Hip Shift Unit Ratios
The unit ratios are factored by the actual rafter widths to determine the offsets detailed in the diagram above. 6^{15}/_{16}/12 Hip Rafter, 8/12 Side tan 60.25512° ÷ (tan 60.25512° + tan 29.74488°) = .75385 6^{15}/_{16}/12 Hip Rafter, 14/12 Side tan 29.74488° ÷ (tan 60.25512° + tan 29.74488°) = .24615 9^{15}/_{16}/12 Hip Rafter, 14/12 Side tan 45.19422° ÷ (tan 45.19422° + tan 89.80578°) = .00340 9^{15}/_{16}/12 Hip Rafter, 9^{15}/_{16}/12 Side tan 89.80578° ÷ (tan 45.19422° + tan 89.80578°) = .99660 7^{15}/_{16}/12 Hip Rafter, 9^{15}/_{16}/12 Side tan 52.91319° ÷ (tan 52.91319° + tan 82.08681°) = .15532 7^{15}/_{16}/12 Hip Rafter, 8/12 Side tan 82.08681° ÷ (tan 52.91319° + tan 82.08681°) = .84468 |
The Hip Rafters are shifted and dropped as per the drawing above.
The lengths conform to the theoretical framing lines with the following exception.
Length reduction for 9^{15}/_{16}/12 Hip Rafter This Hip rafter has been positioned to act as a Common (Jack) Rafter. The 89.85038° angle at the corner is rounded to 90°. Width on adjoining 7^{15}/_{16}/12 Hip Rafter in plan = .23298" (see Section View) Angle in plan = 90° - 52.91319° = 37.08681° Hip Rafter Run = .23298" ÷ sin 37.08681° = .38635" Reduction of Hip Rafter Length = .38635" ÷ cos 39.61624° = .50153" (approx. 1/2") Hip Rafter Length = 19.34208' - (.50153" ÷ 12) = 19.30029' = 19' - 3 ^{5}/_{8}" |
Example Side Cut Calculation for
6^{15}/_{16}/12
Hip Rafter, 8/12 Side
The Side Cut Angle at the Hip Rafter peak on the 8/12 side is 26.31492° The width on the 8/12 side of the Hip Rafter is 1.13078" (see Section View) 1.13078" ÷ sin 26.31492° = 2.55080" We want a closer match to the .75" of the Ridge Board. Angle in plan between Ridge Board and Hip Run = 180° - 60.25512° = 119.74488° 90° - 60.25512° = 29.74488° is covered by the Hip Rafter (The projection of the 29.74488° Plan Angle to the Hip Rafter produces the 26.31492° Side Cut Angle at the Hip Rafter peak) The remaining 90° will be divided equally between the Ridge Board and Hip Rafter. Plan Angle at Hip Rafter = 45° + 29.74488° = 74.74488° Blade Bevel Angle along Plumb Cut = 90° - 74.74488° = 15.25512° Hip Rafter Side Cut Angle = arctan (cos Hip-Valley Pitch Angle ÷ tan Blade Bevel along Plumb Line) = arctan (cos 30.06349° ÷ tan 15.25512°) = 72.50931° The edges are approximately: Ridge Board = .75" ÷ sin 45° = 1.06066 Hip Rafter = 1.13078" ÷ 72.50931° = 1.18560" Since the Hip Rafter edge actually slopes away from level; the current 1/8" difference will be about 1/32" as viewed in plan. Blade Bevel Angles along Rafter Plumb Lines 6^{15}/_{16}/12 Hip Rafter, 14/12 Side arctan (cos 30.06349° ÷ tan 56.56514°) = 29.74488° 7^{15}/_{16}/12 Hip Rafter, 8/12 Side (NOTE: The "natural" Side Cut Angle value of 6.61620° has been changed to 64.06753° for a better match with the edge of the Ridge Board) arctan (cos 33.43752° ÷ tan 64.06753°) = 22.08681° 7^{15}/_{16}/12 Hip Rafter, 9^{15}/_{16}/12 Side arctan (cos 33.43752° ÷ tan 32.24437°) = 52.91319° 9^{15}/_{16}/12 Hip Rafter (= Common Rafter) arctan (cos 39.61624° ÷ tan 30.21300°) = 52.91326° 14/12 Common Rafter arctan (cos 49.39871° ÷ tan 32.87839°) = 45.19422° |