I am framing a large and complicated roof system with king-post truss, simple rafter trusses, and a quad valley between everything. We have finally gotten to the valleys and I have been trying to mathematically determine the jack rafter foot cuts and sq rule mortises for a 4-foot layout up the valleys. I would love to layout and cut everything on the ground. Interestingly today, we discovered that in order to get a 55 degree bevel(top face of Jack) on a 45 degree plumb cut, I set my big saw to 44.5 degrees and it works. CAn someone explain that ? I am now trying to layout the mortises for the 4-foot jack rafters running up the valley and can't find a math solution for this. Any help? (12/12 pitch; valley 7x16; jacks 6x8; 30'8 sq plan)

John: Here are some examples of Jack Rafter Compound Angle Calculations . The formulas will solve the miter, bevel and mortise angles for not only jack rafters but any stick rectangular in section.

A couple of more links:
Roof Framing and Joinery Math
Online Math Notes and Diagrams

Thanks so very much for both your replies. By trial and much error today we arrved at the 30 degree backing angle. It seems confusing to set the saw on 30 degree in order to get a 35.2 degree bevel plane with the top of the jack rafter, which is coming down and approaching the valley on a 45 degree angle. From one of those websites I also confirmed my calculated 83.125 inch spacing between jacks along the valley - 4 foot on centers. I am feeling much more confident that this will work. I'll try to post some photos when erected. Thanks again.

John, your numbers look OK. I wasn't certain yesterday exactly what "mortise layout" you were after and didn't have time to explain the math anyway.

The easiest way to think of your saw bevel along the miter line created by the Jack Rafter Side Cut angle (Hawkindale Angle P2) is to think of the intersecting planes. The surface of your jack rafter and the plane formed by backing the valley are equivalent planes. The plane formed by cutting the compound angle on the jack rafter matches the the side (plumb) face of the valley. The blade bevel to make the cut from the face of the jack rafter set in the plane of the roof must be the same as the bevel to back the valley rafter: Hawkindale Angle C5.

There are different methods of analysing the relationships in a compound angle. Here's a link to a Trigonometric Solution of a Compound Angle . The angles may also be determined by Development . This development assumes the angles on the faces of the stick are known. If a miter angle and the blade bevel (or the dihedral angle) are known the angles may be solved as per this Development of Miter Angle on Upper Shoulder of a Hip-Valley Rafter . (Note the title! The stick being developed is NOT a jack rafter - however, the process is the same. The givens for a jack rafter would be the plumb cut angle and the 45° Deck Angle).

The formula for the rafter mortise spacing along the Valley is:
Spacing O.C. along Ridge Line/sin Jack Rafter Side Cut Angle
or Spacing O.C. along Ridge Line/sin P2
Substituting the numbers in the formulas:
48"/sin 35.26439° = 83.13844", or about 83-1/8".
The easy way is to use this Excel Unadjusted Hip-Valley Dimensions "cheat sheet" ... here's a link to a web based Hip-Valley Framing Calculator .

All:
Awesome work Joe, and thanks for sharing.
I will comment after I have had a chance to review all the links you posted.

Keep up the good work,
Curtis

Joe: Wouldn't the saw bevel for jack rafter be c5 or backing angle if sawing from the top of the jack not the face? Or am I not reading that right.

Curtis

Yes, the blade bevel would be the backing angle if cutting from the top of the jack rafter face. That's what I meant by "face of the jack rafter set in the plane of the roof". Ditto for purlins, except the miter line is created by the complement of angle P2.
Here's another Jack Rafter and Purlin Compound Angle Formulas page. The "Sheathing Angle" referenced is Hawkindale 90°–P2, complementary to the Jack Rafter Side Cut Angle.

Joe: Thanks for clearing up that muddy description.

This is a a great thread, thanks for contributing.
Curtis

Love this stuff! Here's my 2 cents worth, regarding those jack rafter angles (better late than never)...assuming the plan view shows deck angles being 45 degrees as with most building hip or valley intersections , I always lay the jack on its side, draw the proper roof pitch plumb line, set the saw to 45 degrees and cut along that plumb line. No number crunching involved. Works, of course, for all roof pitches. Oh sure, what about that other line on the top of the jack? The layout line on the top of the jack can be developed by drawing a perpendicular line to the plumb line connecting to the edge (on the waste side of the plumb line). Measure that line length (call it 3" in this case). Now on the top surface of the jack, from where that 3" line intersected the top surface, draw a square line down, mark a point at 3" (remember that we know the saw angle has to be 45 since we are cutting along the plumb line, thus the line lengths must also be the same). Connect the dots from the top of the plumb line and the 3" point and you have the jacks top and bottom layout line angle. (I usually use a compass to transfer those line lengths) If I am dying to crunch numbers and check it I use my favorite trig formula ...the tangent of the saw angle equals the sine of the mitre (plumb line side of jack) divided by the tangent of the bevel (top or bottom side of jack). That formula can be applied for most situations, purlins, hip and valley ends. I find that most of angles involved in valley systems can be developed right on the timber as long as you remember to refer it to a plumb line, level line , or plan view deck angle. (We don't need no stinkin Hawkindales)...actually its great stuff but it is good to be able to visualize it all and not rely on remembering R1, P2, C5, Soh-Cah, whatever...time to have a beer!