 from a recent job and decided to cut the 6x6 into Hucks jig for hip roof framing angles. Mathematically, it's called a Trirectangular Tetrahedron. Whatever you call it, Huck's done a great job in developing a jig that's useful on the jobsite. It will give you the miter angles for Purlin Rafters, Frieze Blocks, Bird Blocks or Square Tail Fascia Miter Angles.

Hopefully, this drawings will clear up the purpose of Hucks jig. It's a piece of wood representing the tetrahedron slice of the jack rafter side cut. For a 8:12 pitched roof you draw the jack rafter miter on the side of the material and then set your saw to 45° to cut the cheek cut for the jack rafter. Hucks jig is a slice from the jack rafter side cut.

Tetrahedron Slice unfolded showing the hip roof framing angles from Hucks jig.

1. Rafter Slope Angle
2. Jack Rafter Bevel Angle
3. Purlin, Frieze Block and Square Tail Fascia Miter Angle
4. Hip Rafter Slope Angle
5. Hip Rafter Backing Angle
6. Hip Rafter Square Tail Miter Angle