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