Valleys with Centered Trough Lines
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Main Pitch Angle = 45°
Shed Pitch Angle = arctan [(6 + 7/12) ÷ (18 + 3/12)] = 19.83593° Ridge intercept angle in Plan = 90° Main Backing Angle = 41.69405° Shed Backing Angle = 6.61190° Shed Common Rafter Depth = 11.5 Projection to Valley Rafter = 11.5 ÷ cos 6.61190° = 11.57700 Main Common Rafter Depth = 11.57700 × cos 41.69405° = 8.64463 Required Valley Stock Depth = 11.57700 + 1.5 × tan 41.69405° = 12.91317 Main Common Rafter Depth = 11.5 Projection to Valley Rafter = 11.5 ÷ cos 41.69405° = 15.40095 Shed Common Rafter Depth = 15.40095 × cos 6.61190° = 15.29851 Required Valley Stock Depth = 15.40095 + 1.5 × tan 41.69405° = 16.73712 Valley Depth = 15.5 Greatest depth of material removed by Main Backing Angle = 1.5 × tan 41.69405° = 1.33617 Valley Depth along Trough Line = 15.5 – 1.33617 = 14.16383 Main Common Rafter Depth = 14.16383 × cos 41.69405° = 10.57623 Shed Common Rafter Depth = 14.16383 × cos 6.61190° = 14.06962 |
Valley Trough determined by Proportion
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Main Plan Angle = 19.83593°
Shed Plan Angle = 70.16407° Main Unit Offset Ratio tan 19.83593° ÷ (tan 19.83593° + tan 70.16407°) = .11514 Shed Unit Offset Ratio tan 70.16407° ÷ (tan 19.83593° + tan 70.16407°) = .88486 Main Side Dimension = 3 × .11514 = .34542 Shed Side Dimension = 3 × .88486 = 2.65458 Offset = ( 2.65458 – .34542 ) ÷ 2 = 1.15457 Trough Depth = .34542 × tan 41.69405° = .30770 or Trough Depth = 2.65458 × tan 6.61190° = .30770 Valley Depth = 15.5 – .30770 = 15.19230 Main Common Depth = 15.19230 × cos 41.69405° = 11.34420 Shed Common Depth = 15.19230 × cos 6.61190° = 15.09125 |
Backing optimized for Equal Common Rafter Depths
Determining the Minimum Valley Width
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Main Common Rafter Depth projected to Valley Rafter
= 11.5 ÷ cos 41.69405° = 15.40095 Shed Common Rafter Depth projected to Valley Rafter = 11.5 ÷ cos 6.61190° = 11.57700 Difference between Valley Rafter Sides = 15.40095 – 11.57700 = 3.82395 If a bevel is cut at the 41.69405° Main Side Backing Angle Valley Rafter Width to achieve required dimensions for sides = 3.82395 ÷ tan 41.69405° = 4.29280 |
Backing optimized for Equal Common Rafter Depths
Upper Shoulder Offset
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The Valley is assigned a width of 6
Main Common Rafter Depth projected to Valley Rafter = 11.5 ÷ cos 41.69405° = 15.40095 Shed Common Rafter Depth projected to Valley Rafter = 11.5 ÷ cos 6.61190° = 11.57700 Difference between Valley Rafter Sides = 15.40095 – 11.57700 = 3.82395 Depth removed from Valley if cut at Main Side Backing Angle = 6 × tan 41.69405° = 5.34467 Depth of Valley "Sleeper Section" = 5.34467 – 3.82395 = 1.52072 Let the Sleeper Section Width = w w × tan 41.69405° + w × tan 6.61190° = 1.52072 w ( tan 41.69405° + tan 6.61190° ) = 1.52072 w = 1.52072 ÷ ( tan 41.69405° + tan 6.61190° ) w = 1.51062 Main Side Offset = 6 – 1.51062 = 4.48938 Trough Depth at Offset = 4.48938 × tan 41.69405° = 3.99906 Trough to Bottom Shoulder of Valley = 15.40095 – 3.99906 = 11.40190 |
Backing optimized for Equal Common Rafter Depths
Bottom Shoulder Offset
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The Valley is assigned a width of 6
From the Hip/Valley Sections Calculator : ENTRIES : Select Enter PITCH as ANGLE (Degrees) MAJOR PITCH = 45° MINOR PITCH = 19.83593° Plan Angle between Eaves/Ridges = 90° Common Depth = 11.5 Hip/Valley Width = 6 The sketch above is dimensioned as per the values returned in the Valley match Commons fields. The 11.40190 Trough to bottom of Valley measurement is to a theoretical flat on the bottom shoulder. This shoulder of this Valley requires backing. Let the Depth of the Offset = d d ÷ tan 41.69405° + d ÷ tan 6.61190° = 6 d × tan 6.61190° + d × tan 41.69405° = 6 ( tan 41.69405° × tan 6.61190° ) d ( tan 6.61190° + tan 41.69405° ) = 6 ( tan 41.69405° × tan 6.61190° ) d = 6 ( tan 41.69405° × tan 6.61190° ) ÷ ( tan 6.61190° + tan 41.69405° ) d = .61541 Dimension from 12/12 Main Side of Valley = .61541 ÷ tan 41.69405° = .69086 Dimension from Shed Side of Valley = .61541 ÷ tan 6.61190° = 5.30918 CHECK : .69086 + 5.30918 = 6, the Valley Rafter width Valley Stock Depth = 15.40095 + .61541 = 16.01636 |
