Back to Irregular Hip Roof Dimensions
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Development of Backing Angles
Development of Proportioned Plumb Backing Angles
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° |